Answers To Nicholas F Britton Essential Mathematical Biology Chapter 1

Problem 1

Consider the nonlinear equation for population growth
$$
x_{n+1}=\frac{\lambda x_{n}}{1+x_{n}}
$$
where $\lambda>0$
a) Does this equation exhibit over-, under-or exact compensation?
b) Does it have a non-trivial steady state?
c) Find the stability of the steady state(s) of the equation, and discuss any bifurcations that take place.
d) Draw cobweb maps for each case that you have identified.
e) Check your answers by using the substitution $y_{n}=1 / x_{n}$ to solve the equation exactly. (Hint: try a solution in the form $y_{n}=$ $\left.A r^{n}+B .\right)$

Problem 2

A population is assumed to be governed by the equation $N_{n+1}=$ $f\left(N_{n}\right)$, where the function $f$ is as shown in Figure $1.5$, and there is a single steady state $N^{*}$. A variant of the cobwebbing method uses the graphs of $f$ and its reflection $\bar{f}$ to produce successive iterates $N_{n}$, as in part (a) of the figure.
a) What is the point marked $S$ in part (a) of the figure?
b) A somewhat different situation is shown in part (b) of the figure. Interpret the points $A, B$ and $C$ and use cobwebbing to decide what happens to the population $N_{n}$.

Problem 3

a) Derive the condition for linearised stability of the steady solution $N_{n}=N^{*}$ of the equation $N_{n+1}=h\left(N_{n}\right)$.
b) Consider an organism such as the Pacific salmon that has a two year life cycle, so that after one year newborn individuals become immature young, and after two years mature adults. Let the numbers of young and adults in year $n$ be $y_{n}$ and $a_{n}$, and let
$$
a_{n+1}=f\left(y_{n}\right), \quad y_{n+1}=g\left(a_{n}\right)
$$
where $f$ and $g$ are both increasing functions, $f(0)=g(0)=0$, $f(y)<y$ for all $y>0$, and $g(a) \rightarrow g_{0}$, a constant, as $a \rightarrow \infty$ Explain these hypotheses in biological terms.
c) Show that the fixed points of this system are the intersections of the curves $y=g(a)$ and $y=f^{-1}(a)$. Derive the condition for stability in the form $g^{\prime}\left(a^{*}\right)<\left(f^{-1}\right)^{\prime}\left(a^{*}\right)$ and deduce from geometrical considerations that these points, ordered in the obvious way, are generically alternately stable and unstable. If there are three fixed points $S_{0}=(0,0), S_{1}=\left(y_{1}^{*}, a_{1}^{*}\right)$ and $S_{2}=\left(y_{2}^{*}, a_{2}^{*}\right)$, in that order, show that $S_{1}$ is generically unstable and state three possibilities for the asymptotically stable behaviour of the system as $n \rightarrow \infty$.

Problem 4

It has been suggested that a means of controlling insect pests is to introduce and maintain a number of sterile insects in the population. One model for the resulting population dynamics is given by
$$
N_{n+1}=f\left(N_{n}\right)=R_{0} N_{n} \frac{N_{n}}{N_{n}+S} \frac{1}{1+a N_{n}}
$$
where $R_{0}>1, a>0$ and $S$ is the constant sterile insect population. The idea is that $S$ is under our control, and we wish to choose it to accomplish certain ends. The problem is therefore one in control theory.
a) Explain the $N_{n} /\left(N_{n}+S\right)$ factor in the model.
b) Find the equation satisfied by the steady states $N^{*}$ of the model, and sketch the relationship between $N^{*}$ and $S$ in the form of a graph of $S$ as a function of $N^{*}$.
c) What is the least value $S_{c}$ of $S$ that will drive the insects to extinction?
d) Sketch cobweb maps for $S<S_{c}$ and $S>S_{c}$. What kind of bifurcation occurs as $S$ passes through $S_{c} ?$

Problem 5

In this section we derived the logistic equation for limited population growth using an empirical approach. An alternative is the ecosystems or resource-based approach which goes back at least to both Lotka and Volterra, two of the founders of mathematical ecology. The ecosystems approach will be dealt with more fully in Chapter 2 , but we outline the method here.

