Mathematical Biology

2021

• Paper 1, Section I, 6E

  Mathematical Biology | Part II, 2021

  (a) Consider a population of size $N(t)$ whose per capita rates of birth and death are $b e^{-a N}$ and $d$, respectively, where $b>d$ and all parameters are positive constants.

  (i) Write down the equation for the rate of change of the population.

  (ii) Show that a population of size $N^{*}=\frac{1}{a} \log \frac{b}{d}$ is stationary and that it is asymptotically stable.

  (b) Consider now a disease introduced into this population, where the number of susceptibles and infectives, $S$ and $I$, respectively, satisfy the equations

  $$\begin{aligned} &\frac{d S}{d t}=b e^{-a S} S-\beta S I-d S \\ &\frac{d I}{d t}=\beta S I-(d+\delta) I \end{aligned}$$

  (i) Interpret the biological meaning of each term in the above equations and comment on the reproductive capacity of the susceptible and infected individuals.

  (ii) Show that the disease-free equilibrium, $S=N^{*}$ and $I=0$, is linearly unstable if

  $$N^{*}>\frac{d+\delta}{\beta}$$

  (iii) Show that when the disease-free equilibrium is unstable there exists an endemic equilibrium satisfying

  $$\beta I+d=b e^{-a S}$$

  and that this equilibrium is linearly stable.

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• Paper 2, Section I, E

  Mathematical Biology | Part II, 2021

  Consider a stochastic birth-death process in a population of size $n(t)$, where deaths occur in pairs for $n \geqslant 2$. The probability per unit time of a birth, $n \rightarrow n+1$ for $n \geqslant 0$, is $b$, that of a pair of deaths, $n \rightarrow n-2$ for $n \geqslant 2$, is $d n$, and that of the death of a lonely singleton, $1 \rightarrow 0$, is $D$.

  (a) Write down the master equation for $p_{n}(t)$, the probability of a population of size $n$ at time $t$, distinguishing between the cases $n \geqslant 2, n=0$ and $n=1$.

  (b) For a function $f(n), n \geqslant 0$, show carefully that

  $$\frac{d}{d t}\langle f(n)\rangle=b \sum_{n=0}^{\infty}\left(f_{n+1}-f_{n}\right) p_{n}-d \sum_{n=2}^{\infty}\left(f_{n}-f_{n-2}\right) n p_{n}-D\left(f_{1}-f_{0}\right) p_{1}$$

  where $f_{n}=f(n)$.

  (c) Deduce the evolution equation for the mean $\mu(t)=\langle n\rangle$, and simplify it for the case $D=2 d$.

  (d) For the same value of $D$, show that

  $$\frac{d}{d t}\left\langle n^{2}\right\rangle=b(2 \mu+1)-4 d\left(\left\langle n^{2}\right\rangle-\mu\right)-2 d p_{1}$$

  Deduce that the variance $\sigma^{2}$ in the stationary state for $b, d>0$ satisfies

  $$\frac{3 b}{4 d}-\frac{1}{2}<\sigma^{2}<\frac{3 b}{4 d}$$

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• Paper 3, Section I, E

  Mathematical Biology | Part II, 2021

  The population density $n(a, t)$ of individuals of age $a$ at time $t$ satisfies the partial differential equation

  $$\frac{\partial n}{\partial t}+\frac{\partial n}{\partial a}=-d(a) n(a, t)$$

  with the boundary condition

  $$n(0, t)=\int_{0}^{\infty} b(a) n(a, t) d a$$

  where $b(a)$ and $d(a)$ are, respectively, the per capita age-dependent birth and death rates.

  (a) What is the biological interpretation of the boundary condition?

  (b) Solve equation (1) assuming a separable form of solution, $n(a, t)=A(a) T(t)$.

  (c) Use equation (2) to obtain a necessary condition for the existence of a separable solution to the full problem.

  (d) For a birth rate $b(a)=\beta e^{-\lambda a}$ with $\lambda>0$ and an age-independent death rate $d$, show that a separable solution to the full problem exists and find the critical value of $\beta$ above which the population density grows with time.

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• Paper 3, Section II, 13E

  Mathematical Biology | Part II, 2021

  Consider an epidemic spreading in a population that has been aggregated by age into groups numbered $i=1, \ldots, M$. The $i$ th age group has size $N_{i}$ and the numbers of susceptible, infective and recovered individuals in this group are, respectively, $S_{i}, I_{i}$ and $R_{i}$. The spread of the infection is governed by the equations

  $$\begin{aligned} \frac{d S_{i}}{d t} &=-\lambda_{i}(t) S_{i} \\ \frac{d I_{i}}{d t} &=\lambda_{i}(t) S_{i}-\gamma I_{i} \\ \frac{d R_{i}}{d t} &=\gamma I_{i} \end{aligned}$$

  where

  $$\lambda_{i}(t)=\beta \sum_{j=1}^{M} C_{i j} \frac{I_{j}}{N_{j}},$$

  and $C_{i j}$ is a matrix satisfying $N_{i} C_{i j}=N_{j} C_{j i}$, for $i, j=1, \ldots, M$.

  (a) Describe the biological meaning of the terms in equations (1) and (2), of the matrix $C_{i j}$ and the condition it satisfies, and of the lack of dependence of $\beta$ and $\gamma$ on $i$.

  State the condition on the matrix $C_{i j}$ that would ensure the absence of any transmission of infection between age groups.

  (b) In the early stages of an epidemic, $S_{i} \approx N_{i}$ and $I_{i} \ll N_{i}$. Use this information to linearise the dynamics appropriately, and show that the linearised system predicts

  $$\mathbf{I}(t)=\exp [\gamma(\mathbf{L}-\mathbf{1}) t] \mathbf{I}(0),$$

  where $\mathbf{I}(t)=\left[I_{1}(t), \ldots, I_{M}(t)\right]$ is the vector of infectives at time $t, \mathbf{1}$ is the $M \times M$ identity matrix and $\mathbf{L}$ is a matrix that should be determined.

  (c) Deduce a condition on the eigenvalues of the matrix $\mathbf{C}$ that allows the epidemic to grow.

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• Paper 4, Section I, E

  Mathematical Biology | Part II, 2021

  A marine population grows logistically and disperses by diffusion. It is moderately predated on up to a distance $L$ from a straight coast. Beyond that distance, predation is sufficiently excessive to eliminate the population. The density $n(x, t)$ of the population at a distance $x<L$ from the coast satisfies

  $$\frac{\partial n}{\partial t}=r n\left(1-\frac{n}{K}\right)-\delta n+D \frac{\partial^{2} n}{\partial x^{2}}$$

  subject to the boundary conditions

  $$\frac{\partial n}{\partial x}=0 \text { at } x=0, \quad n=0 \text { at } x=L$$

  (a) Interpret the terms on the right-hand side of $(*)$, commenting on their dependence on $n$. Interpret the boundary conditions.

  (b) Show that a non-zero population is viable if $r>\delta$ and

  $$L>\frac{\pi}{2} \sqrt{\frac{D}{r-\delta}}$$

  Interpret these conditions.

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• Paper 4, Section II, E

  Mathematical Biology | Part II, 2021

  The spatial density $n(x, t)$ of a population at location $x$ and time $t$ satisfies

  $$\frac{\partial n}{\partial t}=f(n)+D \frac{\partial^{2} n}{\partial x^{2}}$$

  where $f(n)=-n(n-r)(n-1), 0<r<1$ and $D>0$.

  (a) Give a biological example of the sort of phenomenon that this equation describes.

  (b) Show that there are three spatially homogeneous and stationary solutions to $(*)$, of which two are linearly stable to homogeneous perturbations and one is linearly unstable.

  (c) For $r=\frac{1}{2}$, find the stationary solution to $(*)$ subject to the conditions

  $$\lim _{x \rightarrow-\infty} n(x)=1, \quad \lim _{x \rightarrow \infty} n(x)=0 \quad \text { and } \quad n(0)=\frac{1}{2}$$

  (d) Write down the differential equation that is satisfied by a travelling-wave solution to $(*)$ of the form $n(x, t)=u(x-c t)$. Let $n_{0}(x)$ be the solution from part (c). Verify that $n_{0}(x-c t)$ satisfies this differential equation for $r \neq \frac{1}{2}$, provided the speed $c$ is chosen appropriately. [Hint: Consider the change to the equation from part (c).]

  (e) State how the sign of $c$ depends on $r$, and give a brief qualitative explanation for why this should be the case.

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2020

• Paper 1, Section I, 6B

  Mathematical Biology | Part II, 2020

  Consider a bivariate diffusion process with drift vector $A_{i}(\mathbf{x})=a_{i j} x_{j}$ and diffusion matrix $b_{i j}$ where

  $$a_{i j}=\left(\begin{array}{cc} -1 & 1 \\ -2 & -1 \end{array}\right), \quad b_{i j}=\left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right)$$

  $\mathbf{x}=\left(x_{1}, x_{2}\right)$ and $i, j=1,2$.

  (i) Write down the Fokker-Planck equation for the probability $P\left(x_{1}, x_{2}, t\right)$.

  (ii) Plot the drift vector as a vector field around the origin in the region $\left|x_{1}\right|<1$, $\left|x_{2}\right|<1$.

  (iii) Obtain the stationary covariances $C_{i j}=\left\langle x_{i} x_{j}\right\rangle$ in terms of the matrices $a_{i j}$ and $b_{i j}$ and hence compute their explicit values.

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• Paper 2, Section I, 6B

  Mathematical Biology | Part II, 2020

  Consider the system of predator-prey equations

  $$\begin{aligned} &\frac{d N_{1}}{d t}=-\epsilon_{1} N_{1}+\alpha N_{1} N_{2} \\ &\frac{d N_{2}}{d t}=\epsilon_{2} N_{2}-\alpha N_{1} N_{2} \end{aligned}$$

  where $\epsilon_{1}, \epsilon_{2}$ and $\alpha$ are positive constants.

  (i) Determine the non-zero fixed point $\left(N_{1}^{*}, N_{2}^{*}\right)$ of this system.

  (ii) Show that the system can be written in the form

  $$\frac{d x_{i}}{d t}=\sum_{j=1}^{2} K_{i j} \frac{\partial H}{\partial x_{j}}, \quad i=1,2$$

  where $x_{i}=\log \left(N_{i} / N_{i}^{*}\right)$ and a suitable $2 \times 2$ antisymmetric matrix $K_{i j}$ and scalar function $H\left(x_{1}, x_{2}\right)$ are to be identified.

  (iii) Hence, or otherwise, show that $H$ is constant on solutions of the predator-prey equations.

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• Paper 3, Section I, B

  Mathematical Biology | Part II, 2020

  Consider a model for the common cold in which the population is partitioned into susceptible $(S)$, infective $(I)$, and recovered $(R)$ categories, which satisfy

  $$\begin{aligned} \frac{d S}{d t} &=\alpha R-\beta S I \\ \frac{d I}{d t} &=\beta S I-\gamma I \\ \frac{d R}{d t} &=\gamma I-\alpha R \end{aligned}$$

  where $\alpha, \beta$ and $\gamma$ are positive constants.

  (i) Show that the sum $N \equiv S+I+R$ does not change in time.

  (ii) Determine the condition, in terms of $\beta, \gamma$ and $N$, for an endemic steady state to exist, that is, a time-independent state with a non-zero number of infectives.

  (iii) By considering a reduced set of equations for $S$ and $I$ only, show that the endemic steady state identified in (ii) above, if it exists, is stable.

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• Paper 3, Section II, 13B

  Mathematical Biology | Part II, 2020

  The larva of a parasitic worm disperses in one dimension while laying eggs at rate $\lambda>0$. The larvae die at rate $\mu$ and have diffusivity $D$, so that their density, $n(x, t)$, obeys

  $$\frac{\partial n}{\partial t}=D \frac{\partial^{2} n}{\partial x^{2}}-\mu n, \quad(D>0, \mu>0)$$

  The eggs do not diffuse, so that their density, $e(x, t)$, obeys

  $$\frac{\partial e}{\partial t}=\lambda n$$

  At $t=0$ there are no eggs and $N$ larvae concentrated at $x=0$, so that $n(x, 0)=N \delta(x)$.

  (i) Determine $n(x, t)$ for $t>0$. Show that $n(x, t) \rightarrow 0$ as $t \rightarrow \infty$.

  (ii) Determine the limit of $e(x, t)$ as $t \rightarrow \infty$.

  (iii) Provide a physical explanation for the remnant density of the eggs identified in part (ii).

  [You may quote without proof the results

  $$\begin{aligned} \int_{-\infty}^{\infty} \exp \left(-x^{2}\right) d x &=\sqrt{\pi} \\ \int_{-\infty}^{\infty} \frac{\exp (i k x)}{k^{2}+\alpha^{2}} d k &=\pi \exp (-\alpha|x|) / \alpha, \quad \alpha>0 \end{aligned}$$

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• Paper 4, Section I, B

  Mathematical Biology | Part II, 2020

  Consider a population process in which the probability of transition from a state with $n$ individuals to a state with $n+1$ individuals in the interval $(t, t+\Delta t)$ is $\lambda n \Delta t$ for small $\Delta t$.

  (i) Write down the master equation for the probability, $P_{n}(t)$, of the state $n$ at time $t$ for $n \geqslant 1$

  (ii) Assuming an initial distribution

  $$P_{n}(0)= \begin{cases}1, & \text { if } n=1 \\ 0, & \text { if } n>1\end{cases}$$

  show that

  $$P_{n}(t)=\exp (-\lambda t)(1-\exp (-\lambda t))^{n-1}$$

  (iii) Hence, determine the mean of $n$ for $t>0$.

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• Paper 4, Section II, 14B

  Mathematical Biology | Part II, 2020

  Consider the stochastic catalytic reaction

  $$E \leftrightharpoons E S, \quad E S \rightarrow E+P$$

  in which a single enzyme $E$ complexes reversibly to $E S$ (at forward rate $k_{1}$ and reverse rate $k_{1}^{\prime}$ ) and decomposes into product $P$ (at forward rate $k_{2}$ ), regenerating enzyme $E$. Assume there is sufficient substrate $S$ so that this catalytic cycle can continue indefinitely. Let $P(E, n)$ be the probability of the state with enzyme $E$ and $n$ products and $P(E S, n)$ the probability of the state with complex $E S$ and $n$ products, these states being mutually exclusive.

  (i) Write down the master equation for the probabilities $P(E, n)$ and $P(E S, n)$ for $n \geqslant 0$

  (ii) Assuming an initial state with zero products, solve the master equation for $P(E, 0)$ and $P(E S, 0)$.

  (iii) Hence find the probability distribution $f(\tau)$ of the time $\tau$ taken to form the first product.

  (iv) Obtain the mean of $\tau$.

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2019

• Paper 1, Section $I$, $\mathbf{6 C}$

  Mathematical Biology | Part II, 2019

  An animal population has annual dynamics, breeding in the summer and hibernating through the winter. At year $t$, the number of individuals alive who were born a years ago is given by $n_{a, t}$. Each individual of age $a$ gives birth to $b_{a}$ offspring, and after the summer has a probability $\mu_{a}$ of dying during the winter. [You may assume that individuals do not give birth during the year in which they are born.]

  Explain carefully why the following equations, together with initial conditions, are appropriate to describe the system:

  $$\begin{aligned} n_{0, t} &=\sum_{a=1}^{\infty} n_{a, t} b_{a} \\ n_{a+1, t+1} &=\left(1-\mu_{a}\right) n_{a, t}, \end{aligned}$$

  Seek a solution of the form $n_{a, t}=r_{a} \gamma^{t}$ where $\gamma$ and $r_{a}$, for $a=1,2,3 \ldots$, are constants. Show $\gamma$ must satisfy $\phi(\gamma)=1$ where

  $$\phi(\gamma)=\sum_{a=1}^{\infty}\left(\prod_{i=0}^{a-1}\left(1-\mu_{i}\right)\right) \gamma^{-a} b_{a}$$

  Explain why, for any reasonable set of parameters $\mu_{i}$ and $b_{i}$, the equation $\phi(\gamma)=1$ has a unique solution. Explain also how $\phi(1)$ can be used to determine if the population will grow or shrink.

