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This work deals with some typographical mistakes into the above-referenced paper. Although they do not affect the main results, it is necessary to make due corrections. We affirm that the results and conclusions obtained are correct and the errors have no further implications in the aforementioned paper.

The main purpose of this work is to analyze a Gause type predator-prey model in which two ecological phenomena are considered: the Allee effect affecting the prey growth function and the formation of group defence by prey in order to avoid the predation. We prove the existence of a separatrix curves in the phase plane, determined by the stable manifold of the equilibrium point associated to the Allee effect, implying that the solutions are highly sensitive to the initial conditions. Trajectories starting at one side of this separatrix curve have the equilibrium point (0,0) as their ω-limit, while trajectories starting at the other side will approach to one of the following three attractors: a stable limit cycle, a stable coexistence point or the stable equilibrium point (K,0) in which the predators disappear and prey attains their carrying capacity. We obtain conditions on the parameter values for the existence of one or two positive hyperbolic equilibrium points and the existence of a limit cycle surrounding one of them. Both ecological processes under study, namely the nonmonotonic functional response and the Allee effect on prey, exert a strong influence on the system dynamics, resulting in multiple domains of attraction. Using Liapunov quantities we demonstrate the uniqueness of limit cycle, which constitutes one of the main differences with the model where the Allee effect is not considered. Computer simulations are also given in support of the conclusions.

In this work, a bidimensional differential equation system obtained by modifying the well-known predator–prey Rosenzweig–MacArthur model is analyzed by considering prey growth influenced by the Allee effect. One of the main consequences of this modification is a separatrix curve that appears in the phase plane, dividing the behavior of the trajectories. The results show that the equilibrium in the origin is an attractor for any set of parameters. The unique positive equilibrium, when it exists, can be either an attractor or a repeller surrounded by a limit cycle, whose uniqueness is established by calculating the Lyapunov quantities. Therefore, both populations could either reach deterministic extinction or long-term deterministic coexistence. The existence of a heteroclinic curve is also proved. When this curve is broken by changing parameter values, then the origin turns out to be an attractor for all orbits in the phase plane. This implies that there are plausible conditions where both populations can go to extinction. We conclude that strong and weak Allee effects on prey population exert similar influences on the predator–prey model, thereby increasing the risk of ecological extinction.

This work deals with the analysis of a predator–prey model derived from the Leslie–Gower type model, where the most common mathematical form to express the Allee effect in the prey growth function is considered. It is well-known that the Leslie–Gower model has a unique globally asymptotically stable equilibrium point. However, it is shown here the Allee effect significantly modifies the original system dynamics, as the studied model involves many non-topological equivalent behaviors. None, one or two equilibrium points can exist at the interior of the first quadrant of the modified Leslie–Gower model with strong Allee effect on prey. However, a collapse may be seen when two positive equilibrium points exist. Moreover, we proved the existence of parameter subsets for which the system can have: a cusp point (Bogdanov–Takens bifurcation), homoclinic curves (homoclinic bifurcation), Hopf bifurcation and the existence of two limit cycles, the innermost stable and the outermost unstable, in inverse stability as they usually appear in the Gause-type predator–prey models. In contrast, the system modelling an special of weak Allee effect, may include none or just one positive equilibrium point and no homoclinic curve; the latter implies a significant difference between the mathematical properties of these forms of the phenomenon, although both systems show some rich and interesting dynamics.

The aim of this paper is to correct two mistakes in [Appl. Math. Modell. (2011) 366–381], which are: the function defining the time rescaling given and the inclusion of a parameter outside of model. For a modified Leslie–Gower type predator–prey model considering the Allee effect on prey, a change of variables and a new time rescaling generating a diffeomorphism is proved; a topologically equivalent system to the original one is obtained, which is the same studied in the mentioned paper; we claim that the results and conclusions obtained are correct and the errors have not further implications.

In this work we analyze a predator-prey model proposed by A. Kent et al. in Ecol. Model.162, 233 (2003), in which two aspect of the model are considered: an effect of emigration or immigration on prey population to constant rate and a prey threshold level for predators. We prove that the system when the immigration effect is introduced in the model has a dynamics that is similar to the Rosenzweig-MacArthur model. Also, when emigration is considered in the model, we show that the behavior of the system is strongly dependent on this phenomenon, this due to the fact that trajectories are highly sensitive to the initial conditions, in similar way as when Allee effect is assumed on prey. Furthermore, we determine constraints in the parameters space for which two stable attractor exist, indicating that the extinction of both population is possible in addition with the coexistence of oscillating of populations size in a unique stable limit cycle. We also show that the consideration of a threshold level of prey population for the predator is not essential in the dynamics of the model.