Lotka–Volterra equations

Example

Let $\alpha, \beta, \gamma, \delta > 0$ be constants. Let $x(t) > 0$ be the size of a prey population and $y(t) > 0$ the size of a predator population at time $t$. The Lotka–Volterra equations are the autonomous system $$x' = \alpha x - \beta x y, \qquad y' = \delta x y - \gamma y.$$ The equilibrium points in the closed quadrant $x, y \ge 0$ are $(0, 0)$ and $(\gamma / \delta, \alpha / \beta)$. The function $V(x, y) = \delta x - \gamma \ln x + \beta y - \alpha \ln y$ is a first integral on the open quadrant $x, y > 0$.

Remarks

Along a solution, $\frac{d}{dt} V = (\delta x - \gamma)(\alpha - \beta y) + (\beta y - \alpha)(\delta x - \gamma) = 0$. The linearisation at $(0, 0)$ has the eigenvalues $\alpha$ and $-\gamma$, so the origin is a saddle. At $(\gamma/\delta, \alpha/\beta)$ the eigenvalues are $\pm i \sqrt{\alpha \gamma}$. The level curves of $V$ in the open quadrant are closed curves around $(\gamma/\delta, \alpha/\beta)$. So each solution other than this equilibrium is periodic: the two populations oscillate.

Explanations

L1 human

Prey grow when there are few predators, and predators grow when there is much prey. This feedback makes both populations rise and fall in cycles, with the predator peak after the prey peak.