• # Paper 1, Section I, 6E

(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

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

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

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

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 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

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 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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• # Paper 1, Section I, 6B

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

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

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

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

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

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

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

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, $\mathbf{6 C}$

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 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

(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.

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• # Paper 4, Section I, C

(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

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

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

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, $\mathbf{6 C}$

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 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

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

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

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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• # Paper 1, Section I, B

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

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

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

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

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

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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• # Paper 1, Section I, B

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. 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

(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

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

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) 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

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

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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• # Paper 1, Section I, E

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

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

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

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

(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

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}