It is well known that predator-prey interactions are strongly dependent on the functional response as well as the population growth rates of prey and predators.In this work, the study will be based on a Leslie-Gower type predation model, described by a twodimensional system of ordinary differential equations (ODEs), assuming the prey population is affected by a strong Allee effect and that predators have an alternative food.The functional response will be assumed linear, which is prey-dependent and monotonously increasing. In turn, the equation of growth of predators will also be considered of the logistic type, where the environmental carrying capacity for predators is assumed proportional to the prey population size. Among the most important results obtained is that for a same set of parameters, there are different behaviors of the system solutions, since two attractor equilibrium poin...

It is well known that predator-prey interactions are strongly dependent on the functional response as well as the population growth rates of prey and predators.In this work, the study will be based on a Leslie-Gower type predation model, described by a twodimensional system of ordinary differential equations (ODEs), assuming the prey population is affected by a strong Allee effect and that predators have an alternative food.The functional response will be assumed linear, which is prey-dependent and monotonously increasing. In turn, the equation of growth of predators will also be considered of the logistic type, where the environmental carrying capacity for predators is assumed proportional to the prey population size. Among the most important results obtained is that for a same set of parameters, there are different behaviors of the system solutions, since two attractor equilibrium poin...

Population dynamics is a relevant topic in Biomathematics, being the study of the long-term behavior of interaction models between species, one of its central problems. A large part of these relationships are described by ordinary differential equations (ODE), having as main objectives the study of the stability of their solutions. In this document we mainly describe the dynamic behavior of the Volterra predation model. In addition, we make a review of some derived predation models and a brief review of the dynamical properties of models describing other interactions between species such as: competition, mutualism, amensalism, and commensalism; also described by nonlinear ODE systems of the second order of Kolmogorov-type. For each of these models, the non-existence of limit cycles can be demonstrated and in most of  them, there is a globally stable equilibrium point. In one of them, th...

Predation models are a great source of study from both an ecological and a mathematical point of view, especially for the analysis of trophic chains. The determination of the dynamics of the systems that describe them, as well as the verification of the nature of these properties by the interacting species, is a topic that is sometimes not always correlated. It is widely known that the incorporation of some mathematical descriptions of ecological phenomena strongly modifies the properties of many of these models. This implies that the systems describing such models are structurally unstable. In this work, we include collaboration or cooperation between predators, a social behavior that describes the help made to capture their favorite prey. It is described by a power function with an exponent between 0 and 1, to indicate the possible interference between them, despite their collaborati...

The interactions between predators and their prey are one of the most important aspects in the dynamics of food chains or trophic webs. Usually, this relationship in the real world is influenced by various behaviors, both from prey and  predators. Collaboration or cooperation between predators is one of those behaviors, which has received less attention than other behaviors of predators, such as competition between them. In this work, we will model the cooperation between predators to capture (or consume) their favorite prey using a recent proposition that modifies the linear functional response of the Leslie-Gower model. We show that this modified model has richer dynamics than the original, obtaining varied results. Among the main ones, populations can oscillate around a point where population sizes are fixed.

The interactions between predators and their prey in the real world are usually affected by diverse ecological phenomena acting on both the prey and the predators. Collaboration or cooperation between predators is one of those behaviors, which has received less attention from researchers than competition among consumers. These contacts are important aspects of the dynamics of food chains and trophic webs. In this work, we will study the influence of collaboration or cooperation (hunting cooperation) between predators to consume (or capture) their favorite prey, which are affected by an effect Allee weak. We extend the results obtained in a previously published model, considering only collaboration between predators and in which the Allee effect is absent.

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.

In several previous works, different predation models have been modified by considering the use of prey refuges, for which a partial analysis of their dynamics is done. In some of them, it is stated that the use of refuge has a stabilizing effect on the predator-prey interaction. One of the purposes of this paper is to show that some of these new systems, derived from the Lotka-Volterra model, this assertion is not fulfilled. In this work, several of the models studied have more than one positive equilibrium point, and the behavior of the solutions is highly dependent on the initial conditions

In various previous works, different predation models have been analyzed considering the use of refuge by the prey population, without carrying out an exhaustive analysis of its dynamics. In some of them it is stated that the use of shelters or dens by a fraction of the prey population has a stabilizing effect on predator-prey interaction. One of the objectives of this work is to show that some of these new systems considering refuge may have the same topological portrait in the phase plane as the original; but in others the dynamics change strongly. In our research we will introduce modifications to the well-known Volterra model, considering various ways to express the number of prey in refuge. In several of them,  local dynamics equivalent to the original model are obtained, confirming that results reported as new were previously known based on the original system, without takin...

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.