Lotka–Volterra equations
The classical Lotka–Volterra model is a foundational mathematical model of predator–prey interaction.
The equations
\[\frac{dN}{dt}=\alpha N-\beta NP,\]\[\frac{dP}{dt}=\delta NP-\gamma P.\]Here \(\alpha\) is the prey growth rate, \(\beta\) measures predation, \(\gamma\) is predator mortality, and \(\delta\) measures predator growth generated by prey consumption.
Equilibria
The model has the extinction equilibrium \((0,0)\) and a positive coexistence equilibrium
\[N^*=\frac{\gamma}{\delta},\qquad P^*=\frac{\alpha}{\beta}.\]Oscillatory behaviour
In the idealised model, predator and prey populations can cycle around the coexistence equilibrium. Prey increase first, predators subsequently increase, increased predation reduces prey, and predator numbers then fall as food becomes scarce.
Key idea. The classical Lotka–Volterra equations demonstrate how simple nonlinear interactions between two species can generate population cycles.