Analysis of a Predator-Prey Model with Distributed Delay

Gunasundari Chandrasekar,Bahri Belkacem Cherif,Senthilkumaran Murugaiah,Arul Joseph Gnanaprakasam,Salah Mahmoud Boulaaras,Maria Alessandra Ragusa

doi:10.1155/2021/9954409

Abstract

In this paper, we consider a predator-prey model, where we assumed that the model to be an infected predator-free equilibrium one. The model includes a distributed delay to describe the time between the predator’s capture of the prey and its conversion to biomass for predators. When the delay is absent, the model exhibits asymptotic convergence to an equilibrium. Therefore, any nonequilibrium dynamics in the model when the delay is included can be attributed to the delay’s inclusion. We assume that the delay is distributed and model the delay using integrodifferential equations. We established the well-posedness and basic properties of solutions of the model with nonspecified delay. Then, we analyzed the local and global dynamics as the mean delay varies.

Highlights

In applied engineering and complex system sciences, mathematical models that display deterministic chaotic dynamical behaviour are of interest
Their research focused on switches of stability of the coexistence equilibrium, the occurrence of periodic solutions, and subsequent bifurcation dynamics as the length of the delay increased
We study the effects of incorporating distributed delay in the system (1) for infected predator-free equilibrium

Summary

Introduction

In applied engineering and complex system sciences, mathematical models that display deterministic chaotic dynamical behaviour are of interest. The majority of encounters in nature are admittedly delayed or isolated, as both predator and prey function stochastically in absorbing available resources. Its variations and extensions are studied in modern day population dynamics theory (see, for example, [1–14]) This model is based on the assumption that in real-world ecosystems prey populations do not grow exponentially in the absence of a predator, but rather their size is eventually limited by the absence of resources. They established the existence of stability switches due to Hopf bifurcations. In [4], we have modified the system (1) with discrete delay They investigated the stability properties and the existence of Hopf bifurcation.

Infected Predator-Free Equilibrium

Well Posedness and Basic Properties of the Model

Stability Results with General Delay

Global Dynamics

Conclusion

Conflicts of Interest