Why focus on the parameter b₁?

In the original paper, b₁ sets the half-saturation level for herbivory on the resource (inversely related to attack efficiency at low prey density). Murdoch and Oaten showed such half-saturation constants strongly affect predator–prey stability; Hastings and Powell found that sweeping b₁ while holding a₁, a₂, b₂, d₁, and d₂ fixed moves the chain through stable equilibria, periodic orbits, and chaos.

What does the “Chaos preset (1991)” button do?

It loads the nondimensional parameter set from Table 1 of Hastings & Powell (1991) with b₁ = 3 in the chaotic regime, sets initial populations to (0.1, 0.1, 0.1), and advances the integrator silently for several hundred steps so the visible trace is closer to the long-term attractor rather than a short transient.

How is this different from the Lotka–Volterra simulator?

Lotka–Volterra is a two-species linear-mass-action model with neutral closed orbits in the plane. This model has three species, logistic resource limitation, saturating predation, and mortality on both consumers—enough structure for period doubling and chaos, which cannot occur in the classical two-species ODE picture.

Can I trust this for real ecosystem forecasting?

No. It is a teaching model with nondimensional populations, no spatial structure, stage structure, or stochasticity. It illustrates that simple food chains can be dynamically rich, not that a particular lake or forest behaves exactly this way.

Three-Species Food Chain (Hastings–Powell)

The Hastings–Powell (1991) model is a classic nondimensional three-species food chain: resource x (plants), consumer y (herbivores), and top predator z. The resource grows logistically, x′ = x(1 − x), while both trophic links use Holling type II saturating functional responses fᵢ(u) = aᵢ u / (1 + bᵢ u). The full system is x′ = x(1 − x) − f₁(x) y, y′ = f₁(x) y − f₂(y) z − d₁ y, and z′ = f₂(y) z − d₂ z. Unlike two-species Lotka–Volterra caricatures, this continuous-time chain can exhibit equilibrium, limit cycles, and chaotic attractors for biologically plausible parameters—especially when the herbivory half-saturation parameter b₁ is varied while other rates are held fixed. Hastings and Powell showed that slow turnover of the top predator (small d₂ relative to d₁) combined with cycling in lower trophic pairs can produce a strange attractor: top-predator crashes release herbivores and resources, then predation rebuilds until herbivores crash again, with sensitive dependence on timing. This simulator integrates the ODEs with fourth-order Runge–Kutta, offers a one-click chaos preset from the paper (a₁ = 5, b₁ = 3, a₂ = 0.1, b₂ = 2, d₁ = 0.4, d₂ = 0.01), plots x, y, z versus time or phase projections (y, z) and (x, y), and runs a short warmup on reset to approach the attractor.

Who it's for: Undergraduate ecology, dynamical systems, or mathematical biology students who know two-species models and want a minimal continuous-time example of chaos in a food web.

Key terms

Food chain
Hastings–Powell model
Holling type II functional response
Chaotic attractor
Trophic cascade
Predator–prey
Runge–Kutta integration
Population dynamics

How it works

Nondimensional three-level food chain (plants x → herbivores y → predators z) with Holling type II functional responses, after Hastings & Powell (1991). Logistic resource growth, saturating predation, and slow top-predator turnover can produce chaotic long-term dynamics — explore especially the half-saturation parameter b₁.

Key equations

x' = x(1−x) − f₁(x)y · y' = f₁(x)y − f₂(y)z − d₁y · z' = f₂(y)z − d₂z · fᵢ(u)=aᵢu/(1+bᵢu)

Frequently asked questions

Why focus on the parameter b₁?: In the original paper, b₁ sets the half-saturation level for herbivory on the resource (inversely related to attack efficiency at low prey density). Murdoch and Oaten showed such half-saturation constants strongly affect predator–prey stability; Hastings and Powell found that sweeping b₁ while holding a₁, a₂, b₂, d₁, and d₂ fixed moves the chain through stable equilibria, periodic orbits, and chaos.
What does the “Chaos preset (1991)” button do?: It loads the nondimensional parameter set from Table 1 of Hastings & Powell (1991) with b₁ = 3 in the chaotic regime, sets initial populations to (0.1, 0.1, 0.1), and advances the integrator silently for several hundred steps so the visible trace is closer to the long-term attractor rather than a short transient.
How is this different from the Lotka–Volterra simulator?: Lotka–Volterra is a two-species linear-mass-action model with neutral closed orbits in the plane. This model has three species, logistic resource limitation, saturating predation, and mortality on both consumers—enough structure for period doubling and chaos, which cannot occur in the classical two-species ODE picture.
Can I trust this for real ecosystem forecasting?: No. It is a teaching model with nondimensional populations, no spatial structure, stage structure, or stochasticity. It illustrates that simple food chains can be dynamically rich, not that a particular lake or forest behaves exactly this way.