Biological control of infection pervasive via pest: a study of prey-predator model incorporating prey refuge under fuzzy impreciseness

The prey-predator systems exist in almost all ecological systems and the refuge stage is highly concerned with such ecological systems in which the prey goes out of reach of the attacking region of the predator. In this paper, dynamical behaviour of predator and prey species with Holling type III response function is investigated. We add refuge in prey species and also additional predators are introduced into the system. Firstly, the crisp model is formulated with these effects and the boundedness, existence, conditions of local and global stability, local bifurcation and Hopf bifurcation are discussed. The crisp model is transformed into the fuzzy model to incorporate the unpredictable behaviour and using the Root Mean Square (RMS) defuzzification method, the fuzzy model is reduced to the defuzzified model in order to avoid the complexity of the analysis of the solution. Further, numerical simulations are performed with the help of MATLAB, MATCONT package and we examine solution curves and phase portraits in two different cases, in the presence and in the absence of additional predators along with the Hopf bifurcation scenarios. Finally, the sensitivities of the controlling parameters are plotted graphically and presented their significances in controlling the infection of the prey.