Predator–prey models
Predator–prey models describe populations that affect one another in opposite ways: predators benefit from consuming prey, while prey lose individuals through predation.
Start with the two populations
Let
\[N(t)=\text{prey population},\qquad P(t)=\text{predator population}.\]The two populations cannot usually be modelled independently because the growth rate of each depends on the other.
The biological feedback
If prey become abundant, predators have more food. Predator survival or reproduction can then improve, so predator numbers rise. As predator numbers rise, predation pressure on prey becomes stronger, so prey numbers fall.
When prey become scarce, predators have less food and may decline. Reduced predator pressure then allows prey to recover.
The classical Lotka–Volterra model
The simplest predator–prey model is
\[\boxed{\frac{dN}{dt}=rN-aNP},\] \[\boxed{\frac{dP}{dt}=eaNP-mP}.\]| Symbol | Meaning |
|---|---|
| \(r\) | prey per-capita growth rate in the absence of predators |
| \(a\) | attack or encounter coefficient |
| \(e\) | conversion efficiency: how prey consumption contributes to predator growth |
| \(m\) | predator per-capita mortality rate in the absence of prey |
Understand the prey equation term by term
The prey equation is
\[\frac{dN}{dt}=rN-aNP.\]The first term, \(rN\), represents exponential prey growth without predators.
The second term, \(aNP\), represents prey lost to predator encounters.
Understand the predator equation term by term
The predator equation is
\[\frac{dP}{dt}=eaNP-mP.\]The term \(eaNP\) represents predator gains generated from successful prey consumption. The term \(mP\) represents predator losses when prey are absent.
The conversion factor \(e\) is needed because consuming one prey individual does not normally create one new predator individual.
The interaction signs are opposite
The same encounter term enters the two equations with different effects:
\[-aNP\quad\text{for prey},\qquad +eaNP\quad\text{for predators}.\]This captures the defining asymmetry of predation: the interaction harms the prey and benefits the predator.
Per-capita form gives more intuition
For prey,
\[\frac{1}{N}\frac{dN}{dt}=r-aP.\]So prey per-capita growth decreases as predator abundance increases.
For predators,
\[\frac{1}{P}\frac{dP}{dt}=eaN-m.\]So predator per-capita growth increases as prey abundance increases.
When does the prey population increase?
Prey increase when
\[r-aP\gt0,\]which means
\[P\lt\frac{r}{a}.\]Prey decrease when predator abundance is above this threshold.
When does the predator population increase?
Predators increase when
\[eaN-m\gt0,\]which means
\[N\gt\frac{m}{ea}.\]Predators decline when prey abundance is below this threshold.
The nullclines
A nullcline is where one population is momentarily not changing.
For prey, the positive nullcline is
\[\boxed{P=\frac{r}{a}}.\]For predators, the positive nullcline is
\[\boxed{N=\frac{m}{ea}}.\]These are especially simple: one is horizontal and the other is vertical.
Phase-plane interpretation
The coexistence equilibrium
At a positive equilibrium, both nullcline conditions hold at the same time:
\[\boxed{N^*=\frac{m}{ea},\qquad P^*=\frac{r}{a}}.\]This equilibrium represents a state where prey and predator populations are both positive and neither is changing instantaneously.
Why cycles appear in the classical model
Suppose prey start above their predator-growth threshold while predator numbers are low. Prey rise first. The increased food supply then allows predators to rise.
Predator abundance eventually becomes high enough that prey begin to decline. Later, prey become too scarce to support the large predator population, so predators decline. With predator pressure reduced, prey begin to increase again.
Population cycles through time
Closed orbits in the ideal classical model
For the undamped classical Lotka–Volterra equations, trajectories in the positive phase plane form closed orbits around the coexistence equilibrium.
This means the model can repeat cycles indefinitely. Their amplitude depends on the initial condition rather than converging toward a single attracting cycle.
Why unlimited prey growth is unrealistic
In the classical model, if predators disappear, prey satisfy
\[\frac{dN}{dt}=rN,\]so prey grow exponentially without limit.
A more realistic extension often gives prey logistic growth:
\[\boxed{\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)-aNP}.\]This introduces intraspecific competition among prey and a carrying capacity \(K\).
Predators may not consume prey indefinitely faster
The term \(aNP\) assumes that each predator's consumption rate increases linearly with prey density. Real predators have finite handling time and may become satiated.
This motivates functional responses, where the predation term becomes a nonlinear function of prey abundance.
For example, a saturating predation rate may take the form
\[\frac{aN}{1+ahN},\]where \(h\) represents handling time.
What is a functional response?
A functional response describes how the rate at which one predator consumes prey changes as prey density changes.
| Type | Basic behaviour |
|---|---|
| Type I | consumption increases approximately linearly with prey density |
| Type II | consumption increases but saturates because of handling time |
| Type III | consumption is low at very low prey density, increases rapidly at intermediate density, then saturates |
Why Type II can change stability
When predation saturates, high prey abundance does not produce proportionally larger prey removal. Combined with logistic prey growth, this can produce stable equilibria, damped oscillations or persistent cycles depending on parameter values.
The qualitative behaviour therefore depends strongly on the form chosen for the predator response.
Predator numerical response
The functional response describes prey consumption per predator. A related idea is the numerical response: how predator abundance changes when prey availability changes.
Predators may reproduce more successfully, survive longer, immigrate into prey-rich areas or leave prey-poor areas.
Predator–prey models are not always about killing
The same mathematical structure can sometimes approximate host–parasitoid, consumer–resource or other antagonistic interactions in which one population gains from exploiting another.
The biological interpretation of the parameters must therefore be specified rather than inferred only from the equations.
Real systems can include several prey or predators
A predator may consume multiple prey species, or prey may face several predators. Food-web models extend the same idea to larger networks of interacting populations.
The interaction structure can then be represented using systems of differential equations or interaction matrices.
Spatial effects
Predators and prey are not always well mixed. Prey refuges, predator territories, migration and patch structure can strongly change encounter rates.
Spatial predator–prey models may use coupled patches, reaction–diffusion equations or individual-based models.
Stochastic predator–prey dynamics
When populations are finite, births, deaths and predation events are random. Even if the deterministic model predicts persistent cycles, a stochastic trajectory can hit zero and cause extinction.
This is especially important when one population becomes very small during a cycle.
What the classical model teaches well
Despite its limitations, the Lotka–Volterra model is valuable because it shows clearly how feedback between species can generate dynamics that neither species would produce alone.
It also introduces nullclines, phase planes, coexistence equilibria and coupled nonlinear differential equations in a biologically intuitive setting.