We study the dynamics of a modified Leslie–Gower one prey–two predators model with competition between predator populations. The model describes complex dynamics in the permanence, global stability and bifurcation. It is shown that there are eight possible equilibrium states. Two equilibrium states, i.e., the extinction of all of the species state and the extinction of both predators state are always unstable, while the other equilibrium states are conditionally locally and globally asymptotically stable. We also analyzed numerically the effect of competition between predators. Our numerical simulations showed that the competition rate of the second-predator may induce the transcritical bifurcation, the saddle-node bifurcation as well as the bi-stability phenomenon. Such numerical results are consistent with the analytical results.

This research focuses on the dynamical of a Leslie-Gower predator-prey model with competition on predator populations. The model represents an interaction between one prey and two predator populations. The analysis shows that there are four equilibrium points, namely the extinction of predator populations point, the extinction of the first predator population point, the extinction of the second predator and the interior point. The existence of the interior equilibrium point is investigated by using Cardan criteria. Local stability analysis shows that both predator populations have never been extinct together. The second and third equilibrium point is local asymptotically stable under some conditions. Numerical simulations are carried out to investigate the stability of the interior point as well as to show that more than one equilibrium point may be asymptotically stable together for a set of parameter.