Numerical bifurcation analysis for a prey-predator type interactions with a time lag and habitat complexity

Abstract

In this paper, a two-component generic prey-predator system incorporated with habitat complexity in predator functional response, and with constant time delay in predator gestation is considered. Although the role of time delay on the system dynamics is widely studied in the literature, only a few researchers have addressed the effect of habitat complexity in the prey-predator type interactions. In the first part of the paper the equilibria and stability analysis of the mathematical model is mentioned. In the second part, particular attention is paid on the numerical bifurcation analysis of the prey and predator densities based on two system parameters:(i) the strength of homogeneous habitat complexity and (ii) predator attack rate with and without time delay. It is found that dynamics with time delay in predator gestation are found to be much richer compared to that without time delay. The system stability may change from stable to unstable through a Hopf bifurcation and the solution branches emanating from these Hopf points are usually stable and supercritical. However, delay driven system may lead unstable orbits arising from Hopf bifurcations. It is also found that increasing the strength of habitat complexity may lead the stability change from unstable to stable.

A mathematical model of population dynamics revisited with fear factor, maturation delay, and spatial coefficients

“…The periodic orbits emanating from the Hopf points in the absence (𝜂 = 0) and presence (𝜂 = 0.03) of time delay are shown with straight and dashed blue lines respectively and are both stable. The critical threshold obtained from theoretical formulations is found as 𝜂 c = 0.0092; see Equation (20). The initial condition is given as respect to parameter c, which represents the efficiency ratio by which predator species convert consumed prey species to new predator species.…”

“…Since the role of fear in the birth and death rate of prey species and that of intra-specific competition are well recognized to have a clear effect on the prey-predator interactions, it is crucial to examine these dynamics with incorporated time delay. 2,20,21 Considering delay factor in mathematical modeling traces its roots to seminal work on delay logistic equation by Hutchinson,22 who provided important insights into creating biologically more realistic mathematical models of prey-predator dynamics in theoretical ecology. 21,23,24 It is not surprising that time delay has a vital role in population dynamics as indeed most of the biological events do not take place instantly and require some time lag which usually leads to dramatic fluctuations in the density of populations.…”

A Stochastic Framework for Solving the Prey-Predator Delay Differential Model of Holling Type-III

“…Consequently, the time delay conditions are used in various ordinary differential systems. In various mechanisms, the use of time delay provides the system's destabilization using the co-occurrence state via Hopf bifurcation, along with the dynamics of the strong oscillation [30,31]. Few investigations have been presented in the literature along with the discussion of the Allee effects that make the system's stability with time delay [32].…”