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  "body": "\n## Example\n\nLet $\\alpha, \\beta, \\gamma, \\delta \u003e 0$ be constants. Let $x(t) \u003e 0$ be the size of a prey population and $y(t) \u003e 0$ the size of a predator population at time $t$.\nThe Lotka–Volterra equations are the [[autonomous system|07BNRFQ0PJ]]\n$$x' = \\alpha x - \\beta x y, \\qquad y' = \\delta x y - \\gamma y.$$\nThe [[equilibrium points|0G4X36AE0X]] in the closed quadrant $x, y \\ge 0$ are $(0, 0)$ and $(\\gamma / \\delta, \\alpha / \\beta)$.\nThe function $V(x, y) = \\delta x - \\gamma \\ln x + \\beta y - \\alpha \\ln y$ is a [[first integral|0SS1P1YC9A]] on the open quadrant $x, y \u003e 0$.\n\n## Remarks\n\nAlong a solution, $\\frac{d}{dt} V = (\\delta x - \\gamma)(\\alpha - \\beta y) + (\\beta y - \\alpha)(\\delta x - \\gamma) = 0$.\nThe [[linearisation|09ZWYRQ41J]] at $(0, 0)$ has the eigenvalues $\\alpha$ and $-\\gamma$, so the origin is a saddle. At $(\\gamma/\\delta, \\alpha/\\beta)$ the eigenvalues are $\\pm i \\sqrt{\\alpha \\gamma}$.\nThe 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|01DSNP0B4R]]: the two populations oscillate.\n\n## Explanations\n\n### L1 human\n\nPrey grow when there are few predators, and predators grow when there is much prey.\nThis feedback makes both populations rise and fall in cycles, with the predator peak after the prey peak.\n",
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