Qualitative study of delayed predator--prey system in ecology

S Shivam; Teekam Singh; Mukesh Kumar

S Shivam Department of Mathematics, Graphic Era (Deemed to be) University, Dehradun 248002, Uttarakhand, India.
Teekam Singh School of Computer Science, University of Petroleum and Energy Studies, Dehradun 248007, Uttarakhand, India.
Mukesh Kumar Department of Mathematics, Graphic Era (Deemed to be) University, Dehradun 248002, Uttarakhand, India.

Abstract

On Earth, biological species like to form communities. Thus, studying this collective behavior in a predator-prey environment will provide more realistic findings. This article examines a predator-prey system in which both the prey and the predator exhibit swarm behavior. We begin by determining the linear stability of the system and then perform Hopf-bifurcation analysis on the consumption rate $(b)$. We then discussed the influence of time delay on the predation term since every biological function takes time to occur. We analyze the system's stability using a discrete-time delay, $\tau$ as a bifurcation parameter and perform Hopf-bifurcation analysis. Finally, numerical simulations are used to verify the analytical conclusions. This study will increase our knowledge of the dynamical behavior of realistic models in ecological systems.