Deterministic population dynamics · 6
Prey–predator models
dN/dt = rN − βNPdP/dt = γβNP − δP
- general form
- dN/dt = f(N) − h(N, P), dP/dt = γ·h(N, P) − δP
- N, P
- prey, predators
- f(N)
- prey growth without predators
- h(N, P)
- the rate predators kill prey (functional response)
- γ
- how many new predators one eaten prey makes
- δ
- predator death rate
Phase plane
Time series
Both populations from (N(0), P(0)); a tap in the phase plane moves this start.
Effect of enrichment K
Where the prey end up for every carrying capacity K from 1 to 100, with the other values as set above.
The theory, in short
- General form. Prey: growth without predators minus predation; predators: converted predation minus deaths.
- Lotka–Volterra (exponential prey, Holling I predation): e1 = (δ/(γβ), r/β) is a centre; orbits are closed curves of V; cycles never decay or grow (neutral, Lyapunov stable but not asymptotically stable).
- Volterra (logistic prey): predators can live only if K > δ/(γβ); then e2 is stable: a spiral for most values, a node (no swing at all) when K is just above δ/(γβ) or the prey grows very fast.
- Rosenzweig–MacArthur (logistic prey, Holling II predation): N* = δN0/(γβ − δ); e2 is stable for K < KH = N0 + 2N*, where the top of the prey nullcline lies left of N*; beyond KH (a Hopf bifurcation) a stable limit cycle appears.
- The paradox of enrichment. Raising K (more food for the prey) destabilises the RM system: the opposite of the Volterra model.
- Predators lag prey. Predator peaks follow prey peaks (a quarter of a period for small LV cycles).
- Classification of equilibria from the Jacobian's eigenvalues, as in the README's table (spiral, node, saddle, centre; non-hyperbolic cases need the nonlinear terms).