In the ecological literature, many models for the predator-prey interactions consider the monotonic functional responses to describe the action of the predators. However, there exist antipredator behaviors which are best represented by non-monotonic functions. The mathematical results on the predator-prey models provide very useful information to understand the complex food webs; they also help to the insight of the mechanisms that govern the evolution of ecological systems. The aim of this paper is to show, the dynamics of a modified Leslie-Gower model, assuming a rational non-monotonic functional response or Holling type IV. A principal target is to compare the obtained properties with other cases, in which different non-monotonic functional responses are incorporated. The model is described by an autonomous bi-dimensional ordinary differential equation system (ODEs), assuming that the prey and predator growth functions are the logistic type. The proposed model is not defined in $(0,0)$; considering a topological equivalent system, it is possible that to prove the origin is a non-hyperbolic saddle point. We also have established, there are subsets of the parameter space in which: i) there exists a unique positive equilibrium point, ii) a heteroclinic curve exists. iii) two concentric limit cycles exist, the innermost unstable and the outermost stable. Numerical simulations are given to endorse the analytical results and to exhibit the richness of the dynamics in the system.

After the well-known classification formulated by Crawford S. Holling in 1959 of the functional responses dependent only of the prey populations, various other have been proposed. In this work a simple Leslie-Gower type predator-prey model is analyzed, incorporating the Rosenzweig functional response described by $h\left( x\right) =qx^{\alpha }$, with $0<\alpha <1$. This function does not conform to the types proposed by Holling, since is not bounded. Although this functional response is non-differentiable for $x=0 $, it is proved that the obtained system is Lipschitzian. However, the existence of a separatrix curve $\Sigma $ in the phase plane it is proven, which divides the phase plane en two complemntary sectors. According to the relative position of the initial conditions respect to the curve $\Sigma $ , the trajectories can have differents $\omega $-$limit$, which can be the equilibrium $\left( 0,0\right) $, or else, a positive equilibrium point, or a limit cycle or a heteroclinic curve. These properties show the great diffference of this model with the original and well-known Leslie-Gower model (when $\alpha =1$), since this last has only a unique positive equilibrium, which is globally asymptotically stable. Then, it can concluded that i) a small change in the mathematical expression for the functional response, it produces a strong change on the dynamics of model. ii) \ a slightest deviation in the initial population sizes, respect to the curve $\Sigma $, it can signify the coexistence of populations or the extinction of both. Numerical simulations are given to endorse our analytical results.