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• Paper 2, Section I, C

  Mathematical Biology | Part II, 2019

  An activator-inhibitor system for $u(x, t)$ and $v(x, t)$ is described by the equations

  $$\begin{aligned} \frac{\partial u}{\partial t} &=u v^{2}-a+D \frac{\partial^{2} u}{\partial x^{2}} \\ \frac{\partial v}{\partial t} &=v-u v^{2}+\frac{\partial^{2} v}{\partial x^{2}} \end{aligned}$$

  where $a, D>0$.

  Find the range of $a$ for which the spatially homogeneous system has a stable equilibrium solution with $u>0$ and $v>0$.

  For the case when the homogeneous system is stable, consider spatial perturbations proportional to $\cos (k x)$ to the equilibrium solution found above. Give a condition on $D$ in terms of $a$ for the system to have a Turing instability (a spatial instability).

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• Paper 3, Section I, $\mathbf{6 C}$

  Mathematical Biology | Part II, 2019

  A model of wound healing in one spatial dimension is given by

  $$\frac{\partial S}{\partial t}=r S(1-S)+D \frac{\partial^{2} S}{\partial x^{2}}$$

  where $S(x, t)$ gives the density of healthy tissue at spatial position $x$ at time $t$ and $r$ and $D$ are positive constants.

  By setting $S(x, t)=f(\xi)$ where $\xi=x-c t$, seek a steady travelling wave solution where $f(\xi)$ tends to one for large negative $\xi$ and tends to zero for large positive $\xi$. By linearising around the leading edge, where $f \approx 1$, find the possible wave speeds $c$ of the system. Assuming that the full nonlinear system will settle to the slowest possible speed, express the wave speed as a function of $D$ and $r$.

  Consider now a situation where the tissue is destroyed in some window of length $W$, i.e. $S(x, 0)=0$ for $0<x<W$ for some constant $W>0$ and $S(x, 0)$ is equal to one elsewhere. Explain what will happen for subsequent times, illustrating your answer with sketches of $S(x, t)$. Determine approximately how long it will take for this wound to heal (in the sense that $S$ is close to one everywhere).

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• Paper 3, Section II, C

  Mathematical Biology | Part II, 2019

  (a) A stochastic birth-death process has a master equation given by

  $$\frac{d p_{n}}{d t}=\lambda\left(p_{n-1}-p_{n}\right)+\beta\left[(n+1) p_{n+1}-n p_{n}\right]$$

  where $p_{n}(t)$ is the probability that there are $n$ individuals in the population at time $t$ for $n=0,1,2, \ldots$ and $p_{n}=0$ for $n<0$.

  (i) Give a brief interpretation of $\lambda$ and $\beta$.

  (ii) Derive an equation for $\frac{\partial \phi}{\partial t}$, where $\phi$ is the generating function

  $$\phi(s, t)=\sum_{n=0}^{\infty} s^{n} p_{n}(t)$$

  (iii) Assuming that the generating function $\phi$ takes the form

  $$\phi(s, t)=e^{(s-1) f(t)}$$

  find $f(t)$ and hence show that, as $t \rightarrow \infty$, both the mean $\langle n\rangle$ and variance $\sigma^{2}$ of the population size tend to constant values, which you should determine.

  (b) Now suppose an extra process is included: $k$ individuals are added to the population at rate $\epsilon(n)$.

  (i) Write down the new master equation, and explain why, for $k>1$, the approach used in part (a) will fail.

  (ii) By working with the master equation directly, find a differential equation for the rate of change of the mean population size $\langle n\rangle$.

  (iii) Now take $\epsilon(n)=a n+b$ for positive constants $a$ and $b$. Show that for $\beta>a k$ the mean population size tends to a constant, which you should determine. Briefly describe what happens for $\beta<a k$.

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• Paper 4, Section I, C

  Mathematical Biology | Part II, 2019

  (a) A variant of the classic logistic population model is given by:

  $$\frac{d x(t)}{d t}=\alpha\left[x(t)-x(t-T)^{2}\right]$$

  where $\alpha, T>0$.

  Show that for small $T$, the constant solution $x(t)=1$ is stable.

  Allow $T$ to increase. Express in terms of $\alpha$ the value of $T$ at which the constant solution $x(t)=1$ loses stability.

  (b) Another variant of the logistic model is given by this equation:

  $$\frac{d x(t)}{d t}=\alpha x(t-T)[1-x(t)]$$

  where $\alpha, T>0$. When is the constant solution $x(t)=1$ stable for this model?

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• Paper 4, Section II, C

  Mathematical Biology | Part II, 2019

  A model of an infectious disease in a plant population is given by

  $$\begin{aligned} \dot{S} &=(S+I)-(S+I) S-\beta I S \\ \dot{I} &=-(S+I) I+\beta I S \end{aligned}$$

  where $S(t)$ is the density of healthy plants and $I(t)$ is the density of diseased plants at time $t$ and $\beta$ is a positive constant.

  (a) Give an interpretation of what each of the terms in equations (1) and (2) represents in terms of the dynamics of the plants. What does the coefficient $\beta$ represent? What can you deduce from the equations about the effect of the disease on the plants?

  (b) By finding all fixed points for $S \geqslant 0$ and $I \geqslant 0$ and analysing their stability, explain what will happen to a healthy plant population if the disease is introduced. Sketch the phase diagram, treating the cases $\beta<1$ and $\beta>1$ separately.

  (c) Define new variables $N(t)$ for the total plant population density and $\theta(t)$ for the proportion of the population that is diseased. Starting from equations (1) and (2) above, derive equations for $\dot{N}$ and $\dot{\theta}$ purely in terms of $N, \theta$ and $\beta$. Without carrying out a full fixed point analysis, explain how this system can be used directly to show the same results you had in part (b). [Hint: start by considering the dynamics of $N(t)$ alone.]

  (d) Suppose now that in an attempt to control disease, plants are culled at a rate $k$ per capita, independently of whether the plants are healthy or diseased. Write down the modified versions of equations (1) and (2). Use these to build updated equations for $\dot{N}$ and $\dot{\theta}$. Without carrying out a detailed fixed point analysis, what can you deduce about the effect of culling? Give the range of $k$ for which culling can effectively control the disease.

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2018

• Paper 1, Section I, $\mathbf{6 C}$

  Mathematical Biology | Part II, 2018

  Consider a birth-death process in which the birth and death rates in a population of size $n$ are, respectively, $B n$ and $D n$, where $B$ and $D$ are per capita birth and death rates.

  (a) Write down the master equation for the probability, $p_{n}(t)$, of the population having size $n$ at time $t$.

  (b) Obtain the differential equations for the rates of change of the mean $\mu(t)=\langle n\rangle$ and the variance $\sigma^{2}(t)=\left\langle n^{2}\right\rangle-\langle n\rangle^{2}$ in terms of $\mu, \sigma, B$ and $D$.

  (c) Compare the equations obtained above with the deterministic description of the evolution of the population size, $d n / d t=(B-D) n$. Comment on why $B$ and $D$ cannot be uniquely deduced from the deterministic model but can be deduced from the stochastic description.

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• Paper 2, Section I, C

  Mathematical Biology | Part II, 2018

  Consider a model of an epidemic consisting of populations of susceptible, $S(t)$, infected, $I(t)$, and recovered, $R(t)$, individuals that obey the following differential equations

  $$\begin{aligned} \frac{d S}{d t} &=a R-b S I \\ \frac{d I}{d t} &=b S I-c I \\ \frac{d R}{d t} &=c I-a R \end{aligned}$$

  where $a, b$ and $c$ are constant. Show that the sum of susceptible, infected and recovered individuals is a constant $N$. Find the fixed points of the dynamics and deduce the condition for an endemic state with a positive number of infected individuals. Expressing $R$ in terms of $S, I$ and $N$, reduce the system of equations to two coupled differential equations and, hence, deduce the conditions for the fixed point to be a node or a focus. How do small perturbations of the populations relax to the steady state in each case?

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• Paper 3, Section I, $\mathbf{6 C}$

  Mathematical Biology | Part II, 2018

  Consider a nonlinear model for the axisymmetric dispersal of a population in two spatial dimensions whose density, $n(r, t)$, obeys

  $$\frac{\partial n}{\partial t}=D \boldsymbol{\nabla} \cdot(n \boldsymbol{\nabla} n)$$

  where $D$ is a positive constant, $r$ is a radial polar coordinate, and $t$ is time.

  Show that

  $$2 \pi \int_{0}^{\infty} n(r, t) r d r=N$$

  is constant. Interpret this condition.

  Show that a similarity solution of the form

  $$n(r, t)=\left(\frac{N}{D t}\right)^{1 / 2} f\left(\frac{r}{(N D t)^{1 / 4}}\right)$$

  is valid for $t>0$ provided that the scaling function $f(x)$ satisfies

  $$\frac{d}{d x}\left(x f \frac{d f}{d x}+\frac{1}{4} x^{2} f\right)=0 .$$

  Show that there exists a value $x_{0}$ (which need not be evaluated) such that $f(x)>0$ for $x<x_{0}$ but $f(x)=0$ for $x>x_{0}$. Determine the area within which $n(r, t)>0$ at time $t$ in terms of $x_{0}$.

  [Hint: The gradient and divergence operators in cylindrical polar coordinates act on radial functions $f$ and $g$ as

  $$\left.\boldsymbol{\nabla} f(r)=\frac{\partial f}{\partial r} \hat{\boldsymbol{r}} \quad, \quad \boldsymbol{\nabla} \cdot[g(r) \hat{\boldsymbol{r}}]=\frac{1}{r} \frac{\partial}{\partial r}(r g(r)) .\right]$$

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• Paper 3, Section II, C

  Mathematical Biology | Part II, 2018

  Consider fluctuations of a population described by the vector $\mathbf{x}=\left(x_{1}, x_{2}, \ldots, x_{N}\right)$. The probability of the state $\mathbf{x}$ at time $t, P(\mathbf{x}, t)$, obeys the multivariate Fokker-Planck equation

  $$\frac{\partial P}{\partial t}=-\frac{\partial}{\partial x_{i}}\left(A_{i}(\mathbf{x}) P\right)+\frac{1}{2} \frac{\partial^{2}}{\partial x_{i} \partial x_{j}}\left(B_{i j}(\mathbf{x}) P\right),$$

  where $P=P(\mathbf{x}, t), A_{i}$ is a drift vector and $B_{i j}$ is a symmetric positive-definite diffusion matrix, and the summation convention is used throughout.

  (a) Show that the Fokker-Planck equation can be expressed as a continuity equation

  $$\frac{\partial P}{\partial t}+\nabla \cdot \mathbf{J}=0$$

  for some choice of probability flux $\mathbf{J}$ which you should determine explicitly. Here, $\nabla=\left(\frac{\partial}{\partial x_{1}}, \frac{\partial}{\partial x_{2}}, \ldots, \frac{\partial}{\partial x_{N}}\right)$ denotes the gradient operator.

  (b) Show that the above implies that an initially normalised probability distribution remains normalised,

  $$\int P(\mathbf{x}, t) d V=1$$

  at all times, where the volume element $d V=d x_{1} d x_{2} \ldots d x_{N}$.

  (c) Show that the first two moments of the probability distribution obey

  $$\begin{aligned} \frac{d}{d t}\left\langle x_{k}\right\rangle &=\left\langle A_{k}\right\rangle \\ \frac{d}{d t}\left\langle x_{k} x_{l}\right\rangle &=\left\langle x_{l} A_{k}+x_{k} A_{l}+B_{k l}\right\rangle \end{aligned}$$

  (d) Now consider small fluctuations with zero mean, and assume that it is possible to linearise the drift vector and the diffusion matrix as $A_{i}(\mathbf{x})=a_{i j} x_{j}$ and $B_{i j}(\mathbf{x})=b_{i j}$ where $a_{i j}$ has real negative eigenvalues and $b_{i j}$ is a symmetric positive-definite matrix. Express the probability flux in terms of the matrices $a_{i j}$ and $b_{i j}$ and assume that it vanishes in the stationary state.

  (e) Hence show that the multivariate normal distribution,

  $$P(\mathbf{x})=\frac{1}{Z} \exp \left(-\frac{1}{2} D_{i j} x_{i} x_{j}\right)$$

  where $Z$ is a normalisation and $D_{i j}$ is symmetric, is a solution of the linearised FokkerPlanck equation in the stationary state, and obtain an equation that relates $D_{i j}$ to the matrices $a_{i j}$ and $b_{i j}$.

  (f) Show that the inverse of the matrix $D_{i j}$ is the matrix of covariances $C_{i j}=\left\langle x_{i} x_{j}\right\rangle$ and obtain an equation relating $C_{i j}$ to the matrices $a_{i j}$ and $b_{i j}$.

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• Paper 4, Section I, C

  Mathematical Biology | Part II, 2018

  Consider a model of a population $N_{\tau}$ in discrete time

  $$N_{\tau+1}=\frac{r N_{\tau}}{\left(1+b N_{\tau}\right)^{2}}$$

  where $r, b>0$ are constants and $\tau=1,2,3, \ldots$ Interpret the constants and show that for $r>1$ there is a stable fixed point.

  Suppose the initial condition is $N_{1}=1 / b$ and that $r>4$. Show, using a cobweb diagram, that the population $N_{\tau}$ is bounded as

  $$\frac{4 r^{2}}{(4+r)^{2} b} \leqslant N_{\tau} \leqslant \frac{r}{4 b}$$

  and attains the bounds.

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• Paper 4, Section II, C

  Mathematical Biology | Part II, 2018

  An activator-inhibitor reaction diffusion system is given, in dimensionless form, by

  $$\frac{\partial u}{\partial t}=d \frac{\partial^{2} u}{\partial x^{2}}+\frac{u^{2}}{v}-2 b u, \quad \frac{\partial v}{\partial t}=\frac{\partial^{2} v}{\partial x^{2}}+u^{2}-v$$

  where $d$ and $b$ are positive constants. Which symbol represents the concentration of activator and which the inhibitor? Determine the positive steady states and show, by an examination of the eigenvalues in a linear stability analysis of the spatially uniform situation, that the reaction kinetics are stable if $b<\frac{1}{2}$.

  Determine the conditions for the steady state to be driven unstable by diffusion, and sketch the $(b, d)$ parameter space in which the diffusion-driven instability occurs. Find the critical wavenumber $k_{c}$ at the bifurcation to such a diffusion-driven instability.

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2017

• Paper 1, Section I, B

  Mathematical Biology | Part II, 2017

  A model of insect dispersal and growth in one spatial dimension is given by

  $$\frac{\partial N}{\partial t}=D \frac{\partial}{\partial x}\left(N^{2} \frac{\partial N}{\partial x}\right)+\alpha N, \quad N(x, 0)=N_{0} \delta(x),$$

  where $\alpha, D$ and $N_{0}$ are constants, $D>0$, and $\alpha$ may be positive or negative.

  By setting $N(x, t)=R(x, \tau) e^{\alpha t}$, where $\tau(t)$ is some time-like variable satisfying $\tau(0)=0$, show that a suitable choice of $\tau$ yields

  $$R_{\tau}=\left(R^{2} R_{x}\right)_{x}, \quad R(x, 0)=N_{0} \delta(x)$$

  where subscript denotes differentiation with respect to $x$ or $\tau$.

