Why do classic Lotka–Volterra cycles look so fragile?

Without prey self-limitation the orbits are neutrally stable closed curves: amplitude is set by the initial condition, not by an attracting cycle. Adding K and a saturating f(N) typically creates a true attractor (equilibrium or limit cycle).

What is the half-saturation idea in Holling II?

Handling time h limits kills per predator. f(N)→1/h as N→∞, and f reaches half of that plateau at N=1/(ah). Predators cannot eat arbitrarily fast when prey are abundant.

When does a limit cycle appear?

In the Rosenzweig–MacArthur picture, enriching K or steepening saturation can destabilize the coexistence equilibrium (paradox of enrichment). Type III can restabilize at low prey density because predators ignore rare prey.

Predator–Prey Functional Response

Lotka–Volterra vs Holling type I/II/III: saturating kill rate, prey carrying capacity, nullclines, and limit cycles.

Functional response

Holling type

Prey logistic (finite K)

Presets

Rates and initials

Prey growth r1

Carrying capacity K36

Attack rate a0.1

Handling time h0.4

Conversion e0.7

Predator death d0.45

Initial prey N₀14

Initial predators P₀8

Nullclines are the teaching core: predators persist only where f(N) reaches d/e. Type II saturation plus a large K can push that vertical line left of the prey-nullcline hump (paradox of enrichment). Type III is shallow at low N, which often restabilizes the same rich K. Classic LV is type I with the 1−N/K term off — orbits are neutrally closed, not attracting cycles.

Shortcuts

Space / Enter — play / pause
R — reset to N₀, P₀

Measured values

Prey N14.00

Predator P8.00

f(N)0.897

Interior equilibrium(8.65, 10.23)

Period hint—

Regimestable coexistence

About this model

Prey N and predator P obey N′=rN(1−N/K)−f(N)P and P′=e f(N)P−dP. The functional response f is Holling type I (linear aN), type II (saturating aN/(1+ahN)), or type III (sigmoid aN²/(1+ahN²)). Classic Lotka–Volterra is the type-I limit with no prey carrying capacity (K→∞), producing neutrally stable cycles. Saturating predation plus finite K yields a stable focus or a limit cycle (Rosenzweig–MacArthur). Nullclines and the f(N) curve make the mechanism visible.

Who it's for: Ecology, theoretical biology, and nonlinear ODE courses comparing functional responses.

Key terms

Holling functional response
Lotka–Volterra
Rosenzweig–MacArthur
Limit cycle
Nullcline
Carrying capacity
Half-saturation

How it works

Rosenzweig–MacArthur predator–prey with a selectable Holling functional response. Prey may grow logistically (finite K) or as classic Lotka–Volterra (no 1−N/K). Nullclines, the f(N) curve, and enrichment vs type-III restabilization are the lesson — not a three-species chain and not the parameter-free LV page in Math.

Key equations

N′ = r N (1 − N/K) − f(N) P

N′ = r N − f(N) P (classic LV, no logistic)

P′ = e f(N) P − d P

f_I = a N · f_II = a N / (1 + a h N) · f_III = a N² / (1 + a h N²)

plateau 1/h · half-sat II: 1/(a h) · half-sat III: 1/√(a h) · P′=0 when f(N)=d/e

Frequently asked questions

Why do classic Lotka–Volterra cycles look so fragile?: Without prey self-limitation the orbits are neutrally stable closed curves: amplitude is set by the initial condition, not by an attracting cycle. Adding K and a saturating f(N) typically creates a true attractor (equilibrium or limit cycle).
What is the half-saturation idea in Holling II?: Handling time h limits kills per predator. f(N)→1/h as N→∞, and f reaches half of that plateau at N=1/(ah). Predators cannot eat arbitrarily fast when prey are abundant.
When does a limit cycle appear?: In the Rosenzweig–MacArthur picture, enriching K or steepening saturation can destabilize the coexistence equilibrium (paradox of enrichment). Type III can restabilize at low prey density because predators ignore rare prey.

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