Mathematical analysis of predator-prey model with two preys and one predator

Hamid A. Adamu

The Mathematics of ecology involves the study of populations that interact, thereby affecting each other's growth rates. This paper investigates a special case of such interaction.  To simplify the model, the paper make some assumptions that simplify the complication of the model. It specifically investigates the predator-prey model with two preys and one predator where the interaction between the species is analsysed both in two and three dimensions. Three dimensional Lotka Volterra model have been examined where a new assumption is added and  the solutions of the model have been  categorised into three stages representing the three coordinates system. These stages highlight the behaviour and the relationship of the preys with their individual predators. The relationship between the species is obtained in terms of mathematics equations where the equilibrium points of 3D Lotka Volterra model are obtained.  Different interpretations arise from those equations. It have been found among other results that the  prey population- x(t) will grows exponentially in the absence of predators-y(t) under the assumption that there is no threat to the prey other than the specific predator. This unbounded growth of x(t)  is what biologically expected in the absence of the middle-level population y(t). However, this stage is the best for the prey population because it is free from predation and the z(t) population, which is the second predator, is left without source of food. In general, the population of all the systems become extinct in the absence of x(t). Moreover, the stability analysis is examined by finding the eigenvalues of the Jacobean matrix. The relationship between the species is presented in plots using MAPLE software.