  Consider a similarity solution of the form $R(x, \tau)=F(\xi) / \tau^{\frac{1}{4}}$ where $\xi=x / \tau^{\frac{1}{4}}$. Show that $F$ must satisfy

  $$-\frac{1}{4}(F \xi)^{\prime}=\left(F^{2} F^{\prime}\right)^{\prime} \quad \text { and } \quad \int_{-\infty}^{+\infty} F(\xi) d \xi=N_{0}$$

  [You may use the fact that these are solved by

  $$F(\xi)= \begin{cases}\frac{1}{2} \sqrt{\xi_{0}^{2}-\xi^{2}} & \text { for }|\xi|<\xi_{0} \\ 0 & \text { otherwise }\end{cases}$$

  where $\left.\xi_{0}=\sqrt{4 N_{0} / \pi} .\right]$

  For $\alpha<0$, what is the maximum distance from the origin that insects ever reach? Give your answer in terms of $D, \alpha$ and $N_{0}$.

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• Paper 2, Section I, B

  Mathematical Biology | Part II, 2017

  A bacterial nutrient uptake model is represented by the reaction system

  $$\begin{array}{rll} 2 S+E & \stackrel{k_{1}}{\longrightarrow} & C \\ C & \stackrel{k_{2}}{\longrightarrow} 2 S+E \\ C \stackrel{k_{3}}{\longrightarrow} & E+2 P \end{array}$$

  where the $k_{i}$ are rate constants. Let $s, e, c$ and $p$ represent the concentrations of $S, E, C$ and $P$ respectively. Initially $s=s_{0}, e=e_{0}, c=0$ and $p=0$. Write down the governing differential equation system for the concentrations.

  Either by using the differential equations or directly from the reaction system above, find two invariant quantities. Use these to simplify the system to

  $$\begin{aligned} \dot{s} &=-2 k_{1} s^{2}\left(e_{0}-c\right)+2 k_{2} c \\ \dot{c} &=k_{1} s^{2}\left(e_{0}-c\right)-\left(k_{2}+k_{3}\right) c . \end{aligned}$$

  By setting $u=s / s_{0}$ and $v=c / e_{0}$ and rescaling time, show that the system can be written as

  $$\begin{aligned} u^{\prime} &=-2 u^{2}(1-v)+2(\mu-\lambda) v \\ \epsilon v^{\prime} &=\quad u^{2}(1-v)-\mu v \end{aligned}$$

  where $\epsilon=e_{0} / s_{0}$ and $\mu$ and $\lambda$ should be given. Give the initial conditions for $u$ and $v$.

  [Hint: Note that $2 X$ is equivalent to $X+X$ in reaction systems.]

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• Paper 3, Section I, B

  Mathematical Biology | Part II, 2017

  A stochastic birth-death process has a master equation given by

  $$\frac{d p(n, t)}{d t}=\lambda[p(n-1, t)-p(n, t)]+\beta[(n+1) p(n+1, t)-n p(n, t)]$$

  where $p(n, t)$ is the probability that there are $n$ individuals in the population at time $t$ for $n=0,1,2, \ldots$ and $p(n, t)=0$ for $n<0$.

  Give the corresponding Fokker-Planck equation for this system.

  Use this Fokker-Planck equation to find expressions for $\frac{d}{d t}\langle x\rangle$ and $\frac{d}{d t}\left\langle x^{2}\right\rangle$.

  [Hint: The general form for a Fokker-Planck equation in $P(x, t)$ is

  $$\frac{\partial P}{\partial t}=-\frac{\partial}{\partial x}(A P)+\frac{1}{2} \frac{\partial^{2}}{\partial x^{2}}(B P)$$

  You may use this general form, stating how $A(x)$ and $B(x)$ are constructed. Alternatively, you may derive a Fokker-Plank equation directly by working from the master equation.]

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• Paper 3, Section II, B

  Mathematical Biology | Part II, 2017

  In a discrete-time model, adults and larvae of a population at time $n$ are represented by $a_{n}$ and $b_{n}$ respectively. The model is represented by the equations

  $$\begin{aligned} a_{n+1} &=(1-k) a_{n}+\frac{b_{n}}{1+a_{n}} \\ b_{n+1} &=\mu a_{n} \end{aligned}$$

  You may assume that $k \in(0,1)$ and $\mu>0$. Give an explanation of what each of the terms represents, and hence give a description of the population model.

  By combining the equations to describe the dynamics purely in terms of the adults, find all equilibria of the system. Show that the equilibrium with the population absent $(a=0)$ is unstable exactly when there exists an equilibrium with the population present $(a>0)$.

  Give the condition on $\mu$ and $k$ for the equilibrium with $a>0$ to be stable, and sketch the corresponding region in the $(k, \mu)$ plane.

  What happens to the population close to the boundaries of this region?

  If this model was modified to include stochastic effects, briefly describe qualitatively the likely dynamics near the boundaries of the region found above.

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• Paper 4, Section I, B

  Mathematical Biology | Part II, 2017

  Consider an epidemic model with host demographics (natural births and deaths).

  The system is given by

  $$\begin{aligned} &\frac{d S}{d t}=-\beta I S-\mu S+\mu N \\ &\frac{d I}{d t}=+\beta I S-\nu I-\mu I \end{aligned}$$

  where $S(t)$ are the susceptibles, $I(t)$ are the infecteds, $N$ is the total population size and the parameters $\beta, \mu$ and $\nu$ are positive. The basic reproduction ratio is defined as $R_{0}=\beta N /(\mu+\nu) .$

  Show that the system has an endemic equilibrium (where the disease is present) for $R_{0}>1$. Show that the endemic equilibrium is stable.

  Interpret the meaning of the case $\nu \gg \mu$ and show that in this case the approximate period of (decaying) oscillation around the endemic equilibrium is given by

  $$T=\frac{2 \pi}{\sqrt{\mu \nu\left(R_{0}-1\right)}}$$

  Suppose now a vaccine is introduced which is given to some proportion of the population at birth, but not enough to eradicate the disease. What will be the effect on the period of (decaying) oscillations?

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• Paper 4, Section II, B

  Mathematical Biology | Part II, 2017

  An activator-inhibitor system is described by the equations

  $$\begin{aligned} &\frac{\partial u}{\partial t}=u(c+u-v)+\frac{\partial^{2} u}{\partial x^{2}} \\ &\frac{\partial v}{\partial t}=v(a u-b v)+d \frac{\partial^{2} v}{\partial x^{2}} \end{aligned}$$

  where $a, b, c, d>0$.

  Find and sketch the range of $a, b$ for which the spatially homogeneous system has a stable stationary solution with $u>0$ and $v>0$.

  Considering spatial perturbations of the form $\cos (k x)$ about the solution found above, find conditions for the system to be unstable. Sketch this region in the $(a, b)$-plane for fixed $d$ (for a value of $d$ such that the region is non-empty).

  Show that $k_{c}$, the critical wavenumber at the onset of the instability, is given by

  $$k_{c}=\sqrt{\frac{2 a c}{d-a}}$$

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2016

• Paper 1, Section I, B

  Mathematical Biology | Part II, 2016

  Consider an epidemic model where susceptibles are vaccinated at per capita rate $v$, but immunity (from infection or vaccination) is lost at per capita rate $b$. The system is given by

  $$\begin{aligned} &\frac{d S}{d t}=-r I S+b(N-I-S)-v S \\ &\frac{d I}{d t}=r I S-a I \end{aligned}$$

  where $S(t)$ are the susceptibles, $I(t)$ are the infecteds, $N$ is the total population size and all parameters are positive. The basic reproduction ratio $R_{0}=r N / a$ satisfies $R_{0}>1$.

  Find the critical vaccination rate $v_{c}$, in terms of $b$ and $R_{0}$, such that the system has an equilibrium with the disease present if $v<v_{c}$. Show that this equilibrium is stable when it exists.

  Find the long-term outcome for $S$ and $I$ if $v>v_{c}$.

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• Paper 2, Section I, B

  Mathematical Biology | Part II, 2016

  (a) The populations of two competing species satisfy

  $$\begin{aligned} \frac{d N_{1}}{d t} &=N_{1}\left[b_{1}-\lambda\left(N_{1}+N_{2}\right)\right] \\ \frac{d N_{2}}{d t} &=N_{2}\left[b_{2}-\lambda\left(N_{1}+N_{2}\right)\right] \end{aligned}$$

  where $b_{1}>b_{2}>0$ and $\lambda>0$. Sketch the phase diagram (limiting attention to $\left.N_{1}, N_{2} \geqslant 0\right)$.

  The relative abundance of species 1 is defined by $U=N_{1} /\left(N_{1}+N_{2}\right)$. Show that

  $$\frac{d U}{d t}=A U(1-U)$$

  where $A$ is a constant that should be determined.

  (b) Consider the spatial system

  $$\frac{\partial u}{\partial t}=u(1-u)+D \frac{\partial^{2} u}{\partial x^{2}}$$

  and consider a travelling-wave solution of the form $u(x, t)=f(x-c t)$ representing one species $(u=1)$ invading territory previously occupied by another species $(u=0)$. By linearising near the front of the invasion, show that the wave speed is given by $c=2 \sqrt{D}$.

  [You may assume that the solution to the full nonlinear system will settle to the slowest possible linear wave speed.]

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• Paper 3, Section $\mathbf{I}$, B

  Mathematical Biology | Part II, 2016

  A delay model for a population of size $N_{t}$ at discrete time $t$ is given by

  $$N_{t+1}=\max \left\{\left(r-N_{t-1}^{2}\right) N_{t}, 0\right\}$$

  Show that for $r>1$ there is a non-trivial equilibrium, and analyse its stability. Show that, as $r$ is increased from 1 , the equilibrium loses stability at $r=3 / 2$ and find the approximate periodicity close to equilibrium at this point.

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• Paper 3, Section II, B

  Mathematical Biology | Part II, 2016

  The Fitzhugh-Nagumo model is given by

  $$\begin{aligned} \dot{u} &=c\left(v+u-\frac{1}{3} u^{3}+z(t)\right) \\ \dot{v} &=-\frac{1}{c}(u-a+b v) \end{aligned}$$

  where $\left(1-\frac{2}{3} b\right)<a<1,0<b \leqslant 1$ and $c \gg 1$.

  For $z(t)=0$, by considering the nullclines in the $(u, v)$-plane, show that there is a unique equilibrium. Sketch the phase diagram

  At $t=0$ the system is at the equilibrium, and $z(t)$ is then 'switched on' to be $z(t)=-V_{0}$ for $t>0$, where $V_{0}$ is a constant. Describe carefully how suitable choices of $V_{0}>0$ can represent a system analogous to (i) a neuron firing once, and (ii) a neuron firing repeatedly. Illustrate your answer with phase diagrams and also plots of $v$ against $t$ for each case. Find the threshold for $V_{0}$ that separates these cases. Comment briefly from a biological perspective on the behaviour of the system when $a=1-\frac{2}{3} b+\epsilon b$ and $0<\epsilon \ll 1$.

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• Paper 4, Section I, B

  Mathematical Biology | Part II, 2016

  A stochastic birth-death process is given by the master equation

  $$\frac{d p_{n}}{d t}=\lambda\left(p_{n-1}-p_{n}\right)+\mu\left[(n-1) p_{n-1}-n p_{n}\right]+\beta\left[(n+1) p_{n+1}-n p_{n}\right]$$

  where $p_{n}(t)$ is the probability that there are $n$ individuals in the population at time $t$ for $n=0,1,2, \ldots$ and $p_{n}=0$ for $n<0$. Give a brief interpretation of $\lambda, \mu$ and $\beta$.

  Derive an equation for $\frac{\partial \phi}{\partial t}$, where $\phi$ is the generating function

  $$\phi(s, t)=\sum_{n=0}^{\infty} s^{n} p_{n}(t)$$

  Now assume that $\beta>\mu$. Show that at steady state

  $$\phi=\left(\frac{\beta-\mu}{\beta-\mu s}\right)^{\lambda / \mu}$$

  and find the corresponding mean and variance.

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• Paper 4, Section II, B

  Mathematical Biology | Part II, 2016

  The population densities of two types of cell are given by $U(x, t)$ and $V(x, t)$. The system is described by the equations

  $$\begin{aligned} \frac{\partial U}{\partial t} &=\alpha U(1-U)+\chi \frac{\partial}{\partial x}\left(U \frac{\partial V}{\partial x}\right)+D \frac{\partial^{2} U}{\partial x^{2}} \\ \frac{\partial V}{\partial t} &=V(1-V)-\beta U V+\frac{\partial^{2} V}{\partial x^{2}} \end{aligned}$$

  where $\alpha, \beta, \chi$ and $D$ are positive constants.

  (a) Identify the terms which involve interaction between the cell types, and briefly describe what each of these terms might represent.

  (b) Consider the system without spatial dynamics. Find the condition on $\beta$ for there to be a non-trivial spatially homogeneous solution that is stable to spatially invariant disturbances.

  (c) Consider now the full spatial system, and consider small spatial perturbations proportional to $\cos (k x)$ of the solution found in part (b). Show that for sufficiently large $\chi$ (the precise threshold should be found) the spatially homogeneous solution is stable to perturbations with either small or large wavenumber, but is unstable to perturbations at some intermediate wavenumber.

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2015

• Paper 1, Section I, E

  Mathematical Biology | Part II, 2015

  The population density $n(a, t)$ of individuals of age $a$ at time $t$ satisfies

  $$\frac{\partial n}{\partial t}+\frac{\partial n}{\partial a}=-\mu(a) n(a, t), \quad n(0, t)=\int_{0}^{\infty} b(a) n(a, t) d a$$

  where $\mu(a)$ is the age-dependent death rate and $b(a)$ is the birth rate per individual of age $a$. Show that this may be solved with a similarity solution of the form $n(a, t)=e^{\gamma t} r(a)$ if the growth rate $\gamma$ satisfies $\phi(\gamma)=1$ where

  $$\phi(\gamma)=\int_{0}^{\infty} b(a) e^{-\gamma a-\int_{0}^{a} \mu(s) d s} d a$$

  Suppose now that the birth rate is given by $b(a)=B a^{p} e^{-\lambda a}$ with $B, \lambda>0$ and $p$ is a positive integer, and the death rate is constant in age (i.e. $\mu(a)=\mu)$. Find the average number of offspring per individual.

  Find the similarity solution, and find the threshold $B^{*}$ for the birth parameter $B$ so that $B>B^{*}$ corresponds to a growing population.

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• Paper 2, Section I, E

  Mathematical Biology | Part II, 2015

  An activator-inhibitor system is described by the equations

  $$\begin{aligned} \frac{\partial u}{\partial t} &=2 u+u^{2}-u v+\frac{\partial^{2} u}{\partial x^{2}} \\ \frac{\partial v}{\partial t} &=a\left(u^{2}-v\right)+d \frac{\partial^{2} v}{\partial x^{2}} \end{aligned}$$

  where $a, d>0$.

  Find the range of $a$ for which the spatially homogeneous system has a stable equilibrium solution with $u>0$ and $v>0$.

  For the case when the homogeneous system is stable, consider spatial perturbations proportional to $\cos (k x)$ to the equilibrium solution found above. Show that the system has a Turing instability when

  $$d>\left(\frac{7}{2}+2 \sqrt{3}\right) a$$

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• Paper 3, Section I, E

  Mathematical Biology | Part II, 2015

  The number of a certain type of annual plant in year $n$ is given by $x_{n}$. Each plant produces $k$ seeds that year and then dies before the next year. The proportion of seeds that germinate to produce a new plant the next year is given by $e^{-\gamma x_{n}}$ where $\gamma>0$. Explain briefly why the system can be described by

  $$x_{n+1}=k x_{n} e^{-\gamma x_{n}}$$

  Give conditions on $k$ for a stable positive equilibrium of the plant population size to be possible.

