Dynamics of a Three Species Predator-Prey Delay Differential Model with Allee Effect and Holling Type-II Functional Response

Abstract

In recent years, predator-prey models appearing in various fields of mathematical biology have been proposed and studied extensively. In this paper, we propose a delay differential model for predator-prey systems of one predator and two preys with cooperation/competition among the preys. The growth of both prey populations is subjected to the Allee effect in the logistic terms (the growth rate increases with the population density while it decreases at larger densities), with Holling type I and II functional responses with the predators. Discrete time delays are incorporated into the model to justify time required by predator to interact with the preys. We study the existence of positive equilibrium points and their asymptotic stabilities. Hopfbifurcations are obtained in terms of critical values of time delays. The system may have a stable periodic orbit, depending on parameter values. The presence of Allee terms and time delays in the model improves the stability of the solutions and enriches the dynamics of the model. Some numerical examples and simulations are provided to validate the derived theoretical results.