Abstract
A prey–predator system with logistic growth of prey and hunting cooperation of predators is studied. The introduction of fractional time derivatives and the related persistent memory strongly characterize the model behavior, as many dynamical systems in the applied sciences are well described by such fractional-order models. Mathematical analysis and numerical simulations are performed to highlight the characteristics of the proposed model. The existence, uniqueness and boundedness of solutions is proved; the stability of the coexistence equilibrium and the occurrence of Hopf bifurcation is investigated. Some numerical approximations of the solution are finally considered; the obtained trajectories confirm the theoretical findings. It is observed that the fractional-order derivative has a stabilizing effect and can be useful to control the coexistence between species.
Highlights
A prey–predator system with logistic growth of prey and hunting cooperation of predators is studied
A number of predator–prey models have been proposed and studied, considering different extensions of the original one: the realistic assumption that prey populations are limited by food resources and not just by predation leads to the inclusion of terms representing carrying capacity for the prey population; the characterization of specific behaviors of the predator population results in the introduction of several functional responses for them; the complexity and dishomogeneity in the environment often requires a spatial description [3]
Even if the interest on fractional calculus traces back at least to the 1970s, in the last decades there has been an explosion of research activities on its application to several areas
Summary
Population dynamics are regulated by several factors: availability of resources, predation, diseases, etc. A number of predator–prey models have been proposed and studied (see the excellent reviews [1,2]), considering different extensions of the original one: the realistic assumption that prey populations are limited by food resources and not just by predation leads to the inclusion of terms representing carrying capacity for the prey population; the characterization of specific behaviors of the predator population results in the introduction of several functional responses for them; the complexity and dishomogeneity in the environment often requires a spatial description [3] It is well-known that species diffusing at different rates can generate spatial patterns, observed in several biological contexts.
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