  Winters become milder and now a proportion $s$ of all plants survive each year $(s \in(0,1))$. Assume that plants produce seeds as before while they are alive. Show that a wider range of $k$ now gives a stable positive equilibrium population.

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• Paper 3, Section II, E

  Mathematical Biology | Part II, 2015

  A fungal disease is introduced into an isolated population of frogs. Without disease, the normalised population size $x$ would obey the logistic equation $\dot{x}=x(1-x)$, where the dot denotes differentiation with respect to time. The disease causes death at rate $d$ and there is no recovery. The disease transmission rate is $\beta$ and, in addition, offspring of infected frogs are infected from birth.

  (a) Briefly explain why the population sizes $x$ and $y$ of uninfected and infected frogs respectively now satisfy

  $$\begin{aligned} \dot{x} &=x[1-x-(1+\beta) y] \\ \dot{y} &=y[(1-d)-(1-\beta) x-y] \end{aligned}$$

  (b) The population starts at the disease-free population size $(x=1)$ and a small number of infected frogs are introduced. Show that the disease will successfully invade if and only if $\beta>d$.

  (c) By finding all the equilibria in $x \geqslant 0, y \geqslant 0$ and considering their stability, find the long-term outcome for the frog population. State the relationships between $d$ and $\beta$ that distinguish different final populations.

  (d) Plot the long-term steady total population size as a function of $d$ for fixed $\beta$, and note that an intermediate mortality rate is actually the most harmful. Explain why this is the case.

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• Paper 4, Section I, E

  Mathematical Biology | Part II, 2015

  (i) A variant of the classic logistic population model is given by the HutchinsonWright equation

  $$\frac{d x(t)}{d t}=\alpha x(t)[1-x(t-T)]$$

  where $\alpha, T>0$. Determine the condition on $\alpha$ (in terms of $T$ ) for the constant solution $x(t)=1$ to be stable.

  (ii) Another variant of the logistic model is given by the equation

  $$\frac{d x(t)}{d t}=\alpha\left[x(t-T)-x(t)^{2}\right]$$

  where $\alpha, T>0$. Give a brief interpretation of what this model represents.

  Determine the condition on $\alpha$ (in terms of $T$ ) for the constant solution $x(t)=1$ to be stable in this model.

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• Paper 4, Section II, E

  Mathematical Biology | Part II, 2015

  In a stochastic model of multiple populations, $P=P(\mathbf{x}, t)$ is the probability that the population sizes are given by the vector $\mathbf{x}$ at time $t$. The jump rate $W(\mathbf{x}, \mathbf{r})$ is the probability per unit time that the population sizes jump from $\mathbf{x}$ to $\mathbf{x}+\mathbf{r}$. Under suitable assumptions, the system may be approximated by the multivariate Fokker-Planck equation (with summation convention)

  $$\frac{\partial}{\partial t} P=-\frac{\partial}{\partial x_{i}} A_{i} P+\frac{1}{2} \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} B_{i j} P$$

  where $A_{i}(\mathbf{x})=\sum_{\mathbf{r}} r_{i} W(\mathbf{x}, \mathbf{r})$ and matrix elements $B_{i j}(\mathbf{x})=\sum_{\mathbf{r}} r_{i} r_{j} W(\mathbf{x}, \mathbf{r})$.

  (a) Use the multivariate Fokker-Planck equation to show that

  $$\begin{aligned} \frac{d}{d t}\left\langle x_{k}\right\rangle &=\left\langle A_{k}\right\rangle \\ \frac{d}{d t}\left\langle x_{k} x_{l}\right\rangle &=\left\langle x_{l} A_{k}+x_{k} A_{l}+B_{k l}\right\rangle \end{aligned}$$

  [You may assume that $P(\mathbf{x}, t) \rightarrow 0$ as $|\mathbf{x}| \rightarrow \infty$.]

  (b) For small fluctuations, you may assume that the vector $\mathbf{A}$ may be approximated by a linear function in $\mathbf{x}$ and the matrix $\mathbf{B}$ may be treated as constant, i.e. $A_{k}(\mathbf{x}) \approx$ $a_{k l}\left(x_{l}-\left\langle x_{l}\right\rangle\right)$ and $B_{k l}(\mathbf{x}) \approx B_{k l}(\langle\mathbf{x}\rangle)=b_{k l}$ (where $a_{k l}$ and $b_{k l}$ are constants). Show that at steady state the covariances $C_{i j}=\operatorname{cov}\left(x_{i}, x_{j}\right)$ satisfy

  $$a_{i k} C_{j k}+a_{j k} C_{i k}+b_{i j}=0 .$$

  (c) A lab-controlled insect population consists of $x_{1}$ larvae and $x_{2}$ adults. Larvae are added to the system at rate $\lambda$. Larvae each mature at rate $\gamma$ per capita. Adults die at rate $\beta$ per capita. Give the vector $\mathbf{A}$ and matrix $\mathbf{B}$ for this model. Show that at steady state

  $$\left\langle x_{1}\right\rangle=\frac{\lambda}{\gamma}, \quad\left\langle x_{2}\right\rangle=\frac{\lambda}{\beta} .$$

  (d) Find the variance of each population size near steady state, and show that the covariance between the populations is zero.

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2014

• Paper 1, Section I, B

  Mathematical Biology | Part II, 2014

  A population model for two species is given by

  $$\begin{aligned} &\frac{d N}{d t}=a N-b N P-k N^{2} \\ &\frac{d P}{d t}=-d P+c N P \end{aligned}$$

  where $a, b, c, d$ and $k$ are positive parameters. Show that this may be rescaled to

  $$\begin{aligned} &\frac{d u}{d \tau}=u(1-v-\beta u) \\ &\frac{d v}{d \tau}=-\alpha v(1-u) \end{aligned}$$

  and give $\alpha$ and $\beta$ in terms of the original parameters.

  For $\beta<1$ find all fixed points in $u \geqslant 0, v \geqslant 0$, and analyse their stability. Assuming that both populations are present initially, what does this suggest will be the long-term outcome?

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• Paper 2, Section I, B

  Mathematical Biology | Part II, 2014

  Consider an experiment where two or three individuals are added to a population with probability $\lambda_{2}$ and $\lambda_{3}$ respectively per unit time. The death rate in the population is a constant $\beta$ per individual per unit time.

  Write down the master equation for the probability $p_{n}(t)$ that there are $n$ individuals in the population at time $t$. From this, derive an equation for $\frac{\partial \phi}{\partial t}$, where $\phi$ is the generating function

  $$\phi(s, t)=\sum_{n=0}^{\infty} s^{n} p_{n}(t)$$

  Find the solution for $\phi$ in steady state, and show that the mean and variance of the population size are given by

  $$\langle n\rangle=3 \frac{\lambda_{3}}{\beta}+2 \frac{\lambda_{2}}{\beta}, \quad \operatorname{var}(n)=6 \frac{\lambda_{3}}{\beta}+3 \frac{\lambda_{2}}{\beta} .$$

  Hence show that, for a free choice of $\lambda_{2}$ and $\lambda_{3}$ subject to a given target mean, the experimenter can minimise the variance by only adding two individuals at a time.

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• Paper 2, Section II, B

  Mathematical Biology | Part II, 2014

  An activator-inhibitor system is described by the equations

  $$\begin{aligned} \frac{\partial u}{\partial t} &=\frac{a u}{v}-u^{2}+d_{1} \frac{\partial^{2} u}{\partial x^{2}} \\ \frac{\partial v}{\partial t} &=v^{2}-\frac{v}{u^{2}}+d_{2} \frac{\partial^{2} v}{\partial x^{2}} \end{aligned}$$

  where $a, d_{1}, d_{2}>0$.

  Find the range of $a$ for which the spatially homogeneous system has a stable equilibrium solution with $u>0$ and $v>0$. Determine when the equilibrium is a stable focus, and sketch the phase diagram for this case (restricting attention to $u>0$ and $v>0)$.

  For the case when the homogeneous system is stable, consider spatial perturbations proportional to $\cos (k x)$ of the solution found above. Briefly explain why the system will be stable to spatial perturbations with very small or very large $k$. Find conditions for the system to be unstable to a spatial perturbation (for some range of $k$ which need not be given). Sketch the region satisfying these conditions in the $\left(a, d_{1} / d_{2}\right)$ plane.

  Find $k_{c}$, the critical wavenumber at the onset of instability, in terms of $a$ and $d_{1}$.

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• Paper 3, Section $I$, B

  Mathematical Biology | Part II, 2014

  An epidemic model is given by

  $$\begin{aligned} &\frac{d S}{d t}=-r I S \\ &\frac{d I}{d t}=+r I S-a I \end{aligned}$$

  where $S(t)$ are the susceptibles, $I(t)$ are the infecteds, and $a$ and $r$ are positive parameters. The basic reproduction ratio is defined as $R_{0}=r N / a$, where $N$ is the total population size. Find a condition on $R_{0}$ for an epidemic to be possible if, initially, $S \approx N$ and $I$ is small but non-zero.

  Now suppose a proportion $p$ of the population was vaccinated (with a completely effective vaccine) so that initially $S \approx(1-p) N$. On a sketch of the $\left(R_{0}, p\right)$ plane, mark the regions where an epidemic is still possible, where the vaccination will prevent an epidemic, and where no vaccination was necessary.

  For the case when an epidemic is possible, show that $\sigma$, the proportion of the initially susceptible population that has not been infected by the end of an epidemic, satisfies

  $$\sigma-\frac{1}{(1-p) R_{0}} \log \sigma \approx 1$$

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• Paper 3, Section II, B

  Mathematical Biology | Part II, 2014

  A discrete-time model for breathing is given by

  $$\begin{aligned} V_{n+1} &=\alpha C_{n-k}, \\ C_{n+1}-C_{n} &=\gamma-\beta V_{n+1}, \end{aligned}$$

  where $V_{n}$ is the volume of each breath in time step $n$ and $C_{n}$ is the concentration of carbon dioxide in the blood at the end of time step $n$. The parameters $\alpha, \beta$ and $\gamma$ are all positive. Briefly explain the biological meaning of each of the above equations.

  Find the steady state. For $k=0$ and $k=1$ determine the stability of the steady state.

  For general (integer) $k>1$, by seeking parameter values when the modulus of a perturbation to the steady state is constant, find the range of parameters where the solution is stable. What is the periodicity of the constant-modulus solution at the edge of this range? Comment on how the size of the range depends on $k$.

  This can be developed into a more realistic model by changing the term $-\beta V_{n+1}$ to $-\beta C_{n} V_{n+1}$ in (2). Briefly explain the biological meaning of this change. Show that for both $k=0$ and $k=1$ the new steady state is stable if $0<a<1$, where $a=\sqrt{\alpha \beta \gamma}$.

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• Paper 4, Section I, B

  Mathematical Biology | Part II, 2014

  The concentration $c(x, t)$ of a chemical in one dimension obeys the equations

  $$\frac{\partial c}{\partial t}=\frac{\partial}{\partial x}\left(c^{2} \frac{\partial c}{\partial x}\right), \quad \int_{-\infty}^{\infty} c(x, t) d x=1$$

  State the physical interpretation of each equation.

  Seek a similarity solution of the form $c=t^{\alpha} f(\xi)$, where $\xi=t^{\beta} x$. Find equations involving $\alpha$ and $\beta$ from the differential equation and the integral. Show that these are satisfied by $\alpha=\beta=-1 / 4$.

  Find the solution for $f(\xi)$. Find and sketch the solution for $c(x, t)$.

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2013

• Paper 1, Section I, A

  Mathematical Biology | Part II, 2013

  In a discrete-time model, a proportion $\mu$ of mature bacteria divides at each time step. When a mature bacterium divides it is destroyed and two new immature bacteria are produced. A proportion $\lambda$ of the immature bacteria matures at each time step, and a proportion $k$ of mature bacteria dies at each time step. Show that this model may be represented by the equations

  $$\begin{aligned} &a_{t+1}=a_{t}+2 \mu b_{t}-\lambda a_{t} \\ &b_{t+1}=b_{t}-\mu b_{t}+\lambda a_{t}-k b_{t} \end{aligned}$$

  Give an expression for the general solution to these equations and show that the population may grow if $\mu>k$.

  At time $T$, the population is treated with an antibiotic that completely stops bacteria from maturing, but otherwise has no direct effects. Explain what will happen to the population of bacteria afterwards, and give expressions for $a_{t}$ and $b_{t}$ for $t>T$ in terms of $a_{T}, b_{T}, \mu$ and $k$.

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• Paper 2, Section I, A

  Mathematical Biology | Part II, 2013

  The population density $n(a, t)$ of individuals of age $a$ at time $t$ satisfies

  $$\frac{\partial n(a, t)}{\partial t}+\frac{\partial n(a, t)}{\partial a}=-\mu(a) n(a, t)$$

  with

  $$n(0, t)=\int_{0}^{\infty} b(a) n(a, t) d a$$

  where $\mu(a)$ is the age-dependent death rate and $b(a)$ is the birth rate per individual of age

  Seek a similarity solution of the form $n(a, t)=e^{\gamma t} r(a)$ and show that

  $$r(a)=r(0) e^{-\gamma a-\int_{0}^{a} \mu(s) d s}, \quad r(0)=\int_{0}^{\infty} b(s) r(s) d s .$$

  Show also that if

  $$\phi(\gamma)=\int_{0}^{\infty} b(a) e^{-\gamma a-\int_{0}^{a} \mu(s) d s} d a=1$$

  then there is such a similarity solution. Give a biological interpretation of $\phi(0)$.

  Suppose now that all births happen at age $a^{*}$, at which time an individual produces $B$ offspring, and that the death rate is constant with age (i.e. $\mu(a)=\mu)$. Find the similarity solution and give the condition for this to represent a growing population.

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• Paper 2, Section II, A

  Mathematical Biology | Part II, 2013

  The concentration $c(x, t)$ of insects at position $x$ at time $t$ satisfies the nonlinear diffusion equation

  $$\frac{\partial c}{\partial t}=\frac{\partial}{\partial x}\left(c^{m} \frac{\partial c}{\partial x}\right)$$

  with $m>0$. Find the value of $\alpha$ which allows a similarity solution of the form $c(x, t)=t^{\alpha} f(\xi)$, with $\xi=t^{\alpha} x$.

  Show that

  $$f(\xi)=\left\{\begin{array}{cl} {\left[\frac{\alpha m}{2}\left(\xi^{2}-\xi_{0}^{2}\right)\right]^{1 / m}} & \text { for }-\xi_{0}<\xi<\xi_{0} \\ 0 & \text { otherwise } \end{array}\right.$$

  where $\xi_{0}$ is a constant. From the original partial differential equation, show that the total number of insects $c_{0}$ does not change in time. From this result, find a general expression relating $\xi_{0}$ and $c_{0}$. Find a closed-form solution for $\xi_{0}$ in the case $m=2$.

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• Paper 3, Section I, A

  Mathematical Biology | Part II, 2013

  An immune system creates a burst of $N$ new white blood cells with probability $b$ per unit time. White blood cells die with probability $d$ each per unit time. Write down the master equation for $P_{n}(t)$, the probability that there are $n$ white blood cells at time $t$.

  Given that $n=n_{0}$ initially, find an expression for the mean of $n$.

  Show that the variance of $n$ has the form $A e^{-2 d t}+B e^{-d t}+C$ and find $A, B$ and $C$.

  If the immune system were modified to produce $k$ times as many cells per burst but with probability per unit time divided by a factor $k$, how would the mean and variance of the number of cells change?

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• Paper 3, Section II, A

  Mathematical Biology | Part II, 2013

  An activator-inhibitor system is described by the equations

  $$\begin{aligned} &\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}+u-u v+a u^{2} \\ &\frac{\partial v}{\partial t}=d \frac{\partial^{2} v}{\partial x^{2}}+u^{2}-b u v \end{aligned}$$

  where $a, b, d>0$.

