A predator-prey system with generalized Holling type IV functional response and Allee effects in prey

In this paper, we propose a two-patch SIRS model with a nonlinear incidence rate: and nonconstant dispersal rates, where the dispersal rates of susceptible and recovered individuals depend on the relative disease prevalence in two patches. In an isolated environment, the model admits Bogdanov–Takens bifurcation of codimension 3 (cusp case) and Hopf bifurcation of codimension up to 2 as the parameters vary, and exhibits rich dynamics such as multiple coexistent steady states and periodic orbits, homoclinic orbits and multitype bistability. The long-term dynamics can be classified in terms of the infection rates (due to single contact) and (due to double exposures). In a connected environment, we establish a threshold between disease extinction and uniform persistence under certain conditions. We numerically explore the effect of population dispersal on disease spread when and patch 1 has a lower infection rate, our results indicate: (i) can be nonmonotonic in dispersal rates and ( is the basic reproduction number of patch i ) may fail; (ii) the constant dispersal of susceptible individuals (or infective individuals) between two patches (or from patch 2 to patch 1) will increase (or reduce) the overall disease prevalence; (iii) the relative prevalence-based dispersal may reduce the overall disease prevalence. When and the disease outbreaks periodically in each isolated patch, we find that: (a) small unidirectional and constant dispersal can lead to complex periodic patterns like relaxation oscillations or mixed-mode oscillations, whereas large ones can make the disease go extinct in one patch and persist in the form of a positive steady state or a periodic solution in the other patch; (b) relative prevalence-based and unidirectional dispersal can make periodic outbreak earlier.

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“…First, let , system ( 2.18 ) becomes where Next, following similar steps as in Arsie et al. ( 2022 ), Li et al. ( 2015 ) and Xiang et al.…”

Section: Dynamics In Isolated Environmentmentioning

confidence: 99%

“…Next, following similar steps as in Arsie et al (2022), Li et al (2015) and Xiang et al (2019) and performing a sequence of near-identity transformations and time rescaling (preserving orientations of orbits), we can reduce system (2.19) to the following form:…”

Section: Degenerate Bogdanov-takens Bifurcation Of Codimensionmentioning

confidence: 99%

Relative prevalence-based dispersal in an epidemic patch model

et al. 2023

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“…In recent decades, the Allee efect in the discrete-time predator-prey model [22][23][24][25][26][27][28] has developed quickly. Celik and Duman [29] studied the stability of the positive equilibrium point for the discrete predator-prey model without the Allee efect:…”

Section: Introductionmentioning

confidence: 99%

Bifurcation and Chaos of a Nonlinear Discrete-Time Predator-Prey Model Involving the Nonlinear Allee Effect

Song

2023

Discrete Dynamics in Nature and Society

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The complex dynamics of a nonlinear discretized predator-prey model with the nonlinear Allee effect in prey and both populations are investigated. First, the rigorous results are derived from the existence and stability of the fixed points of the model. Second, we establish a model with the Allee effect in prey undergoing codimension-one bifurcations (flip bifurcation and Neimark–Sacker bifurcation) and codimension-two bifurcation associated with 1 : 2 strong resonance by using center manifold theorem and bifurcation theory, and the direction of bifurcations is also evaluated. In particular, chaos in the sense of Marotto is proved at some certain conditions. Third, numerical simulations are performed to illustrate the effectiveness of the theoretical results and other complex dynamical behaviors, such as the period-3, 4, 6, 8, 9, 30, and 43 orbits, attracting invariant cycles, coexisting chaotic sets, and so forth. Of most interest is the finding of coexisting attractors and multistability. Moreover, a moderate Allee effect in predators can stabilize the dynamical behavior. Finally, the hybrid feedback control strategy is implemented to stabilize chaotic orbits existing in the model.

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“…It was showed that the model undergoes nilpotent cusp bifurcation, nilpotent saddle bifurcation, and degenerate Hopf bifurcation of codimension 3 and exhibits homoclinic and heteroclinic loops and three limit cycles. They also considered the model in Arsie et al [26]; the difference is that a new unfolding of nilpotent saddle of codimension 3 with a fixed invariant line is discovered and fully developed, and the existence of codimension 2 heteroclinic bifurcation is proven. Nevertheless, in González-Olivares et al [27], the number of limit cycles diminishes to only one under the Allee effect.…”

Section: Introductionmentioning

confidence: 99%

“…It was shown that the existence and stability of the equilibria are changed due to the Allee effect, and the system could go through transcritical bifurcation, saddle‐node bifurcation, Hopf bifurcation, cusp bifurcation, Bogdanov–Takens bifurcation, and homoclinic bifurcation. Recently, Arsie et al [25] considered a predator–prey model with Holling type IV functional response and Allee effect on prey. It was showed that the model undergoes nilpotent cusp bifurcation, nilpotent saddle bifurcation, and degenerate Hopf bifurcation of codimension 3 and exhibits homoclinic and heteroclinic loops and three limit cycles.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation analysis in a predator–prey model with strong Allee effect on prey and density‐dependent mortality of predator

Shang,

Qiao

2023

Math Methods in App Sciences

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The influences of Allee effect and density‐dependent mortality on population growth are of great significance in ecology. In this paper, we first consider a Gause‐type predator–prey model with simplified Holling type IV functional response, strong Allee effect on prey, and density‐dependent mortality of predator. It is shown that the system exhibits rich and complex dynamics like bistability, local and global bifurcations such as transcritical bifurcation, saddle‐node bifurcation, Hopf bifurcation, cusp bifurcation, Bogdanov–Takens bifurcation, Bautin bifurcation, homoclinic bifurcation, saddle‐node bifurcation of limit cycle, and heteroclinic bifurcation. Next, we separately explore the influences of Allee effect and density‐dependent mortality on the dynamics of the same model. The results show that the strong Allee effect induces the occurrence of heteroclinic bifurcation and the reduction of the number of limit cycles, while the density‐dependent mortality stabilizes the system. Since the analytical expressions of the interior equilibria are difficult to derive, the results are verified numerically.

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A predator-prey system with generalized Holling type IV functional response and Allee effects in prey

Cited by 41 publications

References 31 publications

Relative prevalence-based dispersal in an epidemic patch model

Relative prevalence-based dispersal in an epidemic patch model

Bifurcation and Chaos of a Nonlinear Discrete-Time Predator-Prey Model Involving the Nonlinear Allee Effect

Bifurcation analysis in a predator–prey model with strong Allee effect on prey and density‐dependent mortality of predator

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