Disease control prey–predator model incorporating prey refuge under fuzzy uncertainty

Abstract

Ecological system depends on prey predator interaction. Sometimes diseases are spread among prey or predator or both species. In this paper, a disease control prey–predator model with Holling type-II functional response incorporating refuge in the susceptible prey is proposed. The impact of additional food for predator species in the system is investigated. Using some assumptions, the crisp model is formulated and it is converted into fuzzy model and defuzzified models by using basic defuzzification method. In the theoretical section, the condition of boundedness, existence of equilibrium points, global stability of interior equilibrium point and Hopf bifurcation on a refuge parameter are investigated. Theoretical results are verified in our numerical simulation section. Using MATLAB package we examined the behaviour of the species in the presence and absence of additional food in both crisp and fuzzy environments. The phase trajectories for different initial conditions in both environments and variations in population of the species are presented. Using MATCONT package, we present the bifurcation scenarios when the additional food parameter and refuge parameter vary. We compute the existence of Hopf point (H) and branch point (BP) in the model for suitable supply of additional food and intensity of refuge parameter. Finally the sensitivities of the parameters are plotted graphically. This theoretical and numerical study may be useful to control the infectious diseases in real world ecological systems.