This paper deals with a continuous-time predator-prey model of Leslie-Gower type considering the use of a physical refuge by a fraction of the prey population. The fraction of hidden prey is assumed to be dependent on the presence of predators in the environment. The conditions for the existence of equilibrium points and their local stability are established. In particular, it is shown that the point (0; 0) has a great importance in the dynamics of the model, since it determines a separating curve Σ that divides the behavior of the trajectories. Those trajectories that are above this curve have as their w- limit the point (0; 0), so the extinction of both populations may be possible depending on the initial conditions.

The interactions between two species are basic in the study of complex food chains, particularly the relation among the predators and their prey.The analysis of simple models, described by continuous-time systems, in which some ecological phenomena are incorporated giving lights about this interesting interrelationship.In this work, a Leslie-Gower type predator-prey model is analyzed, considering two aspects: the prey defends from the predation, forming group defense and the predators have an alternative food. So, a rational Holling type IV functional response and a modification of the predators carrying capacity are assumed, to describe each phenomenon.We determine conditions on the parameter space for the existence of the equilibria and their nature.Using the Lyapunov quantities method, we also establish conditions on the parameter values for which there exist a unique positive equilib...

The interactions between two species are basic in the study of complex food chains, particularly the relation among the predators and their prey.The analysis of simple models, described by continuous-time systems, in which some ecological phenomena are incorporated giving lights about this interesting interrelationship.In this work, a Leslie-Gower type predator-prey model is analyzed, considering two aspects: the prey defends from the predation, forming group defense and the predators have an alternative food. So, a rational Holling type IV functional response and a modification of the predators carrying capacity are assumed, to describe each phenomenon.We determine conditions on the parameter space for the existence of the equilibria and their nature.Using the Lyapunov quantities method, we also establish conditions on the parameter values for which there exist a unique positive equilib...

Interrelationships between two species are a basic theme in Population Dynamics, particularly the interaction between predators and their prey. This importance is due to the fact that it allows a deeper understanding of the behavior of complex food webs. In this paper we extend the analysis of a modified Leslie-Gower predator-prey model by assuming that the functional response is sigmoid or Holling type III and the predator have an alternative food. We show that the system representing the model has up to three positive equilibrium points; we establish conditions to determine the nature of each equilibrium point. In addition, we show the existence of different types of bifurcations, including those of Hopf and the homoclinic. The analytical results are discussed from an ecological perspective.