Abstract
In this paper, we propose a time-delayed predator-prey model with Holling-type II functional response, which incorporates the gestation period and the cost of fear into prey reproduction. The dynamical behavior of this system is both analytically and numerically investigated from the viewpoint of stability, permanence, and bifurcation. We found that there are stability switches, and Hopf bifurcations occur when the delay τ passes through a sequence of critical values. The explicit formulae which determine the direction, stability, and other properties of the bifurcating periodic solutions are given by using the normal form theory and center manifold theorem. We perform extensive numerical simulations to explore the impact of some important parameters on the dynamics of the system. Numerical simulations show that high levels of fear have a stabilizing effect while relatively low levels of fear have a destabilizing effect on the predator-prey interactions which lead to limit-cycle oscillations. We also found that the model with or without a delay-dependent factor can have a significantly different dynamics. Thus, ignoring the delay or not including the delay-dependent factor might result in inaccurate modelling predictions.
Highlights
Predator-prey interaction is a central topic in ecology and evolutionary biology which has been extensively studied from different aspects by many researchers over the last few decades
Most of the existing predator-prey models are based upon classical Lotka–Volterra formalism, and it is usually assumed that the predators can impact the prey population only through direct killing as direct predation is relatively easy to observe in nature
We assume that the prey population follows a logistic growth without any predator population, which can be split into three different parts, namely, the birth rate, the natural death rate, and the density dependent death due to intraspecific competition among preys
Summary
Predator-prey interaction is a central topic in ecology and evolutionary biology which has been extensively studied from different aspects by many researchers over the last few decades. Numerical simulations showed that both strong adaptation of adult prey and the large cost of fear have destabilizing effect, but large population of predators has a stabilizing effect on the system. Sasmal and Takeuchi studied a predator-prey system with fear effect, group defense, and Holling-type IV functional response; they observed that the model can exhibit multistability [14]. Sarkar and Khajanchi [16] proposed and analyzed a prey-predator system introducing the cost of fear into prey reproduction with Holling-type II functional response. Eir mathematical analysis suggests that strong antipredator responses can stabilize the prey-predator interactions by ignoring the existence of periodic behaviors.
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