  Find and sketch the range of $a, b$ for which the spatially homogeneous system has a stable stationary solution with $u>0$ and $v>0$.

  Considering spatial perturbations of the form $\cos (k x)$ about the solution found above, find conditions for the system to be unstable. Sketch this region in the $(d, b)$ plane for fixed $a \in(0,1)$.

  Find $k_{c}$, the critical wavenumber at the onset of the instability, in terms of $a$ and $b$.

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• Paper 4, Section I, A

  Mathematical Biology | Part II, 2013

  A model of two populations competing for resources takes the form

  $$\begin{aligned} \frac{d n_{1}}{d t} &=r_{1} n_{1}\left(1-n_{1}-a_{12} n_{2}\right) \\ \frac{d n_{2}}{d t} &=r_{2} n_{2}\left(1-n_{2}-a_{21} n_{1}\right) \end{aligned}$$

  where all parameters are positive. Give a brief biological interpretation of $a_{12}, a_{21}, r_{1}$ and $r_{2}$. Briefly describe the dynamics of each population in the absence of the other.

  Give conditions for there to exist a steady-state solution with both populations present (that is, $n_{1}>0$ and $n_{2}>0$ ), and give conditions for this solution to be stable.

  In the case where there exists a solution with both populations present but the solution is not stable, what is the likely long-term outcome for the biological system? Explain your answer with the aid of a phase diagram in the $\left(n_{1}, n_{2}\right)$ plane.

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2012

• Paper 1, Section I, $\mathbf{6 C}$

  Mathematical Biology | Part II, 2012

  Krill is the main food source for baleen whales. The following model has been proposed for the coupled evolution of populations of krill and whales, with $x(t)$ being the number of krill and $y(t)$ being the number of whales:

  $$\begin{array}{r} \frac{d x}{d t}=r x\left(1-\frac{x}{K}\right)-a x y \\ \frac{d y}{d t}=s y\left(1-\frac{y}{b x}\right) \end{array}$$

  where $r, s, a, b$ and $K$ are positive constants.

  Give a biological interpretation for the form of the two differential equations.

  Show that a steady state is possible with $x>0$ and $y>0$ and write down expressions for the steady-state values of $x$ and $y$.

  Determine whether this steady state is stable.

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• Paper 2, Section I, $\mathbf{6 C}$

  Mathematical Biology | Part II, 2012

  Consider a birth-death process in which the birth rate per individual is $\lambda$ and the death rate per individual in a population of size $n$ is $\beta n$.

  Let $P(n, t)$ be the probability that the population has size $n$ at time $t$. Write down the master equation for the system, giving an expression for $\partial P(n, t) / \partial t$.

  Show that

  $$\frac{d}{d t}\langle n\rangle=\lambda\langle n\rangle-\beta\left\langle n^{2}\right\rangle$$

  where $\langle.\rangle$ denotes the mean.

  Deduce that in a steady state $\langle n\rangle \leqslant \lambda / \beta$.

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• Paper 2, Section II, C

  Mathematical Biology | Part II, 2012

  A population of blowflies is modelled by the equation

  $$\frac{d x}{d t}=R(x(t-T))-k x(t)$$

  where $k$ is a constant death rate and $R$ is a function of one variable such that $R(z)>0$ for $z>0$, with $R(z) \sim \beta z$ as $z \rightarrow 0$ and $R(z) \rightarrow 0$ as $z \rightarrow \infty$. The constants $T, k$ and $\beta$ are all positive, with $\beta>k$. Give a brief biological motivation for the term $R(x(t-T))$, in which you explain both the form of the function $R$ and the appearance of a delay time $T$.

  A suitable model for $R(z)$ is $\beta z \exp (-z / d)$, where $d$ is a positive constant. Show that in this case there is a single steady state of the system with non-zero population, i.e. with $x(t)=x_{s}>0$, with $x_{s}$ constant.

  Now consider the stability of this steady state. Show that if $x(t)=x_{s}+y(t)$, with $y(t)$ small, then $y(t)$ satisfies a delay differential equation of the form

  $$\frac{d y}{d t}=-k y(t)+B y(t-T),$$

  where $B$ is a constant to be determined. Show that $y(t)=e^{s t}$ is a solution of (2) if $s=-k+B e^{-s T}$. If $s=\sigma+i \omega$, where $\sigma$ and $\omega$ are both real, write down two equations relating $\sigma$ and $\omega$.

  Deduce that the steady state is stable if $|B|<k$. Show that, for this particular model for $R,|B|>k$ is possible only if $B<0$.

  By considering $B$ decreasing from small negative values, show that an instability will appear when $|B|>\left[k^{2}+\frac{g(k T)^{2}}{T^{2}}\right]^{1 / 2}$, where $\pi / 2<g(k T)<\pi$.

  Deduce that the steady state $x_{s}$ of (1) is unstable if

  $$\beta>k \exp \left[\left(1+\frac{\pi^{2}}{k^{2} T^{2}}\right)^{1 / 2}+1\right]$$

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• Paper 3, Section I, C

  Mathematical Biology | Part II, 2012

  Consider a model of insect dispersal in two dimensions given by

  $$\frac{\partial C}{\partial t}=\frac{1}{r} \frac{\partial}{\partial r}\left(r D C \frac{\partial C}{\partial r}\right)$$

  where $r$ is a radial coordinate, $t$ is time, $C(r, t)$ is the density of insects and $D$ is a constant coefficient such that $D C$ is a diffusivity.

  Show that under suitable assumptions

  $$2 \pi \int_{0}^{\infty} r C d r=N$$

  where $N$ is constant, and interpret this condition.

  Suppose that after a long time the form of $C$ depends only on $r, t, D$ and $N$ (and is thus independent of any detailed form of the initial condition). Show that there is a solution of the form

  $$C(r, t)=\left(\frac{N}{D t}\right)^{1 / 2} g\left(\frac{r}{(N D t)^{1 / 4}}\right) \text {, }$$

  and deduce that the function $g(\xi)$ satisfies

  $$\frac{d}{d \xi}\left(\xi g \frac{d g}{d \xi}+\frac{1}{4} \xi^{2} g\right)=0$$

  Show that this equation has a continuous solution with $g>0$ for $\xi<\xi_{0}$ and $g=0$ for $\xi \geqslant \xi_{0}$, and determine $\xi_{0}$. Hence determine the area within which $C(r, t)>0$ as a function of $t$.

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• Paper 3, Section II, C

  Mathematical Biology | Part II, 2012

  Consider the two-variable reaction-diffusion system

  $$\begin{gathered} \frac{\partial u}{\partial t}=a-u+u^{2} v+\nabla^{2} u \\ \frac{\partial v}{\partial t}=b-u^{2} v+d \nabla^{2} v \end{gathered}$$

  where $a, b$ and $d$ are positive constants.

  Show that there is one possible spatially homogeneous steady state with $u>0$ and $v>0$ and show that it is stable to small-amplitude spatially homogeneous disturbances provided that $\gamma<\beta$, where

  $$\gamma=\frac{b-a}{b+a} \quad \text { and } \quad \beta=(a+b)^{2}$$

  Now assuming that the condition $\gamma<\beta$ is satisfied, investigate the stability of the homogeneous steady state to spatially varying perturbations by considering the timedependence of disturbances whose spatial form is such that $\nabla^{2} u=-k^{2} u$ and $\nabla^{2} v=-k^{2} v$, with $k$ constant. Show that such disturbances vary as $e^{p t}$, where $p$ is one of the roots of

  $$p^{2}+\left(\beta-\gamma+d k^{2}+k^{2}\right) p+d k^{4}+(\beta-d \gamma) k^{2}+\beta$$

  By comparison with the stability condition for the homogeneous case above, give a simple argument as to why the system must be stable if $d=1$.

  Show that the boundary between stability and instability (as some combination of $\beta, \gamma$ and $d$ is varied) must correspond to $p=0$.

  Deduce that $d \gamma>\beta$ is a necessary condition for instability and, furthermore, that instability will occur for some $k$ if

  $$d>\frac{\beta}{\gamma}\left\{1+\frac{2}{\gamma}+2 \sqrt{\frac{1}{\gamma}+\frac{1}{\gamma^{2}}}\right\} .$$

  Deduce that the value of $k^{2}$ at which instability occurs as the stability boundary is crossed is given by

  $$k^{2}=\sqrt{\frac{\beta}{d}}$$

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• Paper 4, Section I, $\mathbf{6 C}$

  Mathematical Biology | Part II, 2012

  The master equation describing the evolution of the probability $P(n, t)$ that a population has $n$ members at time $t$ takes the form

  $$\frac{\partial P(n, t)}{\partial t}=b(n-1) P(n-1, t)-[b(n)+d(n)] P(n, t)+d(n+1) P(n+1, t)$$

  where the functions $b(n)$ and $d(n)$ are both positive for all $n$.

  From (1) derive the corresponding Fokker-Planck equation in the form

  $$\frac{\partial P(x, t)}{\partial t}=-\frac{\partial}{\partial x}\left\{a_{1}(x) P(x, t)\right\}+\frac{1}{2} \frac{\partial^{2}}{\partial x^{2}}\left\{a_{2}(x) P(x, t)\right\}$$

  making clear any assumptions that you make and giving explicit forms for $a_{1}(x)$ and $a_{2}(x)$.

  Assume that (2) has a steady state solution $P_{s}(x)$ and that $a_{1}(x)$ is a decreasing function of $x$ with a single zero at $x_{0}$. Under what circumstances may $P_{s}(x)$ be approximated by a Gaussian centred at $x_{0}$ and what is the corresponding estimate of the variance?

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2011

• Paper 1, Section I, B

  Mathematical Biology | Part II, 2011

  A proposed model of insect dispersal is given by the equation

  $$\frac{\partial n}{\partial t}=D \frac{\partial}{\partial x}\left[\left(\frac{n_{0}}{n}\right) \frac{\partial n}{\partial x}\right]$$

  where $n(x, t)$ is the density of insects and $D$ and $n_{0}$ are constants.

  Interpret the term on the right-hand side.

  Explain why a solution of the form

  $$n(x, t)=n_{0}(D t)^{-\beta} g\left(x /(D t)^{\beta}\right),$$

  where $\beta$ is a positive constant, can potentially represent the dispersal of a fixed number $n_{0}$ of insects initially localised at the origin.

  Show that the equation (1) can be satisfied by a solution of the form (2) if $\beta=1$ and find the corresponding function $g$.

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• Paper 2, Section I, B

  Mathematical Biology | Part II, 2011

  A population with variable growth and harvesting is modelled by the equation

  $$u_{t+1}=\max \left(\frac{r u_{t}^{2}}{1+u_{t}^{2}}-E u_{t}, 0\right)$$

  where $r$ and $E$ are positive constants.

  Given that $r>1$, show that a non-zero steady state exists if $0<E<E_{m}(r)$, where $E_{m}(r)$ is to be determined.

  Show using a cobweb diagram that, if $E<E_{m}(r)$, a non-zero steady state may be attained only if the initial population $u_{0}$ satisfies $\alpha<u_{0}<\beta$, where $\alpha$ should be determined explicitly and $\beta$ should be specified as a root of an algebraic equation.

  With reference to the cobweb diagram, give an additional criterion that implies that $\alpha<u_{0}<\beta$ is a sufficient condition, as well as a necessary condition, for convergence to a non-zero steady state.

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• Paper 2, Section II, B

  Mathematical Biology | Part II, 2011

  Consider a population subject to the following birth-death process. When the number of individuals in the population is $n$, the probability of an increase from $n$ to $n+1$ in unit time is $\beta n+\gamma$ and the probability of a decrease from $n$ to $n-1$ is $\alpha n(n-1)$, where $\alpha, \beta$ and $\gamma$ are constants.

  Show that the master equation for $P(n, t)$, the probability that at time $t$ the population has $n$ members, is

  $$\frac{\partial P}{\partial t}=\alpha n(n+1) P(n+1, t)-\alpha n(n-1) P(n, t)+(\beta n-\beta+\gamma) P(n-1, t)-(\beta n+\gamma) P(n, t) .$$

  Show that $\langle n\rangle$, the mean number of individuals in the population, satisfies

  $$\frac{d\langle n\rangle}{d t}=-\alpha\left\langle n^{2}\right\rangle+(\alpha+\beta)\langle n\rangle+\gamma$$

  Deduce that, in a steady state,

  $$\langle n\rangle=\frac{\alpha+\beta}{2 \alpha} \pm \sqrt{\frac{(\alpha+\beta)^{2}}{4 \alpha^{2}}+\frac{\gamma}{\alpha}-(\Delta n)^{2}}$$

  where $\Delta n$ is the standard deviation of $n$. When is the minus sign admissable?

  Show how a Fokker-Planck equation of the form

  $$\frac{\partial P}{\partial t}=\frac{\partial}{\partial n}[g(n) P(n, t)]+\frac{1}{2} \frac{\partial^{2}}{\partial n^{2}}[h(n) P(n, t)]$$

  may be derived under conditions to be explained, where the functions $g(n)$ and $h(n)$ should be evaluated.

  In the case $\alpha \ll \gamma$ and $\beta=0$, find the leading-order approximation to $n_{*}$ such that $g\left(n_{*}\right)=0$. Defining the new variable $x=n-n_{*}$, where $g\left(n_{*}\right)=0$, approximate $g(n)$ by $g^{\prime}\left(n_{*}\right) x$ and $h(n)$ by $h\left(n_{*}\right)$. Solve $(*)$ for $P(x)$ in the steady-state limit and deduce leading-order estimates for $\langle n\rangle$ and $(\Delta n)^{2}$.

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• Paper 3, Section I, B

  Mathematical Biology | Part II, 2011

  The dynamics of a directly transmitted microparasite can be modelled by the system

  $$\begin{aligned} \frac{d X}{d t} &=b N-\beta X Y-b X \\ \frac{d Y}{d t} &=\beta X Y-(b+r) Y \\ \frac{d Z}{d t} &=r Y-b Z \end{aligned}$$

  where $b, \beta$ and $r$ are positive constants and $X, Y$ and $Z$ are respectively the numbers of susceptible, infected and immune (i.e. infected by the parasite, but showing no further symptoms of infection) individuals in a population of size $N$, independent of $t$, where $N=X+Y+Z$.

  Consider the possible steady states of these equations. Show that there is a threshold population size $N_{c}$ such that if $N<N_{c}$ there is no steady state with the parasite maintained in the population. Show that in this case the number of infected and immune individuals decreases to zero for all possible initial conditions.

  Show that for $N>N_{c}$ there is a possible steady state with $X=X_{s}<N$ and $Y=Y_{s}>0$, and find expressions for $X_{s}$ and $Y_{s}$.

  By linearising the equations for $d X / d t$ and $d Y / d t$ about the steady state $X=X_{s}$ and $Y=Y_{s}$, derive a quadratic equation for the possible growth or decay rate in terms of $X_{s}$ and $Y_{s}$ and hence show that the steady state is stable.

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• Paper 3, Section II, B

  Mathematical Biology | Part II, 2011

  The number density of a population of amoebae is $n(\mathbf{x}, t)$. The amoebae exhibit chemotaxis and are attracted to high concentrations of a chemical which has concentration $a(\mathbf{x}, t)$. The equations governing $n$ and $a$ are

  $$\begin{aligned} &\frac{\partial n}{\partial t}=\alpha n\left(n_{0}^{2}-n^{2}\right)+\nabla^{2} n-\nabla \cdot(\chi(n) n \nabla a) \\ &\frac{\partial a}{\partial t}=\beta n-\gamma a+D \nabla^{2} a \end{aligned}$$

  where the constants $n_{0}, \alpha, \beta, \gamma$ and $D$ are all positive.

