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

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