Pattern dynamics of a predator–prey reaction–diffusion model with spatiotemporal delay

Xu, Jian; Yang, Gaoxiang; Xi, Hongguang; Su, Jianzhong

doi:10.1007/s11071-015-2132-z

Cited by 29 publications

(10 citation statements)

References 23 publications

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“…On the other hand, competition between predators alone may allow pattern formation in predator-prey systems, which includes ratiodependent functional response, the Beddington-DeAngelis functional response, and their generalizations [3,33,31]. Pattern formation of predator-prey models with time delay in the functional response due to handing time of the predator is also studied [47,44,9].…”

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confidence: 99%

Pattern formation of a predator-prey model with the cost of anti-predator behaviors

Wang

Zou

2018

Mathematical Biosciences and Engineering

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We propose and analyse a reaction-diffusion-advection predatorprey model in which we assume that predators move randomly but prey avoid predation by perceiving a repulsion along predator density gradient. Based on recent experimental evidence that anti-predator behaviors alone lead to a 40% reduction on prey reproduction rate, we also incorporate the cost of antipredator responses into the local reaction terms in the model. Sufficient and necessary conditions of spatial pattern formation are obtained for various functional responses between prey and predators. By mathematical and numerical analyses, we find that small prey sensitivity to predation risk may lead to pattern formation if the Holling type II functional response or the Beddington-DeAngelis functional response is adopted while large cost of anti-predator behaviors homogenises the system by excluding pattern formation. However, the ratio-dependent functional response gives an opposite result where large predator-taxis may lead to pattern formation but small cost of anti-predator behaviors inhibits the emergence of spatial heterogeneous solutions.2010 Mathematics Subject Classification. 92D25, 34C23, 92D40. 775 776 XIAOYING WANG AND XINGFU ZOU A PREDATOR-PREY MODEL WITH FEAR EFFECT 777addition to random movement. Ideally, the avoidance behavior of prey leads to a repulsion of prey to lower gradient of predator density. Therefore, the flux of prey isand the flux of predators iswhere γ(u, v) ≥ 0 represents the repulsion effect of the predator-taxis. Hence, a general reaction-diffusion-advection model with avoidance behaviors of prey is

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confidence: 99%

Pattern formation of a predator-prey model with the cost of anti-predator behaviors

Wang

Zou

2018

Mathematical Biosciences and Engineering

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“…Recently, the pattern formation with three or four variables has been investigated, and it obtains promising results [19,20], and Xu et al made a concrete analysis with three variables [21]. As we all know that amplitude equation is a promising tool to investigate the pattern dynamics of the reaction-diffusion system [2,22], however, the amplitude equation is a complex process [3], and the researcher often chose the amplitude equation [23][24][25][26] to investigate the reaction-diffusion system.…”

Section: Introductionmentioning

confidence: 99%

Turing Instability and Amplitude Equation of Reaction-Diffusion System with Multivariable

Zheng

Shen

2020

Mathematical Problems in Engineering

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In this paper, we investigate pattern dynamics with multivariable by using the method of matrix analysis and obtain a condition under which the system loses stability and Turing bifurcation occurs. In addition, we also derive the amplitude equation with multivariable. This is an effective tool to investigate multivariate pattern dynamics. The example and simulation used in this paper validate our theoretical results. The method presented is a novel approach to the investigation of specific real systems based on the model developed in this paper.

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“…The research on nonlinear mechanisms of pattern formation is of great significance for revealing the fundamental laws of natural phenomena. Recently, many researchers have made a great deal of efforts in understanding the pattern self-organization in predator-prey systems [1][2][3][4][5][6][7]. Since the predator-prey systems exist in multi-level and multi-scale in ecosystems, the predator-prey pattern formation reflects the spatiotemporal complexity of ecosystems [8][9][10] and deserves investigation.…”

Section: Introductionmentioning

confidence: 99%

“…In studying the pattern formation of the predator-prey systems, the reaction-diffusion model and Turing instability theory have been widely employed and lots of interesting results have been obtained. For example, Xu et al investigated the spatiotemporal dynamics of a reaction-diffusion predator-prey system with time delay, determining the effects of spatiotemporal delay and predator rate on the spatiotemporal distributions of the species [2]. Brigatti et al explored the classical Lotka-Volterra model, which can control the spatial scale of predator-prey interaction range and found the pattern formation of ecosystems via determining the instability which is driven by the range of interaction [3].…”

Section: Introductionmentioning

confidence: 99%

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Predator–prey pattern formation driven by population diffusion based on Moore neighborhood structure

Huang

Zhang

et al. 2019

Adv Differ Equ

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Diffusion-driven instability is a basic nonlinear mechanism for pattern formation. Therefore, the way of population diffusion may play a determinative role in the spatiotemporal dynamics of biological systems. In this research, we launch an investigation on the pattern formation of a discrete predator-prey system where the population diffusion is based on the Moore neighborhood structure instead of the von Neumann neighborhood structure widely applied previously. Under pattern formation conditions which are determined by Turing instability analysis, numerical simulations are performed to reveal the spatiotemporal complexity of the system. A pure Turing instability can induce the self-organization of many basic types of patterns as described in the literature, as well as new spiral-spot and labyrinth patterns which show the temporally oscillating and chaotic property. Neimark-Sacker-Turing and flip-Turing instability can lead to the formation of circle, spiral and much more complex patterns, which are self-organized via spatial symmetry breaking on the states that are homogeneous in space and non-periodic in time. Especially, the emergence of spiral pattern suggests that spatial order can generate from temporal disorder, implying that even when the predator-prey dynamics in one site is chaotic, the spatially global dynamics may still be predictable. The results obtained in this research suggest that when the way of population diffusion changes, the pattern formation in the predator-prey systems demonstrates great differences. This may provide realistic significance to explain more general predator-prey coexistence.

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Pattern dynamics of a predator–prey reaction–diffusion model with spatiotemporal delay

Cited by 29 publications

References 23 publications

Pattern formation of a predator-prey model with the cost of anti-predator behaviors

Pattern formation of a predator-prey model with the cost of anti-predator behaviors

Turing Instability and Amplitude Equation of Reaction-Diffusion System with Multivariable

Predator–prey pattern formation driven by population diffusion based on Moore neighborhood structure

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