  (i) Give a biological interpretation of each term in these equations and discuss the sign of $\chi(n)$.

  (ii) Show that there is a non-trivial (i.e. $a \neq 0, n \neq 0$ ) steady-state solution for $n$ and $a$, independent of $\mathbf{x}$, and show further that it is stable to small disturbances that are also independent of $\mathbf{x}$.

  (iii) Consider small spatially varying disturbances to the steady state, with spatial structure such that $\nabla^{2} \psi=-k^{2} \psi$, where $\psi$ is any disturbance quantity. Show that if such disturbances also satisfy $\partial \psi / \partial t=p \psi$, where $p$ is a constant, then $p$ satisfies a quadratic equation, to be derived. By considering the conditions required for $p=0$ to be a possible solution of this quadratic equation, or otherwise, deduce that instability is possible if

  $$\beta \chi_{0} n_{0}>2 \alpha n_{0}^{2} D+\gamma+2\left(2 D \alpha n_{0}^{2} \gamma\right)^{1 / 2}$$

  where $\chi_{0}=\chi\left(n_{0}\right)$.

  (iv) Explain briefly how your conclusions might change if an additional geometric constraint implied that $k^{2}>k_{0}^{2}$, where $k_{0}$ is a given constant.

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• Paper 4, Section I, B

  Mathematical Biology | Part II, 2011

  A neglected flower garden contains $M_{n}$ marigolds in the summer of year $n$. On average each marigold produces $\gamma$ seeds through the summer. Seeds may germinate after one or two winters. After three winters or more they will not germinate. Each winter a fraction $1-\alpha$ of all seeds in the garden are eaten by birds (with no preference to the age of the seed). In spring a fraction $\mu$ of seeds that have survived one winter and a fraction $\nu$ of seeds that have survived two winters germinate. Finite resources of water mean that the number of marigolds growing to maturity from $S$ germinating seeds is $\mathcal{N}(S)$, where $\mathcal{N}(S)$ is an increasing function such that $\mathcal{N}(0)=0, \mathcal{N}^{\prime}(0)=1, \mathcal{N}^{\prime}(S)$ is a decreasing function of $S$ and $\mathcal{N}(S) \rightarrow N_{\max }$ as $S \rightarrow \infty$

  Show that $M_{n}$ satisfies the equation

  $$M_{n+1}=\mathcal{N}\left(\alpha \mu \gamma M_{n}+\nu \gamma \alpha^{2}(1-\mu) M_{n-1}\right)$$

  Write down an equation for the number $M_{*}$ of marigolds in a steady state. Show graphically that there are two solutions, one with $M_{*}=0$ and the other with $M_{*}>0$ if

  $$\alpha \mu \gamma+\nu \gamma \alpha^{2}(1-\mu)>1$$

  Show that the $M_{*}=0$ steady-state solution is unstable to small perturbations in this case.

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2010

• Paper 1, Section I, A

  Mathematical Biology | Part II, 2010

  A delay model for a population $N_{t}$ consists of

  $$N_{t+1}=\frac{r N_{t}}{1+b N_{t-1}^{2}}$$

  where $t$ is discrete time, $r>1$ and $b>0$. Investigate the linear stability about the positive steady state $N^{*}$. Show that $r=2$ is a bifurcation value at which the steady state bifurcates to a periodic solution of period 6 .

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• Paper 2, Section $I$, A

  Mathematical Biology | Part II, 2010

  The population of a certain species subjected to a specific kind of predation is modelled by the difference equation

  $$u_{t+1}=a \frac{u_{t}^{2}}{b^{2}+u_{t}^{2}}, \quad a>0$$

  Determine the equilibria and show that if $a^{2}>4 b^{2}$ it is possible for the population to be driven to extinction if it becomes less than a critical size which you should find. Explain your reasoning by means of a cobweb diagram.

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• Paper 2, Section II, A

  Mathematical Biology | Part II, 2010

  The radially symmetric spread of an insect population density $n(r, t)$ in the plane is described by the equation

  $$\frac{\partial n}{\partial t}=\frac{D_{0}}{r} \frac{\partial}{\partial r}\left[r\left(\frac{n}{n_{0}}\right)^{2} \frac{\partial n}{\partial r}\right]$$

  Suppose $Q$ insects are released at $r=0$ at $t=0$. We wish to find a similarity solution to $(*)$ in the form

  $$n(r, t)=\frac{n_{0}}{\lambda^{2}(t)} F\left(\frac{r}{r_{0} \lambda(t)}\right)$$

  Show first that the PDE $(*)$ reduces to an ODE for $F$ if $\lambda(t)$ obeys the equation

  $$\lambda^{5} \frac{d \lambda}{d t}=C \frac{D_{0}}{r_{0}^{2}}$$

  where $C$ is an arbitrary constant (that may be set to unity), and then obtain $\lambda(t)$ and $F$ such that $F(0)=1$ and $F(\xi)=0$ for $\xi \geqslant 1$. Determine $r_{0}$ in terms of $n_{0}$ and $Q$. Sketch the function $n(r, t)$ at various times to indicate its qualitative behaviour.

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• Paper 3, Section I, A

  Mathematical Biology | Part II, 2010

  A population of aerobic bacteria swims in a laterally-infinite layer of fluid occupying $-\infty<x<\infty,-\infty<y<\infty$, and $-d / 2<z<d / 2$, with the top and bottom surfaces in contact with air. Assuming that there is no fluid motion and that all physical quantities depend only on $z$, the oxygen concentration $c$ and bacterial concentration $n$ obey the coupled equations

  $$\begin{aligned} \frac{\partial c}{\partial t} &=D_{c} \frac{\partial^{2} c}{\partial z^{2}}-k n \\ \frac{\partial n}{\partial t} &=D_{n} \frac{\partial^{2} n}{\partial z^{2}}-\frac{\partial}{\partial z}\left(\mu n \frac{\partial c}{\partial z}\right) \end{aligned}$$

  Consider first the case in which there is no chemotaxis, so $n$ has the spatially-uniform value $\bar{n}$. Find the steady-state oxygen concentration consistent with the boundary conditions $c(\pm d / 2)=c_{0}$. Calculate the Fick's law flux of oxygen into the layer and justify your answer on physical grounds.

  Now allowing chemotaxis and cellular diffusion, show that the equilibrium oxygen concentration satisfies

  $$\frac{d^{2} c}{d z^{2}}-\frac{k n_{0}}{D_{c}} \exp \left(\mu c / D_{n}\right)=0$$

  where $n_{0}$ is a suitable normalisation constant that need not be found.

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• Paper 3, Section II, A

  Mathematical Biology | Part II, 2010

  Consider an epidemic model in which $S(x, t)$ is the local population density of susceptibles and $I(x, t)$ is the density of infectives

  $$\begin{aligned} &\frac{\partial S}{\partial t}=-r I S \\ &\frac{\partial I}{\partial t}=D \frac{\partial^{2} I}{\partial x^{2}}+r I S-a I \end{aligned}$$

  where $r, a$, and $D$ are positive. If $S_{0}$ is a characteristic population value, show that the rescalings $I / S_{0} \rightarrow I, S / S_{0} \rightarrow S,\left(r S_{0} / D\right)^{1 / 2} x \rightarrow x, r S_{0} t \rightarrow t$ reduce this system to

  $$\begin{aligned} &\frac{\partial S}{\partial t}=-I S \\ &\frac{\partial I}{\partial t}=\frac{\partial^{2} I}{\partial x^{2}}+I S-\lambda I \end{aligned}$$

  where $\lambda$ should be found.

  Travelling wavefront solutions are of the form $S(x, t)=S(z), I(x, t)=I(z)$, where $z=x-c t$ and $c$ is the wave speed, and we seek solutions with boundary conditions $S(\infty)=1, S^{\prime}(\infty)=0, I(\infty)=I(-\infty)=0$. Under the travelling-wave assumption reduce the rescaled PDEs to ODEs, and show by linearisation around the leading edge of the advancing front that the requirement that $I$ be non-negative leads to the condition $\lambda<1$ and hence the wave speed relation

  $$c \geqslant 2(1-\lambda)^{1 / 2}, \quad \lambda<1$$

  Using the two ODEs you have obtained, show that the surviving susceptible population fraction $\sigma=S(-\infty)$ after the passage of the front satisfies

  $$\sigma-\lambda \ln \sigma=1,$$

  and sketch $\sigma$ as a function of $\lambda$.

  comment

• Paper 4, Section I, A

  Mathematical Biology | Part II, 2010

  A concentration $u(x, t)$ obeys the differential equation

  $$\frac{\partial u}{\partial t}=D u_{x x}+f(u)$$

  in the domain $0 \leqslant x \leqslant L$, with boundary conditions $u(0, t)=u(L, t)=0$ and initial condition $u(x, 0)=u_{0}(x)$, and where $D$ is a positive constant. Assume $f(0)=0$ and $f^{\prime}(0)>0$. Linearising the dynamics around $u=0$, and representing $u(x, t)$ as a suitable Fourier expansion, show that the condition for the linear stability of $u=0$ can be expressed as the following condition on the domain length

  $$L<\pi\left[\frac{D}{f^{\prime}(0)}\right]^{1 / 2}$$

  comment

2009

• Paper 1, Section I, A

  Mathematical Biology | Part II, 2009

  A discrete model for a population $N_{t}$ consists of

  $$N_{t+1}=\frac{r N_{t}}{\left(1+b N_{t}\right)^{2}},$$

  where $t$ is discrete time and $r, b>0$. What do $r$ and $b$ represent in this model? Show that for $r>1$ there is a stable fixed point.

  Suppose the initial condition is $N_{1}=1 / b$, and that $r>4$. Show, with the help of a cobweb, that the population $N_{t}$ is bounded by

  $$\frac{4 r^{2}}{(4+r)^{2} b} \leqslant N_{t} \leqslant \frac{r}{4 b}$$

  and attains those bounds.

  comment

• Paper 2, Section $I$, A

  Mathematical Biology | Part II, 2009

  Consider the reaction system

  $$A \stackrel{k_{1}}{\longrightarrow} X, \quad B+X \stackrel{k_{2}}{\longrightarrow} Y, \quad 2 X+Y \stackrel{k_{3}}{\longrightarrow} 3 X, \quad X \stackrel{k_{4}}{\longrightarrow} E,$$

  where the $k$ s are the rate constants, and the reactant concentrations of $A$ and $B$ are kept constant. Write down the governing differential equation system for the concentrations of $X$ and $Y$ and nondimensionalise the equations by setting $u=\alpha X$ and $v=\alpha Y, \tau=k_{4} t$ so that they become

  $$\frac{d u}{d \tau}=1-(b+1) u+a u^{2} v, \quad \frac{d v}{d \tau}=b u-a u^{2} v$$

  by suitable choice of $\alpha$. Thus find $a$ and $b$. Determine the positive steady state and show that there is a bifurcation value $b=b_{c}=1+a$ at which the steady state becomes unstable to a Hopf bifurcation. Find the period of the oscillations in the neighbourhood of $b_{c}$.

  comment

• Paper 2, Section II, A

  Mathematical Biology | Part II, 2009

  Travelling bands of microorganisms, chemotactically directed, move into a food source, consuming it as they go. A model for this is given by

  $$b_{t}=\frac{\partial}{\partial x}\left[D b_{x}-\frac{b \chi}{a} a_{x}\right], \quad a_{t}=-k b$$

  where $b(x, t)$ and $a(x, t)$ are the bacteria and nutrient respectively and $D, \chi$, and $k$ are positive constants. Look for travelling wave solutions, as functions of $z=x-c t$ where $c$ is the wave speed, with the boundary conditions $b \rightarrow 0$ as $|z| \rightarrow \infty, a \rightarrow 0$ as $z \rightarrow-\infty$, $a \rightarrow 1$ as $z \rightarrow \infty$. Hence show that $b(z)$ and $a(z)$ satisfy

  $$b^{\prime}=\frac{b}{c D}\left[\frac{k b \chi}{a}-c^{2}\right], \quad a^{\prime}=\frac{k b}{c}$$

  where the prime denotes differentiation with respect to $z$. Integrating $d b / d a$, find an algebraic relationship between $b(z)$ and $a(z)$.

  In the special case where $\chi=2 D$ show that

  $$a(z)=\left[1+K e^{-c z / D}\right]^{-1}, \quad b(z)=\frac{c^{2}}{k D} e^{-c z / D}\left[1+K e^{-c z / D}\right]^{-2}$$

  where $K$ is an arbitrary positive constant which is equivalent to a linear translation; it may be set to 1 . Sketch the wave solutions and explain the biological interpretation.

  comment

• Paper 3, Section I, A

  Mathematical Biology | Part II, 2009

  Consider an organism moving on a one-dimensional lattice of spacing $a$, taking steps either to the right or the left at regular time intervals $\tau$. In this random walk there is a slight bias to the right, that is the probabilities of moving to the right and left, $\alpha$ and $\beta$, are such that $\alpha-\beta=\epsilon$, where $0<\epsilon \ll 1$. Write down the appropriate master equation for this process. Show by taking the continuum limit in space and time that $p(x, t)$, the probability that an organism initially at $x=0$ is at $x$ after time $t$, obeys

  $$\frac{\partial p}{\partial t}+V \frac{\partial p}{\partial x}=D \frac{\partial^{2} p}{\partial x^{2}}$$

  Express the constants $V$ and $D$ in terms of $a, \tau, \alpha$ and $\beta$.

  comment

• Paper 3, Section II, A

  Mathematical Biology | Part II, 2009

  An activator-inhibitor reaction diffusion system in dimensionless form is given by

  $$u_{t}=u_{x x}+\frac{u^{2}}{v}-b u, \quad v_{t}=d v_{x x}+u^{2}-v$$

  where $b$ and $d$ are positive constants. Which is the activitor and which the inhibitor? Determine the positive steady states and show, by an examination of the eigenvalues in a linear stability analysis of the spatially uniform situation, that the reaction kinetics is stable if $b<1$.

  Determine the conditions for the steady state to be driven unstable by diffusion. Show that the parameter domain for diffusion-driven instability is given by $0<b<1$, $b d>3+2 \sqrt{2}$, and sketch the $(b, d)$ parameter space in which diffusion-driven instability occurs. Further show that at the bifurcation to such an instability the critical wave number $k_{c}$ is given by $k_{c}^{2}=(1+\sqrt{2}) / d$.

  comment

• Paper 4, Section I, A

  Mathematical Biology | Part II, 2009

  The diffusion equation for a chemical concentration $C(r, t)$ in three dimensions which depends only on the radial coordinate $r$ is

  $$C_{t}=D \frac{1}{r^{2}}\left(r^{2} C_{r}\right)_{r}$$

  The general similarity solution of this equation takes the form

  $$C(r, t)=t^{\alpha} F(\xi), \quad \xi=\frac{r}{t^{\beta}},$$

  where $\alpha$ and $\beta$ are to be determined. By direct substitution into $(*)$ and the requirement of a valid similarity solution, find one relation involving the exponents. Use the conservation of the total number of molecules to determine a second relation. Comment on the relationship between these exponents and the ones appropriate to the similarity solution of the one-dimensional diffusion equation. Show that $F$ obeys

  $$D\left(F^{\prime \prime}+\frac{2}{\xi} F^{\prime}\right)+\frac{1}{2} \xi F^{\prime}+\frac{3}{2} F=0$$

  and that the relevant solution describing the spreading of a delta-function initial condition is $F(\xi)=A \exp \left(-\xi^{2} / 4 D\right)$, where $A$ is a suitable normalisation that need not be found.

  comment

2008

• 1.I.6B

  Mathematical Biology | Part II, 2008

  A gene product with concentration $g$ is produced by a chemical $S$ of concentration $s$, is autocatalysed and degrades linearly according to the kinetic equation

  $$\frac{d g}{d t}=f(g, s)=s+k \frac{g^{2}}{1+g^{2}}-g$$

  where $k>0$ is a constant.

