Existence, stability, and bifurcation analysis of a three-species model of deer, tiger, and leopard

In this paper, we develop a three-species interaction model involving deer, tigers, and leopards. Both tigers and leopards are assumed to prey on deer. The consumption rate of deer by tigers is considered to be influenced by the presence of leopards. The tiger population is also considered to be declining due to natural mortality and intra-species competition. Under conditions of food scarcity, specifically, a shortage of deer, it is further assumed that tigers may attack or kill leopards to seize their prey. The model’s equilibrium points are determined and analyzed for their stability. Conditions for the occurrence of transcritical and Hopf bifurcations are examined, and the direction of the Hopf bifurcation is also investigated. Our findings indicate that intra-species competition among tigers significantly influences the stability of the forest ecosystem. An increase in the growth rate of the deer population improves food availability for both predators, thereby enhancing ecological stability. Conversely, increased aggression by tigers toward leopards may destabilize the system. It is found that higher efficiency of deer capture by tigers can benefit the tiger population but may drive both deer and leopards to extinction. Additionally, increasing the carrying capacity of deer exerts a stabilizing influence on the ecosystem.