Stability and Bifurcation Analysis of a Three-Species Food Chain Model with Fear

Abstract

In the present paper, we investigate the impact of fear in a tri-trophic food chain model. We propose a three-species food chain model, where the growth rate of middle predator is reduced due to the cost of fear of top predator, and the growth rate of prey is suppressed due to the cost of fear of middle predator. Mathematical properties such as equilibrium analysis, stability analysis, bifurcation analysis and persistence have been investigated. We also describe the global stability analysis of the equilibrium points. Our numerical simulations reveal that cost of fear in basal prey may exhibit bistability by producing unstable limit cycles, however, fear in middle predator can replace unstable limit cycles by a stable limit cycle or a stable interior equilibrium. We observe that fear can stabilize the system from chaos to stable focus through the period-halving phenomenon. We conclude that chaotic dynamics can be controlled by the fear factors. We apply basic tools of nonlinear dynamics such as Poincaré section and maximum Lyapunov exponent to identify the chaotic behavior of the system.

The Dynamics of A Square Root Prey-Predator Model with Fear

“…1, pp: 139-146 140 Therefore, the indirect effect is also important for understanding the behavior of the prey-predator system and needs to be investigated in more details. Accordingly, a number of researchers have recently taken into account studying the prey-predator system with the effects of fear to understand the effect of this factor on the dynamical behavior of the system [13][14][15][16]. A predator-prey model incorporating the cost of fear into prey reproduction was proposed and studied by Wang et al [14].…”

“…They showed that high levels of fear can stabilize the predator-prey system by excluding the existence of periodic solutions. However, Panday et al [15] investigated the impact of fear in a tritrophic food chain model with Holling type II of functional responses. They concluded that chaotic dynamics can be controlled by fear factors.…”

“…Time series of the trajectory of system 1 that approaches asymptotically to the axial equilibrium point ( )for data(15) with .…”

Dynamics of a discrete-time system with Z-type control

“…If the population sizes of interacting species are controlled anyhow, then the ecological balance may be maintained. Different biological phenomena like imposition of a population floor [21,22], addition of refugia [23], omnivory [24], intraspecific density dependence [25], toxic inhibition [26], spatial effect [27], dispersal [28][29][30], predator switching [31,32], Allee effect [33], additional predator [34], additional food [35], harvesting of predator [36], fear effect [37,38], etc., may increase the ecological stability. Although a system may be stable by incorporating these phenomena, these are not always capable to achieve desired population abundance.…”

Bifurcation Based‐Delay Feedback Control Strategy for a Fractional‐Order Two‐Prey One‐Predator System

“…Bifurcation theory has been greatly investigated in prey and predator systems [20,21]. It is proverbial that Hopf bifurcations in inter-order ordinary differential systems have been profoundly studied [22,23]. In the wake of development of the fractional-order calculus, Hopf bifurcations of delayed fractional-order models have recently aroused gigantic attention [24][25][26][27].…”