A prey-predator system with disease in prey and cooperative hunting strategy in predator

A predator-prey model is proposed in this work where the prey population is infected by a disease. Here, healthy prey species show defence mechanism while they are attacked by the predator. Moreover as the infected prey are already physically weak, so, predator apply cooperative hunting strategy while consume infected prey to get more food. It helps the predator population to grow with a higher rate. But calculation gives that if they start to hunt the infected prey with a larger cooperative hunting rate, then ultimately predator population decrease with time. Boundedness and positivity of the system variables show that the proposed model system is well-posed. Routh-Hurwitz criterion provides the local stability conditions of the equilibrium points. Also, the system becomes permanent under certain parametric restrictions. The numerical results, verified using MATLAB, support the analytical findings. Numerical simulations give that the parameter denoting cooperative hunting rate can change the system dynamics and we can get oscillating behaviour by regulating this parameter. Moreover transcritical and saddle-node bifurcations occur by regulating the death rate of predator around the critical points. Occurrence of Bogdanov-Takens, generalized Hopf and Cusp bifurcations have also been observed here.