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

Shi, Qingyan; Shi, Junping; Song, Yongli

doi:10.1007/s10884-021-09995-z

Cited by 19 publications

(9 citation statements)

References 48 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…( 52) and Lemma 4.1 respectively, and ℓ n is defined as in Eq. (53). Then the following two statements are true.…”

Section: Ifmentioning

confidence: 88%

“…Furter and Grinfeld [22] obtained that this average kernel can induce spatial nonhomogeneous patterns even for single population model, see [53] for more general models.…”

mentioning

confidence: 95%

“…However, for reaction-diffusion system without nonlocal effect, the bifurcating periodic solutions near the threshold are always spatially homogeneous, while nonhomogeneous periodic solutions can also occur through Hopf bifurcation, but they are always unstable, and consequently hard to simulate. Note that, nonlocal effect could also induced spatially nonhomogeneous steady states through steady state bifurcation [22,53]. Therefore, nonlocal effect could be used as a new mechanism for pattern formation.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Double Hopf bifurcation in nonlocal reaction-diffusion systems with spatial average kernel

Shen¹,

Chen²,

Wei³

2023

DCDS-B

View full text Add to dashboard Cite

<p style='text-indent:20px;'>In this paper, we consider a general reaction-diffusion system with nonlocal effects and Neumann boundary conditions, where a spatial average kernel is chosen to be the nonlocal kernel. By virtue of the center manifold reduction technique and normal form theory, we present a new algorithm for computing normal forms associated with the codimension-two double Hopf bifurcation. The theoretical results are applied to a predator-prey model, and complex dynamic behaviors such as spatially nonhomogeneous periodic oscillations and spatially nonhomogeneous quasi-periodic oscillations could occur.</p>

show abstract

“…( 52) and Lemma 4.1 respectively, and ℓ n is defined as in Eq. (53). Then the following two statements are true.…”

Section: Ifmentioning

confidence: 88%

“…Furter and Grinfeld [22] obtained that this average kernel can induce spatial nonhomogeneous patterns even for single population model, see [53] for more general models.…”

mentioning

confidence: 95%

mentioning

confidence: 99%

See 1 more Smart Citation

Double Hopf bifurcation in nonlocal reaction-diffusion systems with spatial average kernel

Shen¹,

Chen²,

Wei³

2023

DCDS-B

View full text Add to dashboard Cite

show abstract

“…Similar result on the Hopf bifurcation for Rosenzweig-MacArthur predator-prey model can be found in Chen and Yu 15 . See also the recent work of Shi et al 16 on the existence of Turing and Hopf bifurcations for a general two-species system, where they demonstrated that the spatial average intraspecific competition can induce the spatially nonhomogeneous periodic patterns. We also refer to Wu and Song 17 and Yi et al 18 for related works on the Rosenzweig-MacArthur predator-prey model.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2021

View full text Add to dashboard Cite

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

show abstract

“…In recent years, predator-prey model has always been a core topic of concerns in ecology and biomathematics [10,15,16,24,40]. In the real world, besides random movement [9,11,22,31], the spatial movements of predator and prey may actually be more directional, for example, predator chases prey (prey-taxis) [8,17,25,34], or prey avoids predator (predator-taxis) [35,36,39]. The existing researches about the directed movements of predator and prey, have also provided new insights into the emergence of spatial non-uniform distribution of species [2,5,32].…”

Section: Introductionmentioning

confidence: 99%

Multi-stable spatial patterns and spatiotemporal staggered periodic patterns in a predator-taxis model with delay

Xing

Jiang

Cao

2022

Preprint

View full text Add to dashboard Cite

The effects of predator-taxis and conversion time delay on formations of spatiotemporal patterns in a predator-prey model are explored. Firstly, the well-posedness, which implies global existence of classical solutions, is proved.Then, we establish critical conditions for the destabilization of coexistence equilibrium through Turing/Turing-Turing bifurcations via describing the first Turing bifurcation curve, and theoretically predict possible bi-stable/multi-stable spatially heterogeneous patterns. Next, we demonstrate that coexistence equilibrium can also be destabilized through Hopf, Hopf-Hopf, Turing-Hopf bifurcations, and possible stable/bi-stable spatially inhomogeneous staggered periodic patterns, bi-stable spatially inhomogeneous synchronous periodic patterns, are theoretically predicted.Finally, numerical experiments also support theoretical predictions and partially extend them. In a word, theoretical analyses indicate that, on the one hand, large predator-taxis can eliminate spatial patterns caused by self-diffusion; on the other hand, the joint effects of predator-taxis and conversion time delay can induce complex survival patterns, e.g., bi-stable spatially heterogeneous staggered/synchronous periodic patterns, thus diversify populations' survival patterns. Mathematics Subject Classification (2020) 35B32 · 35K57 · 92C15

show abstract

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

Cited by 19 publications

References 48 publications

Double Hopf bifurcation in nonlocal reaction-diffusion systems with spatial average kernel

Double Hopf bifurcation in nonlocal reaction-diffusion systems with spatial average kernel

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

Multi-stable spatial patterns and spatiotemporal staggered periodic patterns in a predator-taxis model with delay

Contact Info

Product

Resources

About