Two predators, one prey model that integrates the effect of supplementary food resources due to one predator's kleptoparasitism under the possibility of retribution by the other predator.

In ecology, foraging requires animals to expend energy in order to obtain resources. The cost of foraging can be reduced through kleptoparasitism, the theft of a resource that another individual has expended effort to acquire. Thus, kleptoparasitism is one of the most significant feeding techniques in ecology. The phenomenon of kleptoparasitism has garnered significant attention from scholars due to its substantial impact on the food chain. However, the proportionate amount of mathematical modelling to facilitate the analysis has made limited progress in the literature. This circumstance motivated us to develop mathematical models that could explain the population dynamics of the prey-predator food chain. This study explores a scenario with two predators and one prey, where one predator is a kleptoparasite and the other is a host. The energy depletion caused by the predator's counterattack subsequent to kleptoparasitism, notwithstanding the nonlethal nature of this antagonism, is an additional component incorporated into this model. It has been suggested that biologically viable equilibria must meet certain parametric conditions in order to exist and to be stable both locally and globally. This article delves deeply into the occurrences of various one-parametric bifurcations, such as saddle-node bifurcation, transcritical bifurcation, and Hopf bifurcation, as well as two-parametric bifurcations, such as Bautin bifurcation. A subcritical Hopf bifurcation happens when the growth rate of the first predator is relatively low, while a supercritical Hopf bifurcation occurs when the growth rate of the first predator is quite large, allowing for the coexistence of all three species. Numerical simulations have been conducted to validate our theoretical findings.