Paper 2, Section I, A

Mathematical Biology | Part II, 2013

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

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

with

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

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

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

r(a)=r(0)eγa0aμ(s)ds,r(0)=0b(s)r(s)ds.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

ϕ(γ)=0b(a)eγa0aμ(s)dsda=1\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 ϕ(0)\phi(0).

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

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