A Condition for Hopf bifurcation to occur in Equations of Lotka - Volterra Type with Delay

TL;DR

This paper investigates how distributed delays influence the stability of a predator-prey model, demonstrating that increasing delay expectation can induce Hopf bifurcations and lead to oscillatory dynamics.

Contribution

It establishes a condition under which Hopf bifurcation occurs in delayed Lotka-Volterra equations, linking delay expectation to stability changes.

Findings

Delay expectation crossing critical values causes stability loss.

Hopf bifurcation leads to oscillations in predator-prey dynamics.

Distributed delay significantly impacts ecological model behavior.

Abstract

It is known that Lotka - Volterra type differential equations with delays or distributed delays have an important role in modeling ecological systems. In this paper we study the effects of distributed delay on the dynamics of the harvested one predator - two prey model. Using the expectation of the distribution of the delay as a bifurcation parameter, we show that the equilibrium that was asymptotic stable becomes unstable and Hopf bifurcation can occur as the expectation crosses some critical values.