Spatial Nonhomogeneous Periodic Solutions Induced by Nonlocal Prey Competition in a Diffusive Predator–Prey Model

Chen, Shanshan; Wei, Junjie; Yang, Kaiqi

doi:10.1142/s0218127419500433

Cited by 13 publications

(19 citation statements)

References 33 publications

(49 reference statements)

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“…We extend the nonexistence condition for 0-mode Hopf bifurcation in Ref. 13 and establish the existence of Hopf bifurcation and Turing bifurcation, and the branching periodic orbit can be spatially nonhomogeneous when the positive equilibrium is destabilized. Subsequently, by considering the interaction of Hopf and Turing bifurcations, we further study the existence of Turing-Hopf bifurcation by taking into account of the effects of the intrinsic growth rate and the diffusion rate of the predator.…”

Section: Discussionmentioning

confidence: 86%

“…that is, 𝑘 0 takes the integer part of ( for 𝛽 ∈ (0, 1) divides (𝛽, 𝑏)-plane into two regions: B 1 and B 2 , which are defined in (13) Proof. For any 𝛽, 𝑏, 𝑙, 𝑑 2 > 0, the assumption 0 < 𝑑 1 < 𝑟𝑙 2 𝑢 * guarantees that Λ 1 is nonempty.…”

Section: Hopf Bifurcation and Double-hopf Bifurcationmentioning

confidence: 99%

“…It was shown in Ref. 13 that system (4) cannot undergo 0-mode Hopf bifurcation for 𝛽 ≥ 1, and the bifurcating periodic orbit destabilizing the positive equilibrium must be spatially nonhomogeneous. We find that the same conclusion holds true when 𝛽 < 1 by Theorems 2 and 3; see the region B 1 in Figure 1.…”

Section: Corollarymentioning

confidence: 99%

“…To this end, fix parameters 𝑙 = 20, 𝛽 = 0.2, 𝑏 = 0.3215, 𝑑 1 = 0.4, similar to Example 3.2 in Ref. 13. Then (𝑢 * , 𝑣 * ) = (3.7322, 3.7322), following the result in Subsection 2.2, we obtain 𝑘 * = 8, 𝑑 2 1 8 = 1.0610, 𝑐 1 8 = 0.1963, 𝜔 = 0.1009 with 𝜔 defined as in (49).…”

Section: Spatiotemporal Patterns Induced By Nonlocal Interactionmentioning

confidence: 99%

“…For system (1) with the global interaction, Chen et al 13 showed that the stable periodic orbit bifurcating from the positive constant steady state can be spatially nonhomogeneous. Subsequently, Cao and Jiang 14 revealed tristable phenomena of (1) from the study of the double zero singularity.…”

Section: Introductionmentioning

confidence: 99%

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Spatiotemporal patterns in a diffusive predator–prey system with nonlocal intraspecific prey competition

et al. 2021

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We investigate the effect of nonlocal intraspecific prey competition on the spatiotemporal dynamics of a Holling-Tanner predator-prey model with diffusion. We first establish the criteria for Hopf, Turing, double-Hopf, and Turing-Hopf bifurcations, and determine the stable and unstable regions of the positive equilibrium. For Turing-Hopf bifurcation, by analyzing the normal form truncated to the third order, we derive that, with strong nonlocal interaction, the system exhibits the tristable phenomena, that is, the coexistence of a stable spatially nonhomogeneous periodic orbit and two nonconstant stable steady states, as well as the existence of periodic orbits with two spatial wave frequencies induced by the nonlocal interaction. The main analytical difficulty arises from the nonlocal interaction that prevents the direct application of formulas for the coefficients of the normal form. Biologically, the emerging spatiotemporal patterns suggest that the global intraspecific competition can promote the coexistence of the prey and predator by allowing the prey maintain a critical total population size, which may provide an alternative approach in explaining the group formation of some prey species under the risk of predation. Some coexistence patterns are destabilized by introducing another prey species with local intraspecific competition, leading to the

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Section: Discussionmentioning

confidence: 86%

Section: Hopf Bifurcation and Double-hopf Bifurcationmentioning

confidence: 99%

Section: Corollarymentioning

confidence: 99%

Section: Spatiotemporal Patterns Induced By Nonlocal Interactionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Spatiotemporal patterns in a diffusive predator–prey system with nonlocal intraspecific prey competition

et al. 2021

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Normal form and Hopf bifurcation for the memory‐based reaction‐diffusion equation with nonlocal effect

An,

Gu,

Zhang

2024

Math Methods in App Sciences

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In this paper, we provide the normal form for the Hopf bifurcation of a class of the reaction‐diffusion equation with memory‐based diffusion and nonlocal effect, where the delay is present in the differential term, similar to the chemotaxis model with time delay. The eigenvalue problems and the decomposition of the phase space are discussed in detail. Through a series of variable transformations, we obtain the third‐order truncated normal form of the model constrained on the central manifold and its equivalent equation in polar coordinates. Then, with the help of the dynamic analysis for the finite dimensional equations, the key parameters for determining the direction and stability of the Hopf bifurcation are given. These theoretical results are applied to the Bazykin's model, the stability, Turing bifurcation and Hopf bifurcation of the equilibrium are demonstrated through both theoretical and numerical methods.

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Effect of Spatial Average on the Spatiotemporal Pattern Formation of Reaction-Diffusion Systems

Shi

Song

2021

J Dyn Diff Equat

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Spatial Nonhomogeneous Periodic Solutions Induced by Nonlocal Prey Competition in a Diffusive Predator–Prey Model

Cited by 13 publications

References 33 publications

Spatiotemporal patterns in a diffusive predator–prey system with nonlocal intraspecific prey competition

Spatiotemporal patterns in a diffusive predator–prey system with nonlocal intraspecific prey competition

Normal form and Hopf bifurcation for the memory‐based reaction‐diffusion equation with nonlocal effect

Effect of Spatial Average on the Spatiotemporal Pattern Formation of Reaction-Diffusion Systems

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