Predator–prey project
Let \(X(t)\) be prey and \(Y(t)\) predators. The Lotka–Volterra model is
\[\frac{dX}{dt}=aX-bXY,\qquad \frac{dY}{dt}=cXY-dY.\]Prey grow in the absence of predators; predator losses occur without prey; encounters transfer the effect between populations.
Equilibrium
The positive equilibrium is
\[X^*=\frac dc,\qquad Y^*=\frac ab.\]Investigation
Numerically solve the equations, plot both populations against time, then draw the phase-plane trajectory \((X,Y)\). Examine how changing encounter parameters alters the cycles.
Project outcome. Relate coupled nonlinear equations to oscillating interacting populations and phase-plane behaviour.