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 Incoming links: Entropy-shaped Lyapunov function for weak competition

Past exam of the topics in applied mathematics course of the University of Oxford / 2009 / A / 3 / a / iii / Solution 2026-10-31

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The non-axis nullclines are

n + m /2 = 1

and

m + n /2 = 1

, intersecting at

(2/3, 2/3)

. Each biological population grows below its own line and decreases above it. The axes are invariant; along a positive axis the surviving species follows logistic growth towards its unit carrying capacity. The origin remains fixed.

All strictly positive initial states converge to coexistence. An entropy-shaped Lyapunov function for weak competition proves the global conclusion beyond the local arrows: put

k = 2/3

and use

V = n - k - k lo g (n / k) + \frac{1}{q} [m - k - k lo g (m / k)] .

(1)

Each bracket is nonnegative, vanishing only at its equilibrium value, and tends to infinity at zero or infinity. Differentiation gives

\dot{V} = - (n - k)^{2} - (m - k)^{2} - (n - k) (m - k) \leq - \frac{1}{2} [(n - k)^{2} + (m - k)^{2}] .

(2)

Thus positive trajectories remain in a compact level set inside the quadrant and approach its unique zero-derivative state. Therefore

(n, m) \to (2/3, 2/3) for n (0), m (0) > 0.

(3)

The phase portrait shows the stable coexistence point, invading directions near the axis saddles, and the two straight nullclines.

Figure 1.
Competing populations converge to coexistence in the positive quadrant
. **Details**.

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