  First consider the case $s=0$. Show that if $k>2$ there are two positive steady states, and determine their stability. Sketch the reaction rate $f(g, 0)$.

  Now consider $s>0$. Show that there is a single steady state if $s$ exceeds a critical value. If the system starts in the steady state $g=0$ with $s=0$ and then $s$ is increased sufficiently before decreasing back to zero, show that a biochemical switch can be achieved to a state $g=g_{2}$, whose value you should determine.

  comment

• 2.I.6B

  Mathematical Biology | Part II, 2008

  The population dynamics of a species is governed by the discrete model

  $$N_{t+1}=f\left(N_{t}\right)=N_{t} \exp \left[r\left(1-\frac{N_{t}}{K}\right)\right]$$

  where $r$ and $K$ are positive constants.

  Determine the steady states and their eigenvalues. Show that a period-doubling bifurcation occurs at $r=2$.

  Show graphically that the maximum possible population after $t=0$ is

  $$N_{\max }=f(K / r) .$$

  comment

• 2.II.13B

  Mathematical Biology | Part II, 2008

  Consider the nonlinear equation describing the invasion of a population $u(x, t)$

  $$u_{t}=m u_{x x}+f(u)$$

  with $m>0, f(u)=-u(u-r)(u-1)$ and $0<r<1$ a constant.

  (a) Considering time-dependent spatially homogeneous solutions, show that there are two stable and one unstable uniform steady states.

  (b) In the case $r=\frac{1}{2}$, find the stationary 'front' which has

  $$u \rightarrow 1 \text { as } x \rightarrow-\infty \quad \text { and } \quad u \rightarrow 0 \text { as } x \rightarrow \infty$$

  [Hint: $f(u)=F^{\prime}(u)$ where $F(u)=-\frac{1}{4} u^{2}(1-u)^{2}+\frac{1}{6}\left(r-\frac{1}{2}\right) u^{2}(2 u-3)$.]

  (c) Now consider travelling-wave solutions to (1) of the form $u(x, t)=U(z)$ where $z=x-v t$. Show that $U$ satisfies an equation of the form

  $$m \ddot{U}+v \dot{U}=-V^{\prime}(U),$$

  where $\left({ }^{\cdot}\right) \equiv \frac{d}{d z}(\quad)$ and ()$^{\prime} \equiv \frac{d}{d U}(\quad)$.

  Sketch the form of $V(U)$ for $r=\frac{1}{2}, r>\frac{1}{2}$ and $r<\frac{1}{2}$. Using conditions (2), show that

  $$v \int_{-\infty}^{\infty} \dot{U}^{2} d z=F(1)-F(0) .$$

  Deduce how the sign of the travelling-wave velocity $v$ depends on $r$.

  comment

• 3.I.6B

  Mathematical Biology | Part II, 2008

  An allosteric enzyme $E$ reacts with a substrate $S$ to produce a product $P$ according to the mechanism

  $$\begin{gathered} S+E \underset{k_{-1}}{\stackrel{k_{1}}{\rightleftharpoons}} C_{1} \stackrel{k_{2}}{\longrightarrow} E+P \\ S+C_{1} \underset{k_{-3}}{\stackrel{k_{3}}{\rightleftharpoons}} C_{2} \stackrel{k_{4}}{\rightarrow} C_{1}+P \end{gathered}$$

  where $C_{1}$ and $C_{2}$ are enzyme-substrate complexes. With lowercase letters denoting concentrations, write down a system of differential equations based on the Law of Mass Action which model this reaction mechanism.

  The initial conditions are $s=s_{0}, e=e_{0}, c_{1}=c_{2}=p=0 .$ Using $u=s / s_{0}$, $v_{i}=c_{i} / e_{0}, \tau=k_{1} e_{0} t$ and $\epsilon=e_{0} / s_{0}$, show that the nondimensional reaction mechanism reduces to

  $$\frac{d u}{d \tau}=f\left(u, v_{1}, v_{2}\right) \quad \text { and } \quad \epsilon \frac{d v_{i}}{d \tau}=g_{i}\left(u, v_{1}, v_{2}\right) \quad \text { for } \quad i=1,2$$

  finding expressions for $f, g_{1}$ and $g_{2}$.

  comment

• 3.II.13B

  Mathematical Biology | Part II, 2008

  Consider the activator-inhibitor system in the fast-inhibitor limit

  $$\begin{gathered} u_{t}=D u_{x x}-u(u-r)(u-1)-\rho(v-u) \\ 0=v_{x x}-(v-u) \end{gathered}$$

  where $D$ is small, $0<r<1$ and $0<\rho<1$.

  Examine the linear stability of the state $u=v=0$ using perturbations of the form $\exp (i k x+\sigma t)$. Sketch the growth-rate $\sigma$ as a function of the wavenumber $k$. Find the growth-rate of the most unstable wave, and so determine the boundary in the $r$ - $\rho$ parameter plane which separates stable and unstable modes.

  Show that the system is unchanged under the transformation $u \rightarrow 1-u, v \rightarrow 1-v$ and $r \rightarrow 1-r$. Hence write down the equation for the boundary between stable and unstable modes of the state $u=v=1$.

  comment

• 4.I.6B

  Mathematical Biology | Part II, 2008

  A semi-infinite elastic filament lies along the positive $x$-axis in a viscous fluid. When it is perturbed slightly to the shape $y=h(x, t)$, it evolves according to

  $$\zeta h_{t}=-A h_{x x x x}$$

  where $\zeta$ characterises the viscous drag and $A$ the bending stiffness. Motion is forced by boundary conditions

  $$h=h_{0} \cos (\omega t) \quad \text { and } \quad h_{x x}=0 \quad \text { at } \quad x=0, \quad \text { while } \quad h \rightarrow 0 \quad \text { as } \quad x \rightarrow \infty$$

  Use dimensional analysis to find the characteristic length $\ell(\omega)$ of the disturbance. Show that the steady oscillating solution takes the form

  $$h(x, t)=h_{0} \operatorname{Re}\left[e^{i \omega t} F(\eta)\right] \quad \text { with } \quad \eta=x / \ell,$$

  finding the ordinary differential equation for $F$.

  Find two solutions for $F$ which decay as $x \rightarrow \infty$. Without solving explicitly for the amplitudes, show that $h(x, t)$ is the superposition of two travelling waves which decay with increasing $x$, one with crests moving to the left and one to the right. Which dominates?

  comment

2007

• $2 . \mathrm{I} . 6 \mathrm{~B} \quad$

  Mathematical Biology | Part II, 2007

  A field contains $X_{n}$ seed-producing poppies in August of year $n$. On average each poppy produces $\gamma$ seeds, a number that is assumed not to vary from year to year. A fraction $\sigma$ of seeds survive the winter and a fraction $\alpha$ of those germinate in May of year $n+1$. A fraction $\beta$ of those that survive the next winter germinate in year $n+2$. Show that $X_{n}$ satisfies the following difference equation:

  $$X_{n+1}=\alpha \sigma \gamma X_{n}+\beta \sigma^{2}(1-\alpha) \gamma X_{n-1}$$

  Write down the general solution of this equation, and show that the poppies in the field will eventually die out if

  $$\sigma \gamma[(1-\alpha) \beta \sigma+\alpha]<1$$

  comment

• 1.I.6B

  Mathematical Biology | Part II, 2007

  A chemostat is a well-mixed tank of given volume $V_{0}$ that contains water in which lives a population $N(t)$ of bacteria that consume nutrient whose concentration is $C(t)$ per unit volume. An inflow pipe supplies a solution of nutrient at concentration $C_{0}$ and at a constant flow rate of $Q$ units of volume per unit time. The mixture flows out at the same rate through an outflow pipe. The bacteria consume the nutrient at a rate $N K(C)$, where

  $$K(C)=\frac{K_{\max } C}{K_{0}+C}$$

  and the bacterial population grows at a rate $\gamma N K(C)$, where $0<\gamma<1$.

  Write down the differential equations for $N(t), C(t)$ and show that they can be rescaled into the following form:

  $$\begin{aligned} &\frac{d n}{d \tau}=\alpha \frac{c n}{1+c}-n \\ &\frac{d c}{d \tau}=-\frac{c n}{1+c}-c+\beta \end{aligned}$$

  where $\alpha, \beta$ are positive constants, to be found.

  Show that this system of equations has a non-trivial steady state if $\alpha>1$ and $\beta>\frac{1}{\alpha-1}$, and that it is stable.

  comment

• 2.II.13B

  Mathematical Biology | Part II, 2007

  Show that the concentration $C(\mathbf{x}, t)$ of a diffusible chemical substance in a stationary medium satisfies the partial differential equation

  $$\frac{\partial C}{\partial t}=\nabla \cdot(D \nabla C)+F$$

  where $D$ is the diffusivity and $F(\mathbf{x}, t)$ is the rate of supply of the chemical.

  A finite amount of the chemical, $4 \pi M$, is supplied at the origin at time $t=0$, and spreads out in a spherically symmetric manner, so that $C=C(r, t)$ for $r>0, t>0$, where $r$ is the radial coordinate. The diffusivity is given by $D=k C$, for constant $k$. Show, by dimensional analysis or otherwise, that it is appropriate to seek a similarity solution in which

  $$C=\frac{M^{\alpha}}{(k t)^{\beta}} f(\xi), \quad \xi=\frac{r}{(M k t)^{\gamma}} \quad \text { and } \quad \int_{0}^{\infty} \xi^{2} f(\xi) d \xi=1$$

  where $\alpha, \beta, \gamma$ are constants to be determined, and derive the ordinary differential equation satisfied by $f(\xi)$.

  Solve this ordinary differential equation, subject to appropriate boundary conditions, and deduce that the chemical occupies a finite spherical region of radius

  $$r_{0}(t)=(75 M k t)^{1 / 5}$$

  [Note: in spherical polar coordinates

  $$\left.\nabla C \equiv\left(\frac{\partial C}{\partial r}, 0,0\right) \quad \text { and } \quad \nabla \cdot(V(r, t), 0,0) \equiv \frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} V\right) \cdot\right]$$

  comment

• 3.I.6B

  Mathematical Biology | Part II, 2007

  Consider a birth and death process in which births always give rise to two offspring, with rate $\lambda$, while the death rate per individual is $\beta$.

  Write down the master equation (or probability balance equation) for this system.

  Show that the population mean is given by

  $$\langle n\rangle=\frac{2 \lambda}{\beta}\left(1-e^{-\beta t}\right)+n_{0} e^{-\beta t}$$

  where $n_{0}$ is the initial population mean, and that the population variance satisfies

  $$\sigma^{2} \rightarrow 3 \lambda / \beta \quad \text { as } \quad t \rightarrow \infty$$

  comment

• 3.II.13B

  Mathematical Biology | Part II, 2007

  The number density of a population of cells is $n(\mathbf{x}, t)$. The cells produce a chemical whose concentration is $C(\mathbf{x}, t)$ and respond to it chemotactically. The equations governing $n$ and $C$ are

  $$\begin{aligned} \frac{\partial n}{\partial t} &=\gamma n\left(n_{0}-n\right)+D_{n} \nabla^{2} n-\chi \nabla \cdot(n \nabla C) \\ \frac{\partial C}{\partial t} &=\alpha n-\beta C+D_{c} \nabla^{2} C \end{aligned}$$

  (i) Give a biological interpretation of each term in these equations, where you may assume that $\alpha, \beta, \gamma, n_{0}, D_{n}, D_{c}$ and $\chi$ are all positive.

  (ii) Show that there is a steady-state solution that is stable to spatially invariant disturbances.

  (iii) Analyse small, spatially-varying perturbations to the steady state that satisfy $\nabla^{2} \phi=-k^{2} \phi$ for any variable $\phi$, and show that a chemotactic instability is possible if

  $$\chi \alpha n_{0}>\beta D_{n}+\gamma n_{0} D_{c}+\left(4 \beta \gamma n_{0} D_{n} D_{c}\right)^{1 / 2}$$

  (iv) Find the critical value of $k$ at which the instability first appears as $\chi$ is increased.

  comment

• 4.I.6B

  Mathematical Biology | Part II, 2007

  The non-dimensional equations for two competing populations are

  $$\begin{aligned} &\frac{d u}{d t}=u(1-v)-\epsilon_{1} u^{2} \\ &\frac{d v}{d t}=\alpha\left[v(1-u)-\epsilon_{2} v^{2}\right] \end{aligned}$$

  Explain the meaning of each term in these equations.

  Find all the fixed points of this system when $\alpha>0,0<\epsilon_{1}<1$ and $0<\epsilon_{2}<1$, and investigate their stability.

  comment

2006

• 1.I.6B

  Mathematical Biology | Part II, 2006

  A large population of some species has probability $P(n, t)$ of taking the value $n$ at time $t$. Explain the use of the generating function $\phi(s, t)=\sum_{n=0}^{\infty} s^{n} P(n, t)$, and give expressions for $P(n, t)$ and $\langle n\rangle$ in terms of $\phi$.

  A particular population is subject to a birth-death process, so that the probability of an increase from $n$ to $n+1$ in unit time is $\alpha+\beta n$, while the probability of a decrease from $n$ to $n-1$ is $\gamma n$, with $\gamma>\beta$. Show that the master equation for $P(n, t)$ is

  $$\frac{\partial P(n, t)}{\partial t}=(\alpha+\beta(n-1)) P(n-1, t)+\gamma(n+1) P(n+1, t)-(\alpha+(\beta+\gamma) n) P(n, t)$$

  Derive the equation satisfied by $\phi$, and show that in the statistically steady state, when $\phi$ and $P$ are independent of time, $\phi$ takes the form

  $$\phi(s)=\left(\frac{\gamma-\beta}{\gamma-\beta s}\right)^{\alpha / \beta}$$

  Using the equation for $\phi$, or otherwise, find $\langle n\rangle$.

  comment

• 2.I.6B

  Mathematical Biology | Part II, 2006

  Two interacting populations of prey and predators, with populations $u(t), v(t)$ respectively, obey the evolution equations (with all parameters positive)

  $$\begin{aligned} &\frac{d u}{d t}=u\left(\mu_{1}-\alpha_{1} v-\delta u\right) \\ &\frac{d v}{d t}=v\left(-\mu_{2}+\alpha_{2} u\right)-\epsilon \end{aligned}$$

  Give an explanation in terms of population dynamics of each of the terms in these equations.

  Show that if $\alpha_{2} \mu_{1}>\delta \mu_{2}$ there are two non-trivial fixed points with $u, v \neq 0$, provided $\epsilon$ is sufficiently small. Find the trace and determinant of the Jacobian in terms of $u, v$ and show that, when $\delta$ and $\epsilon$ are very small, the fixed point with $u \approx \mu_{1} / \delta$, $v \approx \epsilon \delta / \mu_{1} \alpha_{2}$ is always unstable.

  comment

• 2.II.13B

  Mathematical Biology | Part II, 2006

  Consider the discrete predator-prey model for two populations $N_{t}, P_{t}$ of prey and predators, respectively:

  $$\left.\begin{array}{rl} N_{t+T} & =r N_{t} \exp \left(-a P_{t}\right) \\ P_{t+T} & =s N_{t}\left(1-b \exp \left(-a P_{t}\right)\right) \end{array}\right\},$$

  where $r, s, a, b$ are constants, all assumed to be positive.

  (a) Give plausible explanations of the meanings of $T, r, s, a, b$.

