In recent years, predator–prey systems have increased their applications and have given rise to systems which represent more accurately different biological issues that appear in the context of interacting species. Our aim in this paper is to give a state-of-the-art review of recent predator–prey models which include some interesting characteristics such as Allee effect, fear effect, cannibalism, and immigration. We compare the qualitative results obtained for each of them, particularly regarding the equilibria, local and global stability, and the existence of limit cycles.

In nature, the vast majority of species live in ecosystems that are not isolated, and the same is true for predator-prey ecological systems. With this work, we extend a predator-prey model by considering the inclusion of an immigration term in both species. From a biological point of view, that allows us to achieve a more realistic model. We consider a system with a Holling type Ⅰ functional response and study its global dynamics, which allows to not only determine the behavior in a region of the plane $ \mathbb{R}^2 $, but also to control the orbits that either go or come to infinity. First, we study the local dynamics of the system, by analyzing the singular points and their stability, as well as the possible behavior of the limit cycles when they exist. By using the Poincaré compactification, we determine the global dynamics by studying the global phase portraits in the positive quadrant of the Poincaré disk, which is the region where the system is of interest from a biological point of view.

We study a population model with two preys and one predator, considering a Holling type II functional response for the interaction between first prey and predator and taking into account indirect effect of predation. We perform the stability analysis of equilibria and study the possibility of Hopf bifurcation. We also include a detailed discussion on the problem of persistence. Several numerical simulations are provided in order to illustrate the theoretical results of the paper.

We classify the global dynamics of a family of Kolmogorov systems depending on three parameters which has ecological meaning as it modelizes a predator–prey system. We obtain all their topologically distinct global phase portraits in the positive quadrant of the Poincaré disc, so we provide all the possible distinct dynamics of these systems.

Most ecological interactions occur in open ecosystems, where species can enter or leave. Predator–prey dynamics are no exception. This work generalizes a predator–prey model by incorporating immigration for both species, making it more biologically relevant. We analyze a system with a Holling type II functional response in the positive quadrant of the Poincaré disk, the biologically meaningful region, including its behavior near infinity. We identify several realizable phase portrait configurations and provide sufficient conditions for the occurrence of a Hopf bifurcation giving rise to a small-amplitude limit cycle.