Global Dynamics of a Lotka-Volterra Model with Two Predators Competing for One Prey

In this paper we study the global dynamics of 3–dimensional predator prey Lotka–Volterra systems, which describes two predators competing for food or shared one resource. From theoretical analysis on all parameters of this system, we show that if the resource for prey is limited, then there exist some values of parameters such that two predators and one prey coexist and their population are asymptotic to steady states. Otherwise, at least one of two predator species is extinct. On the other hand, if the resource for prey is unlimited, then there are the complete classification of parameters values such that the system has two possible global dynamics. Either every solution of the system is asymptotic to a closed orbit, or to the equilibrium in the invariant coordinate plane, or every solution of the system is a periodic orbit except the equilibrium in the positive octant of R3. This implies that the principle of competitive exclusion holds for some values of parameters of the Lotka–Volterra system, and it does not hold for the other values of parameters of the Lotka–Volterra system. Hence, there are only two coexistence styles for all three species: periodic oscillation or steady states, which depends on the resource for prey. The results have an importance biological implication in pest control.