Lotka-Volterra Equations

Mathematical Modeling · Axiom Academy

LESSON Lotka-Volterra Equations A detailed mathematical analysis of the classic predator-prey model: equations, equilibria, and conserved quantities 1. The Lotka-Volterra Equations The Lotka-Volterra predator-prey model consists of two coupled ordinary differential equations. Let x(t) denote the prey population and y(t) denote the predator population at time t: Each parameter in the Lotka-Volterra equations has a clear biological interpretation. Understanding these meanings helps connect the mathematics to ecological reality. The intrinsic growth rate of prey in the absence of predators. Units: 1/time. Prey would grow exponentially at rate alpha without predation. The rate at which predator-prey encounters result in prey death. Units: 1/(predators x time). Controls how effectively predators hunt. The rate at which consumed prey are converted into new predators. Units: 1/(prey x time). Represents reproductive efficiency. The natural mortality rate of predators in the absence of prey. Units: 1/time. Predators die exponentially at rate gamma without food. In natural systems, alpha and gamma (intrinsic rates) are often on similar timescales, while beta and delta (interaction rates) are typically much smaller, reflecting that encounters are relatively rare events. Equilibrium points are where both populations remain constant. We find them by setting both derivatives equal to zero and solving the resulting algebraic system:

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