Let the per capita growth rate of a population depend on some resource. Let this resource exist in two states, either free (available for use by the members of the population) or bound (already in use). Let the density of free resource be $R$; then
$$
\frac{d N}{d t}=N G(R)
$$
Let the resource be abiotic (non-biological), and therefore not subject to birth and death. (The archetypal example in the ecosystems approach is a mineral resource, but another possibility is something like nest-sites.) Let the total amount of resource, free and buund, be a constant $C>0$, and let the amount of bound resource depend on the population. Then
$$
R=C-H(N)
$$
It remains to model $G$ and $H$. Clearly $G$ is an increasing function satisfying $G(0)<0 ;$ the simplest model is $G(R)=\alpha R-\beta, \alpha>0$ $\beta>0 . H$ is also an increasing function but satisfies $H(0)=0 ;$ the simplest model is $H(N)=\gamma N, \gamma>0$
a) Show that we still obtain the logistic equation for the growth of the population.
b) Give expressions for the Malthusian parameter $r$ and the carrying capacity $K$.
c) What happens if the total amount $C$ of resource is insufficient?
d) Give an advantage of each approach to modelling limited growth.

Problem 6

A population $N$ is growing according to a logistic differential equation, and $N\left(t_{1}\right)=n_{1}, N\left(t_{1}+\tau\right)=n_{2}, N\left(t_{1}+2 \tau\right)=n_{3}$. Show that the carrying capacity is given by
$$
K=\frac{1 / n_{1}+1 / n_{3}-2 / n_{2}}{1 /\left(n_{1} n_{3}\right)-1 / n_{2}^{2}}
$$

Problem 7

A model for population growth is given by
$$
\frac{d N}{d t}=f(N)=r N\left(\frac{N}{U}-1\right)\left(1-\frac{N}{K}\right)
$$
where $r, K$ and $U$ are positive parameters with $U<K$.
a) Sketch the function $f(N)$.
b) Discuss the behaviour of $N(t)$ as $t \rightarrow \infty$, and show that the model exhibits critical depensation.

Problem 8

For some organisms finding a suitable mate may cause difficulties at low population densities, and a more realistic equation for population growth than a linear one in the absence of intraspecific competition may be $\dot{N}=r N^{2}$, with $r>0$, to be solved with initial conditions $N(0)=N_{0}$
a) Show that this model exhibits depensation.
b) Solve this problem and show that the solution becomes infinite in finite time.
c) The model above is improved to
$$
\frac{d N}{d t}=r N^{2}\left(1-\frac{N}{K}\right)
$$
Without solving this equation find the steady state solutions and say whether they are stable or unstable.
d) Derive the model of part (c) from that of part (b) by a resourcebased method. Assume that the total amount of resource (free and bound) is sufficient to sustain a population. If the amount of free resource is non-negative initially, show, by a graphical argument or otherwise, that it always remains non-negative.

Problem 9

Beverton-Holt equation. The Beverton-Holt stock-recruitment curve, given by equation (1.5.12), is derived from a sub-model in continuous time that describes the dynamics of the larval population between hatching and recruitment to the adult population. Let $L(t)$ be the number of larvae at time $t$, and assume that they are subject to density-dependent mortality through intraspecific competition or other effects between time $n+t_{1}$ and $n+t_{2}$, where $0 \leq t_{1}<t_{2} \leq 1$. In this time, they die according to
$$
\frac{d L}{d t}=-\left(\mu_{1}+\mu_{2} L\right) L
$$
where $\mu_{1}$ and $\mu_{2}$ are positive constants. If $L\left(n+t_{1}\right)$ is proportional to $N_{n}$, the number of adults in the population at time $n$, and $N_{n+1}$ is proportional to $L\left(n+t_{2}\right)$, derive the Beverton-Holt Equation (1.5.12).

Problem 10

A fishery population is modelled by
$$
N_{n+1}=f\left(N_{n}\right)-Y_{n}
$$
where $Y_{n}$ is the catch taken in year $n$, the yield. and $f$ is given by the Ricker model $f(N)=N e^{r(1-N / K)}$, a model for over-compensatory competition, where $r$ and $K$ are positive parameters. Sketch $Y^{*}$ as a function of $N^{*}$.

Problem 11

Models of fisheries in continuous time, where growth and harvesting occur continuously and simultaneously, are easier to analyse than the discrete time stock-recruitment models of this section. A population of fish that would otherwise grow according to the logistic law $\dot{N}=$ $r N(1-N / K)$ is fished at constant effort $E$, leading to a harvest at constant rate $q E N$, where $q$ is the catchability coefficient.
a) Show that the steady state in the presence of fishing is given by $N^{*}=K(1-q E / r)$
b) Find the steady state yield-effort relationship.
c) Find the maximum sustainable yield.

