Abstract
The most important fact in the field of theoretical ecology and evolutionary biology is the strategy of predation for predators and avoidance of prey from predator attack. A lot of experimental works suggest that the reduction of prey depends on both direct predation and fear of predation. We explore the impact of fear effect and mutual interference among predators into a three-species food chain model. In this manuscript, we have considered a tri-topic food chain model with Beddington–DeAngelis functional response between interacting species, incorporating the reduction of prey and intermediate predator growth due to the fear of intermediate and top predator, respectively. We have provided parametric conditions for existence of biologically feasible equilibria as well as their local and global stability. We have established conditions of transcritical, saddle-node and Hopf bifurcation about different equilibria. Finally, we have performed some numerical investigations to justify analytical findings.
Highlights
Analysis of dynamical activities of prey-predator interaction is one of the momentous themes for researchers in mathematical biology and theoretical ecology
Proof To established the global stability of axial equilibria E1, we introduced the Lyapunov function V1 as follows : V1
Fear of intermediate predator on prey and fear of top predator on intermediate predator both has stabilizing effect on the system dynamics (see Fig. 20(b)). In this investigation we have explored the dynamics of a tri-topic Hasting-Powell food web model incorporating the fear effect of the intermediate predator on growth function of prey, fear the effect of top predator on growth function of intermediate predator and replacing the prey dependent type II functional response by the ratio-dependent Beddington-DeAngelis functional response
Summary
25], both prey and predator dependent functional response like Cowley-Martin type functional response [17], Beddington-DeAngelis type functional response [26, 27], Monod-Haldane type functional response [28] and ratio-dependent functional response [20, 29]. Very recently Elliott et al, [45] studied some field experiments on Drosophila melanogaster as prey and mantid as their predator species, to observe the effect of fear on populations robustness in relation to species density They explored that in presence of mantid, the reproductive rate of drasophila reduces in both their breeding in addition to non-breeding seasons. Depending on field experimental data, Wang et al, Zanette et al [18, 36] developed mathematical formulation for prey-predator interaction by introducing the cost of fear for prey because of predator species, where fear shows a crucial function on prey birth rate They observed that strong anti-predator behaviour or correspondingly most important cost of fear may reduce the risk of the existence of oscillatory behaviours and eliminate the scenarios “paradox of enrichment”.
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