The spatially inhomogeneous Hopf bifurcation induced by memory delay in a memory-based diffusion system

Song, Yongli; Peng, Yahong; Zhang, Tonghua

doi:10.1016/j.jde.2021.08.010

Cited by 75 publications

(23 citation statements)

References 27 publications

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“…For d 21 = 0, E * (γ, v γ ) is asymptotically stable for d 11 ≥ 0 and d 22 ≥ 0 provided that a−1 2 < γ < a. For d 21 > 0, the stability and Hopf bifurcation for system (3.1) has been detailedly investigated in [39,40]. In what follows, we are interested in the double Hopf bifurcation induced by the spatial memory diffusion coefficient d 21 and the delay τ .…”

Section: Examplesmentioning

confidence: 99%

“…For this purpose, we need to investigate the critical values of d 21 and τ at which (2.4) with (3.2) has two pairs of purely imaginary roots. In the following, we use the same notations as in [40] and simply introduce some results from [39,40] for the analysis.…”

Section: Examplesmentioning

confidence: 99%

“…we have (u * , v * ) = (1/5, 5/2). For the dynamical classification near the double Hopf bifurcation point, we introduce the results of linear analysis from [39], as shown in Fig. 1…”

Section: Dynamics Near the Double Hopf Bifurcation Point With Differe...mentioning

confidence: 99%

“…The spatially inhomogeneous Hopf bifurcation, double Hopf bifurcation and stability switches are also found in [40]. The algorithm for computing the normal form to investigate the spatially inhomogeneous Hopf bifurcation are developed in [39].…”

Section: Introductionmentioning

confidence: 99%

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Double Hopf bifurcation analysis in the memory-based diffusion system

Song¹,

Peng²,

Zhang³

2022

Preprint

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In this paper, we derive the algorithm for calculating the normal form of the double Hopf bifurcation that appears in a memory-based diffusion system via taking memory-based diffusion coefficient and the memory delay as the perturbation parameters. Using the obtained theoretical results, we study the dynamical classification near the double Hopf bifurcation point in a predator-prey system with Holling type II functional response. We show the existence of different kinds of stable spatially inhomogeneous periodic solutions, the transition from one kind to the other as well as the coexistence of two types of periodic solutions with different spatial profiles by varying the memory-based diffusion coefficient and the memory delay.

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Section: Examplesmentioning

confidence: 99%

Section: Examplesmentioning

confidence: 99%

“…we have (u * , v * ) = (1/5, 5/2). For the dynamical classification near the double Hopf bifurcation point, we introduce the results of linear analysis from [39], as shown in Fig. 1…”

Section: Dynamics Near the Double Hopf Bifurcation Point With Differe...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Double Hopf bifurcation analysis in the memory-based diffusion system

Song¹,

Peng²,

Zhang³

2022

Preprint

Self Cite

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“…At this time, there are usually two types of memory and cognition: (i) Cross-memory-based diffusion: it refers to the directional diffusion of predators (or preys), according to the memory of past predation (or escape) behavior; (ii) Self-memory diffusion: it refers to the directional diffusion of predators (or prey), according to the memory of their own behavior at some time in the past. Song [28] studied the predator-prey model with the first type of cross-memory-based diffusion. Shi & Wang [26] studied a general Lotka-Volterra reaction-diffusion model with selfmemory-based diffusion and cross-memory-based diffusion.…”

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confidence: 99%

Stability and Hopf bifurcation in a prey-predator model with memory-based diffusion

Dai

2022

DCDS-B

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<p style='text-indent:20px;'>In this paper, we consider a predator-prey model with memory-based diffusion. We first analyze the stability of all steady states in detail. Then by analyzing the distribution of eigenvalues, we find that the average memory period can cause the stability change of the positive steady state, and Hopf bifurcation occurs at the positive steady state. Moreover, from the central manifold theorem and the normal form theory, we give the direction and stability of Hopf bifurcation. The results show that, under certain conditions, a family of spatially inhomogeneous periodic solutions will bifurcate from the positive steady state when the average memory period appear.</p>

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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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The spatially inhomogeneous Hopf bifurcation induced by memory delay in a memory-based diffusion system

Cited by 75 publications

References 27 publications

Double Hopf bifurcation analysis in the memory-based diffusion system

Double Hopf bifurcation analysis in the memory-based diffusion system

Stability and Hopf bifurcation in a prey-predator model with memory-based diffusion

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

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