Structural sensitivity of chaotic dynamics in Hastings-Powell's model

TL;DR

This paper investigates how replacing Holling type II responses with Ivlev responses in the Hastings-Powell model affects chaotic dynamics and species extinction, revealing significant differences in stability and coexistence.

Contribution

It demonstrates the structural sensitivity of chaotic dynamics to the type of functional responses, proving bifurcation thresholds and showing the non-extinction of predators with Ivlev responses.

Findings

Existence of two Hopf-bifurcation thresholds.

Unstable limit cycle detected numerically.

No predator extinction with Ivlev functional responses.

Abstract

The classical Hastings-Powell model is well known to exhibit chaotic dynamics in a three-species food chain. Chaotic dynamics appear through period-doubling bifurcation of stable coexistence limit cycle around an unstable interior equilibrium point. A specific choice of parameter value leads to a situation where the chaotic attractor disappears through a collision with an unstable limit cycle. As a result, the top predator goes to extinction. Here we explore the structural sensitivity of this phenomenon by replacing the Holling type II functional responses with Ivlev functional responses. Here we prove the existence of two Hopf-bifurcation thresholds and numerically detect the existence of an unstable limit cycle. The model with Ivlev functional responses does not indicate any possibility of extinction of the top predator. Further, the choice of functional responses depicts a significantly different picture of the coexistence of the three species involved with the model.