10.5 Predator and prey
The Lotka–Volterra system \(x' = ax - bxy\), \(y' = -cy + dxy\) (\(a, b, c, d > 0\)) models prey \(x\) and predators \(y\). Its interior equilibrium \((c/d, a/b)\) has Jacobian \(\begin {bmatrix} 0 & -bc/d\\ ad/b & 0\end {bmatrix}\), with eigenvalues \(\pm i\sqrt {ac}\): a linear centre, non-hyperbolic, so Theorem 10.2.2 is silent. But the system has a conserved quantity.
- (i)
- \(H(x, y) = dx - c\log x + by - a\log y\) is constant along every orbit in the open positive quadrant.
- (ii)
- Every such orbit other than the equilibrium is a closed curve, so every solution is periodic.
- (iii)
- Over a period, the average populations equal the equilibrium values: \(\bar x = c/d\) and \(\bar y = a/b\).
Proof. (i) Using \(x' = x(a - by)\) and \(y' = y(-c + dx)\), \[ \frac {d}{dt}H = \Bigl (d - \frac cx\Bigr )x' + \Bigl (b - \frac ay\Bigr )y' = (dx - c)(a - by) + (by - a)(-c + dx) = 0 . \] (ii) \(H\) is the sum of a strictly convex function of \(x\), minimised at \(c/d\), and a strictly convex function of \(y\), minimised at \(a/b\); so \(H\) is strictly convex on the quadrant, tends to infinity at its boundary and at infinity, and its level sets above the minimum are closed curves around the equilibrium. Each orbit lies in one level set, contains no equilibrium, and cannot stop, so it goes round the whole curve and returns: by Lemma 10.1.2 it is periodic. (iii) \((\log x)' = a - by\); integrating over a period \(T\), \(0 = \log x(T) - \log x(0) = aT - b\int _0^Ty\), so \(\bar y = a/b\); similarly \((\log y)' = -c + dx\) gives \(\bar x = c/d\). □
Part (iii) explains a puzzle that prompted Volterra’s work: why the proportion of predatory fish caught in the Adriatic rose during the First World War, when fishing stopped. Fishing at rate \(e\) replaces \(a\) by \(a - e\) and \(c\) by \(c + e\), so it lowers the average predator population \((a - e)/b\) and raises the average prey \((c + e)/d\); stopping it does the reverse. The model’s closed orbits are, however, structurally unstable, like the centre of Example 10.2.3: adding prey self-limitation turns them into spirals converging to the equilibrium.
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