Problem 12

Consider a population of fish that would grow without harvesting at a rate $\dot{N}=N F(N)$ that is depensatory, so that $F$ increases, for small values of $N . \mathrm{It}$ is then fished at constant effort $E$, leading to a harvest at constant rate $q E N$.
a) Show that, at steady state, $q E^{*} N^{*}=N^{*} F\left(N^{*}\right)$.
b) Sketch $q E^{*} N^{*}$ and $N^{*} F\left(N^{*}\right)$ on the same graph, and deduce the yield-effort relationships for critical and non-critical depensation.

Problem 13

In a mainland-island metapopulation model there is supposed to be a mainland in addition to the island patches, where there is a large population with negligible risk of extinction. The equation for the fraction of occupied patches is modified to
$$
\frac{d p}{d t}=(m+c p)(1-p)-e p
$$
a) Explain the model.
b) Find the (biologically realistic) steady states of the model, and discuss their stability.
A multispecies version of this is the basis of MacArthur and Wilson's dynamic theory of island biogeography.

Problem 14

One of the best-studied delay equations is Hutchinson's ( 1948$)$ equation, a modification of the logistic equation, given by
$$
\frac{d N}{d t}(t)=r N(t)\left(1-\frac{N(t-\tau)}{K}\right)
$$
One way to interpret this is that the per capita growth rate de pends on the availability of a resource, which in turn depends on the population size a time $\tau$ earlier. The population takes a timè $\tau$ to respond to the resource. Find the steady state of this equation, and $_{y}$ investigate its stability.

Problem 15

Hutchinson's equation may be generalised to
$$
\frac{d N}{d t}(t)=r N(t)\left(1-\frac{1}{K} \int_{0}^{\infty} N(t-u) k(u) d u\right)
$$
where $k(u)$ denotes the weight given to the population size a time $u$ earlier, normalised so that $\int_{0}^{\infty} k(u) d u=1 .$ (If $k(u)=\delta(u-\tau)$, we recover the original equation.) In this question we shall take $k(u)=$ $\frac{1}{\tau} \exp \left(-\frac{u}{r}\right)$, which has an average delay $\int_{0}^{\infty} u k(u) d u=\tau$. This is called the weak generic delay kernel.
a) Show that $N=0$ and $N=K$ are steady states of Equation $(1.7 .20)$.
b) Show that the equation linearised about $N=K$ is given by
$$
\frac{d n}{d t}(t)=-r \int_{0}^{\infty} n(t-u) k(u) d u
$$
c) This equation is linear, so that we expect solutions like $n(t)=$ $n_{0} \exp (s t)$. Derive the characteristic equation for $s$.
d) Show that the steady state is linearly stable.
c) Show that $P(t)=\int_{0}^{\infty} N(t-u) k(u) d u$ satisfies $\frac{d P}{d t}=\frac{P-N}{\tau} .$ The equation is equivalent to the system $\dot{N}=r N(1-P / K), \dot{P}=$ $(P-N) / \tau$, and may be analysed by methods to be discussed in Chapter 2 .

Problem 16

Analyse Equation (1.7.20) with strong generic delay, $k(u)=\frac{u}{\tau^{2}} \exp \left(-\frac{u}{\tau}\right)$. Derive a condition for instability of its non-trivial steady state $K$.

Problem 17

Give at least three criticisms of Fibonacci's rabbit model.

Problem 18

What difference does it make to the model and the analysis if the census takes place just after the births instead of just before?

Problem 19

Consider a population of annual plants with the following characteristics. Seeds are produced at the end of the summer. A proportion survive one winter, and a proportion of these germinate the following spring. Of the remainder, a proportion survive a second winter, and a proportion of these germinate the spring following this second winter, but none can germinate later than this.
a) Justify the model
$$
N_{n+2}=\alpha \sigma \gamma N_{n+1}+\beta(1-\alpha) \sigma^{2} \gamma N_{n}
$$
for the population, and interpret the parameters biologically.
b) Write this in terms of a Leslie matrix.
c) Use an eigenvalue equation to show that the condition for the plant population to thrive is
$$
\gamma>\frac{1}{\alpha \sigma+\beta(1-\alpha) \sigma^{2}}
$$
d) Show that this condition is equivalent to the condition $R_{0}>1$, where $R_{0}$ is the basic reproductive ratio, interpreted here to be the expected number of offspring produced by an individual during its lifetime that survive to breed, in the absence of any other mortality.

