Consequences of weak Allee effect on prey in the May–Holling–Tanner predator–prey model

González‐Olivares, Eduardo; Gallego‐Berrío, Lina M.; González‐Yañez, Betsabé; Rojas‐Palma, Alejandro

doi:10.1002/mma.3441

Cited by 6 publications

(3 citation statements)

References 38 publications

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“…However, these phenomena were not considered by the authors as they used model (1). This manuscript also extends some of the results obtained by Arancibia-Ibarra and González-Olivares [25] and González-Olivares et al [26] for a modified Leslie-Gower model with m = 0, that is, with a specific type of weak Allee effect, see second row of Table 1. Leslie-Gower models with strong Allee effect and different type of functional responses have been extensively studied in [27][28][29], see first row of Table 1.…”

Section: Introductionsupporting

confidence: 80%

“…In section 2, we non-dimensionalise the Leslie-Gower model with weak Allee effect and discuss the number of equilibria the model has in the first quadrant. The main mathematical difference between system (1), (2) with m < 0, and (2) with m ≥ 0, is the fact that (2) with m < 0 has at most three positive equilibria 1 in the first quadrant instead of one for system (1) and two for system (2) with m ≥ 0 [25,26,29]. These additional equilibrium gives rise to different type of bifurcations, such as saddle-node bifurcations, Bogdanov-Takens bifurcations, and homoclinic bifurcations.…”

Section: Introductionmentioning

confidence: 99%

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Stability Analysis of a Modified Leslie–Gower Predation Model With Weak Allee Effect in the Prey

Arancibia–Ibarra

Flores

Heijster

2022

Front. Appl. Math. Stat.

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In this manuscript, we study a Leslie–Gower predator-prey model with a hyperbolic functional response and weak Allee effect. The results reveal that the model supports coexistence and oscillation of both predator and prey populations. We also identify regions in the parameter space in which different kinds of bifurcations, such as saddle-node bifurcations, Hopf bifurcations and Bogdanov–Takens bifurcations.

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Section: Introductionsupporting

confidence: 80%

Section: Introductionmentioning

confidence: 99%

Stability Analysis of a Modified Leslie–Gower Predation Model With Weak Allee Effect in the Prey

Arancibia–Ibarra

Flores

Heijster

2022

Front. Appl. Math. Stat.

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“…In turn, our goal in this work is to extend the analysis to include an additional phenomenon of competition among the predators and to address the question of what are the possible Allee thresholds in the model. Indeed, some of the most important recent advances on population models with Allee effect focus on stability of equilibrium points, the number of limit cycles, and local bifurcation analysis; see, for instance, Aguirre et al 35 and González-Olivares et al [36][37][38] However, to the best of our knowledge, only a few works have investigated the features of Allee thresholds and basins of attraction and their relation to bifurcations under system parameters in predation models, 6,[39][40][41] let alone in the presence of multiple Allee effects. This is a relevant challenge.…”

Section: The Model With Double Allee Effectmentioning

confidence: 99%

Allee thresholds and basins of attraction in a predation model with double Allee effect

Julio

Aguirre

2018

Math Methods in App Sciences

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We investigate the nature of Allee thresholds and basins of attraction in a predation model with double Allee effect in the prey and a competition behaviour in the predator. From a mathematical perspective, this implies to find and characterise the corresponding basin boundaries in phase space. This is typically a major challenge since the objects that act as boundaries between 2 different basins are invariant manifolds of the system, which may also undergo topological changes at bifurcations. For this goal, we make an extensive use of analytical tools from dynamical systems theory and numerical bifurcation analysis and determine the full bifurcation diagram. Local bifurcations include saddle‐node, transcritical and Hopf bifurcations, while global phenomena include homoclinic bifurcations, heteroclinic connections, and heteroclinic cycles. We identify the Allee threshold to be either a limit cycle, a homoclinic orbit, or the stable manifold of an equilibrium. This strategy based on bifurcation and invariant manifold analysis allows us to identify the mathematical mechanisms that produce rearrangements of separatrices in phase space. In this way, we give a full geometrical explanation of how the Allee threshold and basins of attraction undergo critical transitions. This approach is complemented with a study of the dynamics near infinity. In this way, we determine the conditions such that the basins of attraction are bounded or unbounded sets in phase space. All in all, these results allow us to show a complete description of phase portraits, extinction thresholds, and basins of attraction of our model under variation of parameters.

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A modified Leslie–Gower predator–prey model with ratio‐dependent functional response and alternative food for the predator

Flores

González‐Olivares

2017

Math Methods in App Sciences

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In this work, a modified Leslie–Gower predator–prey model is analyzed, considering an alternative food for the predator and a ratio‐dependent functional response to express the species interaction. The system is well defined in the entire first quadrant except at the origin (0,0). Given the importance of the origin (0,0) as it represents the extinction of both populations, it is convenient to provide a continuous extension of the system to the origin. By changing variables and a time rescaling, we obtain a polynomial differential equations system, which is topologically equivalent to the original one, obtaining that the non‐hyperbolic equilibrium point (0,0) in the new system is a repellor for all parameter values. Therefore, our novel model presents a remarkable difference with other models using ratio‐dependent functional response. We establish conditions on the parameter values for the existence of up to two positive equilibrium points; when this happen, one of them is always a hyperbolic saddle point, and the other can be either an attractor or a repellor surrounded by at least one limit cycle. We also show the existence of a separatrix curve dividing the behavior of the trajectories in the phase plane. Moreover, we establish parameter sets for which a homoclinic curve exits, and we show the existence of saddle‐node bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation, and homoclinic bifurcation. An important feature in this model is that the prey population can go to extinction; meanwhile, population of predators can survive because of the consumption of alternative food in the absence of prey. In addition, the prey population can attain their carrying capacity level when predators go to extinction. We demonstrate that the solutions are non‐negatives and bounded (dissipativity and permanence of population in many other works). Furthermore, some simulations to reinforce our mathematical results are shown, and we further discuss their ecological meanings. Copyright © 2017 John Wiley & Sons, Ltd.

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Consequences of weak Allee effect on prey in the May–Holling–Tanner predator–prey model

Cited by 6 publications

References 38 publications

Stability Analysis of a Modified Leslie–Gower Predation Model With Weak Allee Effect in the Prey

Stability Analysis of a Modified Leslie–Gower Predation Model With Weak Allee Effect in the Prey

Allee thresholds and basins of attraction in a predation model with double Allee effect

A modified Leslie–Gower predator–prey model with ratio‐dependent functional response and alternative food for the predator

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