Non-smooth dynamics emerging from predator-driven discontinuous prey dispersal

Joydeb Bhattacharyya,Joydev Chattopadhyay

doi:10.1007/s11071-021-06963-6

Joydeb Bhattacharyya, Joydev Chattopadhyay

Open Access

https://doi.org/10.1007/s11071-021-06963-6

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Journal: Nonlinear Dynamics	Publication Date: Oct 13, 2021
Citations: 3	License type: cc-by

Affiliation: Indian Statistical Institute

Abstract

In ecology, the refuge protection of the prey plays a significant role in the dynamics of the interactions between prey and predator. In this paper, we investigate the dynamics of a non-smooth prey–predator mathematical model characterized by density-dependent intermittent refuge protection of the prey. The model assumes the population density of the predator as an index for the prey to decide on when to avail or discontinue refuge protection, representing the level of apprehension of the prey by the predators. We apply Filippov’s regularization approach to study the model and obtain the sliding segment of the system. We obtain the criterion for the existence of the regular or virtual equilibria, boundary equilibrium, tangent points, and pseudo-equilibria of the Filippov system. The conditions for the visibility (or invisibility) of the tangent points are derived. We investigate the regular or virtual equilibrium bifurcation, boundary-node bifurcation and pseudo-saddle-node bifurcation. Further, we examine the effects of dispersal delay on the Filippov system associated with prey vigilance in identifying the predator population density. We observe that the hysteresis in the Filippov system produces stable limit cycles around the predator population density threshold in some bounded region in the phase plane. Moreover, we find that the level of apprehension and vigilance of the prey play a significant role in their refuge-dispersion dynamics.

Highlights

In an ecosystem, the prey population exhibit a variety of mechanisms to avoid predation [27]
Since refuges are safe for the prey but restrict their foraging time or mating opportunities than in open habitat, there is a reduction in the growth rate in the prey population during refuge [1,18,28, 33]
We have considered a two-species predator-prey model in an open habitat under the assumption that a portion of the prey population takes refuge from the predators when the density of the predators goes above a certain threshold level L; below this threshold value, the prey does not go to the refuge and prefers to stay in the open habitat which is more favourable compared to the refuge

Summary

Introduction

The prey population exhibit a variety of mechanisms to avoid predation [27]. The three curves Γi (i = 1, 2, 3) divide the δ − L parameter plane into six regions, and the existence or coexistence of stable regular or virtual interior equilibria ES∗ε (ε = 0, 1) is pointed out in each of the regions (cf Fig. 3).

Results

Conclusion