Limit cycles in a Gause‐type predator–prey model with sigmoid functional response and weak Allee effect on prey

Gallego‐Berrío

González‐Yañez

et al. 2015

Self Cite

In this work, a modified Holling-Tanner predator-prey model is analyzed, considering important aspects describing the interaction such as the following: the predator growth function is of a logistic type, and a weak Allee effect acting in the prey growth function and the functional response is of hyperbolic type. By making a change of variables and a time rescaling, we obtain a polynomial differential equations system topologically equivalent to the original one, in which the nonhyperbolic equilibrium point .0, 0/ is an attractor for all parameter values. An important consequence of this property is the existence of a separatrix curve dividing the behavior of trajectories in the phase plane, and the system exhibits the bistability phenomenon, because the trajectories can have different !-limit sets, as an example, the origin .0, 0/ or a stable limit cycle surrounding an unstable positive equilibrium point. We show that, under certain parameter conditions, a positive equilibrium may undergo saddle-node, Hopf, and Bogdanov-Takens bifurcations; the existence of a homoclinic curve on the phase plane is also proved, which breaks in an unstable limit cycle. Some simulations to reinforce our results are also shown.

Section: Introductionmentioning

confidence: 99%

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

Gallego‐Berrío

González‐Yañez

et al. 2015

Self Cite

“…In equation , the strong Allee effect or critical depensation is modeled, which is characterized by the existence of the critical threshold m > 0 below which the population experiences extinction. If m ≤0, it has a weak Allee effect or pure depensation , which is known by not having a threshold that must be surpassed by a population in order to grow. In Figure , the graph of this function for different values of m is shown.…”

Section: Introductionmentioning

confidence: 99%

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

Gallego‐Berrío

González‐Yañez

et al. 2015

a Communicated by J. Vigo-AguiarIn this work, a modified Holling-Tanner predator-prey model is analyzed, considering important aspects describing the interaction such as the predator growth function is of a logistic type; a weak Allee effect acting in the prey growth function, and the functional response is of hyperbolic type. Making a change of variables and time rescaling, we obtain a polynomial differential equations system topologically equivalent to the original one in which the non-hyperbolic equilibrium point .0, 0/ is an attractor for all parameter values. An important consequence of this property is the existence of a separatrix curve dividing the behavior of trajectories in the phase plane, and the system exhibits the bistability phenomenon, because the trajectories can have different ! limit sets; as example, the origin .0, 0/ or a stable limit cycle surrounding an unstable positive equilibrium point. We show that, under certain parameter conditions, a positive equilibrium may undergo saddle-node, Hopf, and Bogdanov-Takens bifurcations; the existence of a homoclinic curve on the phase plane is also proved, which breaks in an unstable limit cycle. Some simulations to reinforce our results are also shown. Copyright

“…To simplify the calculations, we reduce system to a normal form, following the methodology used by González‐Olivares and Rojas‐Palma, which involves a change of variable and a time rescaling . Let the function

Φ : {double-struckR}^{2} \times R_{0}^{+} \to {double-struckR}^{2} \times R_{0}^{+},

such that

Φ (u, v, τ) = Φ (K u, n K v, \frac{u}{r} τ) = (x, y, t) .

…”

Section: The Modelmentioning

confidence: 99%

“…The equilibrium points of system (1) or vector field X are (0, 0), (K, 0), . To simplify the calculations, we reduce system (1) to a normal form, following the methodology used by González-Olivares and Rojas-Palma, 26,27 which involves a change of variable and a time rescaling. 8 Let the function…”

Section: The Modelmentioning

confidence: 99%

A simple Gause‐type predator–prey model considering social predation

Valenzuela‐Figueroa

Rojas‐Palma

2018

The consumer–resource relationships are among the most fundamental of all ecological relationships and have been the focus of ecology since its beginnings. Usually are described by nonlinear differential equation systems, putting the emphasis in the effect of antipredator behavior (APB) by the prey; nevertheless, a minor quantity of articles has considered the social behavior of predators. In this work, two predator–prey models derived from the Volterra model are analyzed, in which the equation of predators is modified considering cooperation or collaboration among predators. It is well known that competition among predators produces a stabilizing effect on system describing the model, since there exists a wide set in the parameter space where the system has a unique equilibrium point in the phase plane, which is globally asymptotically stable. Meanwhile, the cooperation can originate more complex and unusual dynamics. As we will show, it is possible to prove that for certain subset of parameter values the predator population sizes tend to infinite when the prey population goes to extinct. This apparently contradicts the idea of a realistic model, when it is implicitly assumed that the predators are specialist, ie, the prey is its unique source of food. However, this could be a desirable effect when the prey constitutes a plague. To reinforce the analytical result, numerical simulations are presented.