In this paper, focusing on individualistic generalist predators and prey living in herds which coexist in a common area, we propose a generalization of a previous model, namely, a two-population system that accounts for the prey response to predator attacks. In particular, we suggest a new prey-predator interaction term with a denominator of the Beddington-DeAngelis form and a function in the numerator that behaves as $ N $ for small values of $ N $, and as $ N^{\alpha} $ for large values of $ N $, where $ N $ denotes the number of prey. We can take the savanna biome as a reference example, concentrating on large herbivores inhabiting it and some predators that feed on them. Only two conditionally stable equilibrium points have emerged from the model analysis: the predator-only equilibrium and the coexistence one. Transcritical bifurcations from the former to the latter type of equilibrium, as well as saddle-node bifurcations of the coexistence equilibrium have been identified numerically by using MATLAB. In addition, the model was found to exhibit bistability. Bistability is studied by using the MATLAB toolbox bSTAB, paying particular attention to the basin stability values. Comparison of coexistence equilibria with other prey-predator models in the literature essentially shows that, in this case, prey thrive in greater numbers and predators in smaller numbers. The population changes due to parameter variations were found to be significantly less pronounced.
Citation: Francesca Acotto, Ezio Venturino. How do predator interference, prey herding and their possible retaliation affect prey-predator coexistence?[J]. AIMS Mathematics, 2024, 9(7): 17122-17145. doi: 10.3934/math.2024831
In this paper, focusing on individualistic generalist predators and prey living in herds which coexist in a common area, we propose a generalization of a previous model, namely, a two-population system that accounts for the prey response to predator attacks. In particular, we suggest a new prey-predator interaction term with a denominator of the Beddington-DeAngelis form and a function in the numerator that behaves as $ N $ for small values of $ N $, and as $ N^{\alpha} $ for large values of $ N $, where $ N $ denotes the number of prey. We can take the savanna biome as a reference example, concentrating on large herbivores inhabiting it and some predators that feed on them. Only two conditionally stable equilibrium points have emerged from the model analysis: the predator-only equilibrium and the coexistence one. Transcritical bifurcations from the former to the latter type of equilibrium, as well as saddle-node bifurcations of the coexistence equilibrium have been identified numerically by using MATLAB. In addition, the model was found to exhibit bistability. Bistability is studied by using the MATLAB toolbox bSTAB, paying particular attention to the basin stability values. Comparison of coexistence equilibria with other prey-predator models in the literature essentially shows that, in this case, prey thrive in greater numbers and predators in smaller numbers. The population changes due to parameter variations were found to be significantly less pronounced.
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