Problem 20

In the circulatory system, the red blood cells (RBCs) are constantly being destroyed and replaced. Assume that the spleen filters out and destroys a certain fraction $f$ of the cells daily and that the bone marrow produces a number proportional to the number lost on the previous day. If
$-C_{n}$ is the number of RBCs in circulation on day $n$,
$-M_{n}$ is the number of RBCs produced by the marrow on day $n$, then
$$
\begin{gathered}
C_{n+1}=(1-f) C_{n}+M_{n} \\
M_{n+1}=\gamma f C_{n}
\end{gathered}
$$
where $\gamma$ is a constant.
a) Explain these equations.
b) Find the principal eigenvalue $\lambda_{1}$.
c) The definition of homeostasis is that $C_{n} \rightarrow C^{*}$, a non-zero constant, as $n \rightarrow \infty$. How may the parameters be chosen to achieve this?
d) Why cannot this be a good model for homeostasis?

Problem 21

Let the matrix $L$ have distinct eigenvalues $\lambda_{i}$ with corresponding eigenvectors $\mathbf{v}_{i}$. If $\mathbf{u}_{n+1}=L \mathbf{u}_{n}$ and $\mathbf{u}_{0}=\sum_{i=0}^{\omega} A_{i} \mathbf{v}_{i}$, where the $A_{i}$ are constants, show that
$$
\mathbf{u}_{n}=\sum_{i=0}^{\omega} A_{i} \lambda_{i}^{n} \mathbf{v}_{i}
$$

Problem 22

If $L$ is given by Equation (1.9.26), show that
$$
\operatorname{det}(\lambda I-L)=\lambda^{\omega}-\sum_{i=1}^{\omega} f_{i} \lambda^{\omega-i}
$$
where $f_{i}=l_{i} m_{i}, l_{i}=s_{i} s_{i-1} \ldots s_{1}$.

Problem 23

Consider the population process described by equation (1.9.25), where $L$ is the Leslie matrix (1.9.26). Show that the stable age structure is given by
$$
\mathbf{v}_{1}=\left(\lambda_{1}^{\omega-1} l_{1}, \lambda_{1}^{\omega-2} l_{2}, \cdots, \lambda_{1} l_{\omega-1}, l_{\omega}\right)^{T}
$$
where $l_{i}=s_{i} s_{i-1} \ldots s_{1}, \lambda_{1}$ is the principal eigenvalue of $L$ and $\omega$ is the oldest age-class.

Problem 24

a) Show that the discrete Euler-Lotka function $\hat{f}(r)=\sum_{i=1}^{\omega} f_{i} e^{-r i}$ is a monotonic decreasing function of $r$, satisfying $\hat{f}(r) \rightarrow \infty$ as $r \rightarrow-\infty, \hat{f}(r) \rightarrow 0$ as $r \rightarrow \infty$
b) Deduce that $R_{0}>1$ if and only if $r_{1}>0$, or $\lambda_{1}>1$.
c) Show that $r_{1} \approx(1-\hat{f}(0)) / \hat{f}^{\prime}(0)$ if $r_{1}$ is small.

Problem 25

A life table for the vole Microtus agrestis reared in the laboratory is given on the right. Each age class is eight weeks long.
a) Calculate $R_{0}$
b) Calculate an approximation to Lotka's intrinsic rate of natural increase $r_{1}$
c) How would you improve this approximation?

Problem 26

Generating function method for discrete renewal equations. Let the sequences $b_{n}, f_{n}$ and $g_{n}$ satisfy the Euler renewal equation (1.10.29). Define generating functions by
$$
b(s)=\sum_{i=1}^{\infty} b_{i} s^{i}, \quad f(s)=\sum_{i=1}^{\infty} f_{i} s^{i}, \quad g(s)=\sum_{i=1}^{\infty} g_{i} s^{i}
$$
a) Show that $b(s)=g(s) /(1-f(s))$
b) Show that the roots of $1-f(s)=0$ are the reciprocals of the roots of the Euler eigenvalue equation (1.10.31).
c) If these roots are distinct, show that
$$
1-f(s)=\prod_{i=i}^{\omega}\left(1-\lambda_{i} s\right)
$$
where the $\lambda_{i}$ are the non-zero eigenvalues of $L$.
d) Use partial fractions to deduce that
$$
b_{n}=\sum_{i=1}^{\omega} \frac{g\left(s_{i}\right)}{f^{\prime}\left(s_{i}\right)} \lambda_{i}^{n+1}
$$
where $s_{i}=\frac{1}{\lambda_{i}} .$ This determines the constants $A_{i}$ in (1.10.32), and so gives the solution of the initial value problem (1.10.27).

