Lotka-Volterra model of competition

The Lotka-Volterra model is a model for competition between two species over the same food resource, ignoring other complications such as predators and season. The general form is the Differential system

˙ 𝑥 = 𝛼 𝑥 + 𝛽 𝑥 𝑦 = 𝑥 (𝛼 + 𝛽 𝑦) ˙ 𝑦 = 𝛾 𝑦 + 𝛿 𝑥 𝑦 = 𝑦 (𝛾 + 𝛿 𝑥)

which is linearized by the Jacobian

𝐷 𝑓 (𝑥, 𝑦) = [𝛼 + 𝛽 𝑦 𝛽 𝑥 𝛿 𝑦 𝛾 + 𝛿 𝑥]

Practice problems

2024. Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering, p. 198

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