Lotka-Volterra Predator-Prey Model
The classical model of ecological dynamics: a cyclical struggle between predators and prey. Developed independently by Alfred J. Lotka (1925) and Vito Volterra (1926), these coupled first-order non-linear differential equations describe how populations interact over time. Explore how changes in reproduction and predation rates affect the fragile balance of nature.
The Equations
The Lotka-Volterra equations are a pair of first-order non-linear differential equations:
Where $x$ represents the population of prey (e.g., rabbits) and $y$ represents the population of predators (e.g., foxes).
- $\alpha x$: Exponential growth of prey in the absence of predators.
- $-\beta xy$: The rate at which prey are consumed, proportional to the chance of a predator-prey encounter.
- $\delta xy$: The growth of the predator population, driven by consuming prey.
- $-\gamma y$: The natural death rate of predators in the absence of food.
Phase Space and Dynamics
Without external intervention, the populations oscillate periodically. A peak in the prey population leads to a delayed peak in the predator population. The increased predator population then causes a sharp decline in prey, which in turn leads to a collapse in the predator population, allowing the cycle to begin anew.
When plotted against each other in phase space ($x$ vs $y$), these periodic oscillations trace closed loops (limit cycles). The size and shape of these orbits are determined by the initial populations and the system parameters.