A Beddington-DeAngelis type one-predator two-prey competitive system with help

In this paper, we present and study a two competing prey and one predator system where during predation both the teams of prey help each other and the rate of predation on both the teams are different. The prey consumption rate per individual of predator is considered as Beddington-DeAngelis type. Beddington-DeAngelis type functional response describes predator feeding rate depending on both the prey and predator density. Ecological significance of considering this type of functional response depends on how frequently a predator encounters another predator while feeding on the same type of prey, resulting in strong intraspecific competition and thereby reducing the predator feeding rate. Firstly we establish persistence and permanence of the system and examine the role of cooperation coefficient. Different conditions for the coexistence of equilibrium solutions are discussed. Then the local asymptotic stability of various equilibrium points and the existence of Hopf bifurcation at the positive equilibrium are carried out to understand the dynamics of the model system. The Kolmogorov analysis of subsystems suggests for the existence of transcritical bifurcation. Sufficient conditions for the nonexistence of periodic orbits of the considered model system have been presented. The analytical results found in this paper are illustrated with the help of numerical examples.