Modelling, dynamics and analysis of multi-species systems with prey refuge

Abstract

Many biological problems can be reduced to the description of a food chain model or a food web. In these systems, the biodiversity and coexistence of all species are vital issues to discuss. Three ecological models have been proposed in case of the existence of a reserved area, in order to understand multi-species interactions so as to prevent the slow extinction of some endangered species and to test the stability when the length of the food chain and size of the web models are increased. It is taken that the environment has been divided into two disjoint regions, namely, unreserved and reserved zones, where a predator is not allowed to enter the latter. The first model describes a four species food chain predator-prey model with prey refuge (prey in the reserved zone, prey in the unreserved zone, predator and top predator), with the predator being entirely dependent on the prey in the unprotected area. The second model addresses the same problem, but in addition, a third component in the chain partially depends on the prey in the unreserved zone. Finally, the last model investigates a four species food web system with a prey refuge and in this case, the fourth component can also feed directly on the prey in the unreserved zone. The boundedness, existence and uniqueness of the solutions of the proposed models are established. The local and global dynamical behaviours are investigated, with the persistence conditions of the models being elicited. The local bifurcation near each of the equilibrium points is obtained. The numerical simulations in MATLABR are used to study the influence of the existence of the reserved zone on the dynamical behaviour of the proposed models. It has been concluded that the role of the reserved area could be beneficial for the survival and stabilising of multi-species interactions.