  (b) Nondimensionalize equations $(*)$ to show that with appropriate rescaling they may be reduced to the form

  $$\left.\begin{array}{l} n_{t+1}=r n_{t} \exp \left(-p_{t}\right) \\ p_{t+1}=n_{t}\left(1-b \exp \left(-p_{t}\right)\right) \end{array}\right\}$$

  (c) Now assume that $b<1, r>1$. Show that the origin is unstable, and that there is a nontrivial fixed point $(n, p)=\left(n_{c}(b, r), p_{c}(b, r)\right)$. Investigate the stability of this point by writing $\left(n_{t}, p_{t}\right)=\left(n_{c}+n_{t}^{\prime}, p_{c}+p_{t}^{\prime}\right)$ and linearizing. Express the linearized equations as a second order recurrence relation for $n_{t}^{\prime}$, and hence show that $n_{t}^{\prime}$ satisfies an equation of the form

  $$n_{t}^{\prime}=A \lambda_{1}^{t}+B \lambda_{2}^{t}$$

  where the quantities $\lambda_{1,2}$ satisfy $\lambda_{1}+\lambda_{2}=1+b n_{c} / r, \lambda_{1} \lambda_{2}=n_{c}$ and $A, B$ are constants. Give a similar expression for $p_{t}^{\prime}$ for the same values of $A, B$.

  Show that when $r$ is just greater than unity the $\lambda_{i}(i=1,2)$ are real and both less than unity, while if $n_{c}$ is just greater than unity then the $\lambda_{i}$ are complex with modulus greater than one. Show also that $n_{c}$ increases monotonically with $r$ and that if the roots are real neither of them can be unity.

  Deduce that the fixed point is stable for sufficiently small $r$ but loses stability for a value of $r$ that depends on $b$ but is certainly less than $e=\exp (1)$. Give an equation that determines the value of $r$ where stability is lost, and an equation that gives the argument of the eigenvalue at this point. Sketch the behaviour of the moduli of the eigenvalues as functions of $r$.

  comment

• 3.I.6B

  Mathematical Biology | Part II, 2006

  The SIR epidemic model for an infectious disease divides the population $N$ into three categories of susceptible $S(t)$, infected $I(t)$ and recovered (non-infectious) $R(t)$. It is supposed that the disease is non-lethal, so that the population does not change in time.

  Explain the reasons for the terms in the following model equations:

  $$\frac{d S}{d t}=p R-r I S, \quad \frac{d I}{d t}=r I S-a I, \quad \frac{d R}{d t}=a I-p R$$

  At time $t=0, S \approx N$ while $I, R \ll 1$.

  (a) Show that if $r N<a$ no epidemic occurs.

  (b) Now suppose that $p>0$ and there is an epidemic. Show that the system has a nontrivial fixed point, and that it is stable to small disturbances. Show also that for both small and large $p$ both the trace and the determinant of the Jacobian matrix are $O(p)$, and deduce that the matrix has complex eigenvalues for sufficiently small $p$, and real eigenvalues for sufficiently large $p$.

  comment

• 3.II.13B

  Mathematical Biology | Part II, 2006

  A chemical system with concentrations $u(x, t), v(x, t)$ obeys the coupled reactiondiffusion equations

  $$\begin{aligned} &\frac{d u}{d t}=r u+u^{2}-u v+\kappa_{1} \frac{d^{2} u}{d x^{2}} \\ &\frac{d v}{d t}=s\left(u^{2}-v\right)+\kappa_{2} \frac{d^{2} v}{d x^{2}} \end{aligned}$$

  where $r, s, \kappa_{1}, \kappa_{2}$ are constants with $s, \kappa_{1}, \kappa_{2}$ positive.

  (a) Find conditions on $r, s$ such that there is a steady homogeneous solution $u=u_{0}$, $v=u_{0}^{2}$ which is stable to spatially homogeneous perturbations.

  (b) Investigate the stability of this homogeneous solution to disturbances proportional to $\exp (i k x)$. Assuming that a solution satisfying the conditions of part (a) exists, find the region of parameter space in which the solution is stable to space-dependent disturbances, and show in particular that one boundary of this region for fixed $s$ is given by

  $$d \equiv \sqrt{\frac{\kappa_{2}}{\kappa_{1}}}=\sqrt{2 s}+\frac{1}{u_{0}} \sqrt{s\left(2 u_{0}^{2}-u_{0}\right)}$$

  Sketch the various regions of existence and stability of steady, spatially homogeneous solutions in the $\left(d, u_{0}\right)$ plane for the case $s=2$.

  (c) Show that the critical wavenumber $k=k_{c}$ for the onset of the instability satisfies the relation

  $$k_{c}^{2}=\frac{1}{\sqrt{\kappa_{1} \kappa_{2}}}\left[\frac{s(d-\sqrt{2 s})}{d(2 \sqrt{2 s}-d)}\right] .$$

  Explain carefully what happens when $d<\sqrt{2 s}$ and when $d>2 \sqrt{2 s}$.

  comment

• 4.I.6B

  Mathematical Biology | Part II, 2006

  A nonlinear model of insect dispersal with exponential death rate takes the form (for insect population $n(x, t)$ )

  $$\tag{*} \frac{\partial n}{\partial t}=-\mu n+\frac{\partial}{\partial x}\left(n \frac{\partial n}{\partial x}\right)$$

  At time $t=0$ the total insect population is $Q$, and all the insects are at the origin. A solution is sought in the form

  $$\tag{†} n=\frac{e^{-\mu t}}{\lambda(t)} f(\eta) ; \quad \eta=\frac{x}{\lambda(t)}, \quad \lambda(0)=0$$

  (a) Verify that $\int_{-\infty}^{\infty} f d \eta=Q$, provided $f$ decays sufficiently rapidly as $|x| \rightarrow \infty$.

  (b) Show, by substituting the form of $n$ given in equation $(†)$ into equation $(*)$, that $(*)$ is satisfied, for nonzero $f$, when

  $$\frac{d \lambda}{d t}=\lambda^{-2} e^{-\mu t} \quad \text { and } \quad \frac{d f}{d \eta}=-\eta$$

  Hence find the complete solution and show that the insect population is always confined to a finite region that never exceeds the range

  $$|x| \leqslant\left(\frac{9 Q}{2 \mu}\right)^{1 / 3}$$

  comment

2005

• $3 . \mathrm{I} . 6 \mathrm{E} \quad$

  Mathematical Biology | Part II, 2005

  Let $x$ be the concentration of a binary master sequence of length $L$ and let $y$ be the total concentration of all mutant sequences. Master sequences try to self-replicate at a total rate $a x$, but each independent digit is only copied correctly with probability $q$. Mutant sequences self-replicate at a total rate $b y$, where $a>b$, and the probability of mutation back to the master sequence is negligible.

  (a) The evolution of $x$ is given by

  $$\frac{d x}{d t}=a q^{L} x$$

  Write down the corresponding equation for $y$ and derive a differential equation for the master-to-mutant ratio $z=x / y$.

  (b) What is the maximum length $L_{\max }$ for which there is a positive steady-state value of $z$ ? Is the positive steady state stable when it exists?

  (c) Obtain a first-order approximation to $L_{\max }$ assuming that both $1-q \ll 1$ and $s \ll 1$, where the selection coefficient $s$ is defined by $b=a(1-s)$.

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• 1.I.6E

  Mathematical Biology | Part II, 2005

  Consider a biological system in which concentrations $x(t)$ and $y(t)$ satisfy

  $$\frac{d x}{d t}=f(y)-x \quad \text { and } \quad \frac{d y}{d t}=g(x)-y$$

  where $f$ and $g$ are positive and monotonically decreasing functions of their arguments, so that $x$ represses the synthesis of $y$ and vice versa.

  (a) Suppose the functions $f$ and $g$ are bounded. Sketch the phase plane and explain why there is always at least one steady state. Show that if there is a steady state with

  $$\frac{\partial \ln f}{\partial \ln y} \frac{\partial \ln g}{\partial \ln x}>1$$

  then the system is multistable.

  (b) If $f=\lambda /\left(1+y^{m}\right)$ and $g=\lambda /\left(1+x^{n}\right)$, where $\lambda, m$ and $n$ are positive constants, what values of $m$ and $n$ allow the system to display multistability for some value of $\lambda$ ?

  Can $f=\lambda / y^{m}$ and $g=\lambda / x^{n}$ generate multistability? Explain your answer carefully.

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• 2.I.6E

  Mathematical Biology | Part II, 2005

  Consider a system with stochastic reaction events

  $$x \stackrel{\lambda}{\longrightarrow} x+1 \quad \text { and } \quad x \stackrel{\beta x^{2}}{\longrightarrow} x-2,$$

  where $\lambda$ and $\beta$ are rate constants.

  (a) State or derive the exact differential equation satisfied by the average number of molecules $<x>$. Assuming that fluctuations are negligible, approximate the differential equation to obtain the steady-state value of $\langle x\rangle$.

  (b) Using this approximation, calculate the elasticity $H$, the average lifetime $\tau$, and the average chemical event size $<r>$ (averaged over fluxes).

  (c) State the stationary Fluctuation Dissipation Theorem for the normalised variance $\eta$. Hence show that

  $$\eta=\frac{3}{4<x>} .$$

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• 2.II.13E

  Mathematical Biology | Part II, 2005

  Consider the reaction-diffusion system

  $$\begin{aligned} &\frac{\partial u}{\partial \tau}=\beta_{u}\left(\frac{u^{2}}{v}-u\right)+D_{u} \frac{\partial^{2} u}{\partial x^{2}} \\ &\frac{\partial v}{\partial \tau}=\beta_{v}\left(u^{2}-v\right)+D_{v} \frac{\partial^{2} v}{\partial x^{2}} \end{aligned}$$

  for an activator $u$ and inhibitor $v$, where $\beta_{u}$ and $\beta_{v}$ are degradation rate constants and $D_{u}$ and $D_{v}$ are diffusion rate constants.

  (a) Find a suitably scaled time $t$ and length $s$ such that

  $$\begin{aligned} \frac{\partial u}{\partial t} &=\frac{u^{2}}{v}-u+\frac{\partial^{2} u}{\partial s^{2}} \\ \frac{1}{Q} \frac{\partial v}{\partial t} &=u^{2}-v+P \frac{\partial^{2} v}{\partial s^{2}} \end{aligned}$$

  and find expressions for $P$ and $Q$.

  (b) Show that the Jacobian matrix for small spatially homogenous deviations from a nonzero steady state of $(*)$ is

  $$J=\left(\begin{array}{cc} 1 & -1 \\ 2 Q & -Q \end{array}\right)$$

  and find the values of $Q$ for which the steady state is stable.

  [Hint: The eigenvalues of a $2 \times 2$ real matrix both have positive real parts iff the matrix has a positive trace and determinant.]

  (c) Derive linearised ordinary differential equations for the amplitudes $\hat{u}(t)$ and $\hat{v}(t)$ of small spatially inhomogeneous deviations from a steady state of $(*)$ that are proportional to $\cos (s / L)$, where $L$ is a constant.

  (d) Assuming that the system is stable to homogeneous perturbations, derive the condition for inhomogeneous instability. Interpret this condition in terms of how far activator and inhibitor molecules diffuse on average before they are degraded.

  (e) Calculate the lengthscale $L_{\text {crit }}$ of disturbances that are expected to be observed when the condition for inhomogeneous instability is just satisfied. What are the dominant mechanisms for stabilising disturbances on lengthscales (i) much less than and (ii) much greater than $L_{\text {crit }}$ ?

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• 3.II.13E

  Mathematical Biology | Part II, 2005

  Protein synthesis by RNA can be represented by the stochastic system

  $$\begin{array}{lll} x_{1} \stackrel{\lambda_{1}}{\longrightarrow} x_{1}+1 & \text { and } & x_{1} \stackrel{\beta_{1} x_{1}}{\longrightarrow} x_{1}-1 \\ x_{2} \stackrel{\lambda_{2} x_{1}}{\longrightarrow} x_{2}+1 & \text { and } & x_{2} \stackrel{\beta_{2} x_{2}}{\longrightarrow} x_{2}-1 \end{array}$$

  in which $x_{1}$ is an environmental variable corresponding to the number of RNA molecules per cell and $x_{2}$ is a system variable, with birth rate proportional to $x_{1}$, corresponding to the number of protein molecules.

  (a) Use the normalized stationary Fluctuation-Dissipation Theorem (FDT) to calculate the (exact) normalized stationary variances $\eta_{11}=\sigma_{1}^{2} /<x_{1}>^{2}$ and $\eta_{22}=$ $\sigma_{2}^{2} /<x_{2}>^{2}$ in terms of the averages $<x_{1}>$ and $<x_{2}>$.

  (b) Separate $\eta_{22}$ into an intrinsic and an extrinsic term by considering the limits when $x_{1}$ does not fluctuate (intrinsic), and when $x_{2}$ responds deterministically to changes in $x_{1}$ (extrinsic). Explain how the extrinsic term represents the magnitude of environmental fluctuations and time-averaging.

  (c) Assume now that the birth rate of $x_{2}$ is changed from the "constitutive" mechanism $\lambda_{2} x_{1}$ in (1) to a "negative feedback" mechanism $\lambda_{2} x_{1} f\left(x_{2}\right)$, where $f$ is a monotonically decreasing function of $x_{2}$. Use the stationary FDT to approximate $\eta_{22}$ in terms of $h=\left|\partial \ln f / \partial \ln x_{2}\right|$. Apply your answer to the case $f\left(x_{2}\right)=k / x_{2}$.

  [Hint: To reduce the algebra introduce the elasticity $H_{22}=\partial \ln \left(R_{2}^{-} / R_{2}^{+}\right) / \partial \ln x_{2}$, where $R_{2}^{-}$and $R_{2}^{+}$are the death and birth rates of $x_{2}$ respectively.]

  (d) Explain the extrinsic term for the negative feedback system in terms of environmental fluctuations, time-averaging, and static susceptibility.

  (e) Explain why the FDT is exact for the constitutive system but approximate for the feedback system. When, generally speaking, does the FDT approximation work well?

  (f) Consider the following three experimental observations: (i) Large changes in $\lambda_{2}$ have no effect on $\eta_{22}$; (ii) When $x_{2}$ is perturbed by $1 \%$ from its stationary average, perturbations are corrected more rapidly in the feedback system than in the constitutive system; (iii) The feedback system displays lower values $\eta_{22}$ than the constitutive system.

  What does (i) imply about the relative importance of the noise terms? Can (ii) be directly explained by (iii), i.e., does rapid adjustment reduce noise? Justify your answers.

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• 4.I.6E

  Mathematical Biology | Part II, 2005

  The output of a linear perceptron is given by $y=\mathbf{w} \cdot \mathbf{x}$, where $\mathbf{w}$ is a vector of weights connecting a fluctuating input vector $x$ to an output unit. The weights are given random initial values and are then updated according to a learning rule that has a time-constant $\tau$ much greater than the fluctuation timescale of the inputs.

  (a) Find the behaviour of $|\mathbf{w}|$ for each of the following two rules (i) $\tau \frac{d \mathbf{w}}{d t}=y \mathbf{x}$ (ii) $\tau \frac{d \mathbf{w}}{d t}=y \mathbf{x}-\alpha y^{2} \mathbf{w}|\mathbf{w}|^{2}$, where $\alpha$ is a positive constant.

  (b) Consider a third learning rule

  $$\text { (iii) } \tau \frac{d \mathbf{w}}{d t}=y \mathbf{x}-\mathbf{w}|\mathbf{w}|^{2} \text {. }$$

  Show that in a steady state the vector of weights satisfies the eigenvalue equation

  $$\mathbf{C w}=\lambda \mathbf{w}$$

  where the matrix $\mathbf{C}$ and eigenvalue $\lambda$ should be identified.

  (c) Comment briefly on the biological implications of the three rules.

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