Stochastic spatial Lotka-Volterra predator-prey models

Abstract

Stochastic, spatially extended models for predator-prey interaction display spatio-temporal structures that are not captured by the Lotka-Volterra mean-field rate equations. These spreading activity fronts reflect persistent correlations between predators and prey that can be analyzed through field-theoretic methods. Introducing local restrictions on the prey population induces a predator extinction threshold, with the critical dynamics at this continuous active-to-absorbing state transition governed by the scaling exponents of directed percolation. Novel features in biologically motivated model variants include the stabilizing effect of a periodically varying carrying capacity that describes seasonally oscillating resource availability; enhanced mean species densities and local fluctuations caused by spatially varying reaction rates; and intriguing evolutionary dynamics emerging when variable interaction rates are affixed to individuals combined with trait inheritance to their offspring. The basic susceptible-infected-susceptible and susceptible-infected-recovered models for infectious disease spreading near their epidemic thresholds are respectively captured by the directed and dynamic isotropic percolation universality classes. Systems with three cyclically competing species akin to spatial rock-paper-scissors games may display striking spiral patterns, yet conservation laws can prevent such noise-induced structure formation. In diffusively coupled inhomogeneous settings, one may observe the stabilization of vulnerable ecologies prone to finite-size extinction or fixation due to immigration waves emanating from the interfaces.

Extracted equations

• ∂n_A/∂t = λ n_A n_B - μ_A n_A + D_A ∇²n_A
• ∂n_B/∂t = -λ n_A n_B - μ_B n_B + D_B ∇²n_B
• Master equation: dP(n,t)/dt = Σ_i [W(n|n-e_i)P(n-e_i,t) - W(n-e_i|n)P(n,t)]

Paper claims vs. our run

• Stochastic spatial models display noise-induced activity fronts not captured by mean-field Lotka-Volterra equations
  not-testable
  no fidelity score recorded
• Persistent correlations between predators and prey emerge in spatially extended systems
  not-testable
  no fidelity score recorded
• Introducing local carrying capacity induces predator extinction threshold at critical lambda_c
  not-testable
  no fidelity score recorded
• Critical dynamics at active-to-absorbing transition governed by directed percolation universality class
  not-testable
  no fidelity score recorded
• Periodically varying carrying capacity stabilizes coexistence
  not-testable
  no fidelity score recorded
• Spatially varying reaction rates enhance mean species densities and local fluctuations
  not-testable
  no fidelity score recorded

Parameters

Run notes

Stub simulator: Monte Carlo runner not yet wired