Double Allee Effects on Prey in a Modified Rosenzweig-MacArthur Predator-Prey Model

González‐Olivares, Eduardo; Huincahue, Jaime

doi:10.1007/978-3-319-03967-1_9

Cited by 2 publications

(4 citation statements)

References 21 publications

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“…On the other hand, the second factor, 1 x+n , represents the presence of a second Allee effect due to a fraction n > 0 of the population being nonfertile. 32,33 As the number of sterile members n of the population is increased, the Allee function A(x) decreases, and hence, the population has greater difficulties to grow.…”

Section: The Model With Double Allee Effectmentioning

confidence: 99%

“…In particular, for an isolated population, parameter m is often interpreted as an Allee threshold, ie, the minimum viable population density that guarantees long‐term non‐extinction. On the other hand, the second factor,

\frac{1}{x + n}

, represents the presence of a second Allee effect due to a fraction n >0 of the population being nonfertile . As the number of sterile members n of the population is increased, the Allee function A ( x ) decreases, and hence, the population has greater difficulties to grow.…”

Section: Introductionmentioning

confidence: 99%

“…However, as far as we know, only a few works have addressed the subject of multiple Allee effects. For instance, González‐Olivares et al use the same expression for A ( x ) as in with the aim of comparing their properties to predation models with simple Allee effect. 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.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: The Model With Double Allee Effectmentioning

confidence: 99%

\frac{1}{x + n}

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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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“…two or more Allee effects act simultaneously on the single population, see e.g. [11], [17] and [18]. For example, the stabilization of vole (Microtus agrestis) populations in southeastern New York depends on the variation in reproductive rate and recruitment of the population, and this effect is referred to as multiple Allee effect, see [36] for more details.…”

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confidence: 99%

Canards and homoclinic orbits in a slow-fast modified May-Holling-Tanner predator-prey model with weak multiple Allee effect

Zhao

Shen

2022

DCDS-B

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<p style='text-indent:20px;'>This paper studies bifurcations of canards and homoclinic orbits in a slow-fast modified May-Holling-Tanner predator-prey model with weak multiple Allee effect. Based on geometric singular perturbation theory (GSPT) and canard theory, canard explosion is observed and the associated bifurcation curve is determined. Due to the canard point, a homoclinic orbit with slow and fast segments and homoclinic to a saddle can also exist, in which, the stable and unstable manifolds of the saddle are connected under certain parameter value. By analyzing the slow divergence integral, it is proved that the cyclicity of canard cycles in this model is at most four. Finally, by calculating the entry-exit function explicitly, a unique, orbitally stable canard relaxation oscillation passing through a transcritical bifurcation point is detected. All these theoretical predictions on the birth of canard explosion, canard limit cycles and homoclinic orbits are verified by numerical simulations.</p>

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Double Allee Effects on Prey in a Modified Rosenzweig-MacArthur Predator-Prey Model

Cited by 2 publications

References 21 publications

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

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

Canards and homoclinic orbits in a slow-fast modified May-Holling-Tanner predator-prey model with weak multiple Allee effect

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