Problem 27

Let $r_{1}$ be the real root of the Euler-Lotka Equation (1.10.33), and let $R_{0}$ be given by Equation (1.10.34).
a) Let $\bar{a}=\left(\sum_{i=1}^{\omega} i f_{i}\right) /\left(\sum_{i=1}^{\omega} f_{i}\right)$ be the average age of giving birth. If $r_{1}$ is small, show that
$$
r_{1} \approx \frac{R_{0}-1}{\bar{a} R_{0}}
$$

Problem 28

Let the net maternity function $f$ on $[0, \omega]$ be continuous, nonnegative (at each point) and not identically zero. Show that the Euler-Lotka function
$$
\tilde{f}(r)=\int_{0}^{\infty} \exp (-r a) f(a) d a
$$
(which is the Laplace transform of the net maternity function), is monotonic decreasing and satisfies $\tilde{f}(r) \rightarrow \infty$ as $r \rightarrow-\infty, \tilde{f}(r) \rightarrow 0$ as $r \rightarrow \infty$. Deduce that the Euler-Lotka Equation (1.10.38) has a unique real root $r_{1}$.

Problem 29

Show that the Euler-Lotka function $\tilde{f}$ given by (1.10.39) crosses the vertical axis at $R_{0}$, and deduce that $R_{0}>1$ if and only if $r_{1}>0$.

Problem 30

Laplace transform method for continuous renewal equations, for readers with a knowledge of Laplace transforms only. Define the Laplace transform $\tilde{h}(s)$ of a function $h(t)$ by
$$
\tilde{h}(s)=\int_{0}^{\infty} h(t) e^{-s t} d t
$$
The inverse Laplace transform of $\tilde{h}(s)$ is defined by
$$
h(t)=\frac{1}{2 \pi i} \int_{c-i \infty}^{c+i \infty} e^{s t} \tilde{h}(s) d s
$$
where the contour of integration is to the right of all the roots of $\tilde{h}(s)$. Let the functions $b, f$ and $g$ satisfy the Euler renewal equation $(1.10 .36)$
a) Show that $\tilde{b}(s)=\tilde{g}(s) /(1-\tilde{f}(s))$
b) The characteristic (Euler-Lotka) equation is given by
$$
1=\int_{0}^{\infty} f(a) e^{-s a} d a=\tilde{f}(s)
$$
and $\tilde{f}$ is the Euler-Lotka function. Show that the characteristic equation has one real root $s_{1}$, and that all the other roots have real part less than $s_{1}$.
c) If the characteristic equation only has simple roots, denoted by $s_{i}$, show by using Cauchy's integral formula that
$$
b(t)=\sum_{i=0}^{\infty} \frac{g\left(s_{i}\right)}{f^{\prime}\left(s_{i}\right)} \exp \left(s_{i} t\right)
$$

Problem 31

Consider a female producing offspring in a population growing steadily at rate $r .$ To maximise her contribution to the population as a whole, she would like not only to produce many children but to produce them early, when they will make up a greater fraction of the population of their age. A birth delayed by time $\tau$ will be worth $e^{-r \tau}$ times as much as a birth not subject to such a delay. The reproductive value of an individual of age $a$ relative to the reproductive value of a new-born individual is defined by
$$
v(a)=\frac{\int_{a}^{\infty} \exp (-r b) l(b) m(b) d b}{\exp (-r a) l(a)}
$$
a) Interpret this definition biologically.
b) What qualitative characteristics would you expect $v$ to have as a function of $a ?$

Problem 32

a) Explain why the survival function $l(a)$ must satisfy McKendrick's equation (1.11.40).
b) Deduce that $l(a)=\exp \left(-\int_{0}^{a} d(b) d b\right)$.

Problem 33

Consider a population growing at Lotka's intrinsic rate of increase $r$.
a) Show that the life expectancy of a cohort, defined to be the mean age at death of individuals born at the same time, is given by $\int_{0}^{\infty} a d(a) l(a) d a$
b) Show that the mean age of those dying simultaneously is given by
$$
\frac{\int_{0}^{\infty} a d(a) e^{-r a} l(a) d a}{\int_{0}^{\infty} d(a) e^{-r a} l(a) d a}
$$
c) Show that these two quantities are different in general.