Bifurcations in Delayed Lotka-Volterra Intraguild Predation Model

Abstract

Omnivory is defined as feeding on more than one trophic level. An example of this is the so-called intraguild predation (IG) which includes a predator and its prey that share a common resource. IG predation models are known to exhibit interesting dynamics including chaos. This work considers a three-species food web model with omnivory, where the interactions between the basal resource, the IG prey, and the IG predator are of Lotka-Volterra type. In the absence of predation, the basal resource follows a delayed logistic equation or popularly known as Hutchinson's equation. Conditions for the existence, stability, and bifurcations of all non-negative equilibrium solutions are given using the delay time as parameter. Results are illustrated using numerical bifurcation analysis.

Extracted equations

• dx/dt = [a0 - a1*x(t-τ) - a2*y(t) - a3*z(t)]*x(t)
• dy/dt = [-b0 + b1*x(t) - b3*z(t)]*y(t)
• dz/dt = [-c0 + c1*x(t) + c2*y(t)]*z(t)

Paper claims vs. our run

• E1 is locally asymptotically stable for τ < π/(2*a0) when A > K and C > K
  not-testable
  no fidelity score recorded
• Hopf bifurcation occurs at τ_c = π/(2*a0) ≈ 1.5708 for Example 1 parameters
  not-testable
  no fidelity score recorded
• Stability switch from stable to unstable equilibrium as τ increases past τ_c
  not-testable
  no fidelity score recorded
• Stable branch of periodic solutions emerges from Hopf bifurcation point
  not-testable
  no fidelity score recorded
• All three species persist in oscillatory dynamics after bifurcation
  not-testable
  no fidelity score recorded

Parameters

Run notes

equations tried: ['dx/dt = [a0 - a1*x(t-τ) - a2*y(t) - a3*z(t)]*x(t)', 'dy/dt = [-b0 + b1*x(t) - b3*z(t)]*y(t)', 'dz/dt = [-c0 + c1*x(t) + c2*y(t)]*z(t)']