Dynamics in a ratio-dependent eco-epidemiological predator-prey model having cross species disease transmission

Abstract

We have proposed and analyzed an eco-epidemiological predator-prey interaction model having disease in both the population with ratio-dependent functional response. The total population has been classified into susceptible prey, infected prey, susceptible predator, and infected predator. The infection propagation is considered directly proportional to the number of individuals come in to contact with one infected individual. The predation ability of infected predator is neglected during the disease. The infection is considered to be weak infection in case of predator. Predator can recover themselves due to their natural immune system or application of external curative stimulants. Though such weak infections are not generally considered for study but these cannot be completely ignored. The positivity of the solutions, the existence of various biologically feasible equilibrium points, their stability are investigated. The numerical analysis is carried out with hypothetical set of parameters to substantiate the analytical findings that our model exhibits. The oscillatory coexistence of the species which is very common in nature is observed for disease free as well as coexisting system. The stability nature of the Hopf-bifurcating periodic solutions of the disease free as well as coexisting equilibrium are determined by computing the Lyapunov coefficients. Further, the system undergoes the Bogdanov-Takens bifurcation in two-parameter space around the disease free as well as coexisting equilibrium. It is also observed that the system will be disease free through proper predational strategies.