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.