Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis

Wang, Jinfeng; Wu, Sainan; Shi, Junping

doi:10.3934/dcdsb.2020162

Cited by 28 publications

(36 citation statements)

References 40 publications

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“…The outcome shows that the addition of attractive prey-taxi does destroy the stability of constant steady states and induce the occurrence of spatial pattern, see Theorem 3.2. Contrast to the results, the prey-taxis induced instability can not occur for two species predator-prey systems as shown in [24,21], where prey-taxis may annihilate the spatial patterns. Some numerical simulations of (1) can be used to verify our analysis.…”

contrasting

confidence: 81%

“…To study the effect of directed motility of mature predators, we discuss the stability/instability change of the positive steady state in terms of χ. We demonstrate the emergence of stationary patterns for strong attractive prey-taxis (χ > 0), this is totally different from two-species predator-prey systems with prey-taxis [21]. From the biological point of view, this in particular means that it is possible to control all species entirely in the habitat by restricting the directed motility of predators to be small.…”

Section: Introductionmentioning

confidence: 77%

“…It is worth noting that there have been extensive works on the existence of globally bounded classical solutions for two-species predator-prey systems with prey-taxis, see e.g., χ sufficiently small and two space dimensions [18,12,31]; while in the three-dimensional setting, one may overcome this restriction by either assuming the prey-taxis to be saturated at larger predator quantities [18,9] or by considering weak solutions instead [29]. It can find that prey-taxis has extremely important significance on the dynamic behavior: attractive prey-taxis has stabilization effect, while repulsive prey-taxis can generate pattern formation, see, for example [24,21]. For three-species predator-prey systems with prey-taxis, the dynamics becomes more complicated, which strongly depends on the interaction between predators and prey [8,23,22].…”

Section: Introductionmentioning

confidence: 99%

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Dynamics and pattern formation in a cross-diffusion model with stage structure for predators

Wang

2022

DCDS-B

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<p style='text-indent:20px;'>This paper is concerned with a predator-prey model with stage structure for the predator, with a cross-diffusion term modeling the effect that mature predators move toward the direction of gradient of prey. It is first shown that the corresponding Neumann initial-boundary value problem in an <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional bounded smooth domain possesses a unique global classical solution which is uniformly-in-time bounded for the weak cross-diffusion. It is further shown that, in the presence of cross-diffusion, the model admits threshold-type dynamics in terms of the cross-diffusion coefficient; that is, the homogenous steady state keeps stability for weak attractive prey-taxis, while the stationary patterns will occur for strong attractive prey-taxis. This implies that such cross diffusion does contribute to the rich dynamics of predator-prey model with stage structure for predators.</p>

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contrasting

confidence: 81%

Section: Introductionmentioning

confidence: 77%

Section: Introductionmentioning

confidence: 99%

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Dynamics and pattern formation in a cross-diffusion model with stage structure for predators

Wang

2022

DCDS-B

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“…That is to say, although the same parameters are fixed in the Fig. 5, the Turing pattern will not appear in the random network system (2). And, the right picture of Fig.…”

Section: Numerical Simulationsmentioning

confidence: 91%

“…Fig 4. There is no Turing instability of the random network system(2). Here parameters are c = 2.3, β = 1, d 1 = 0.5, d 2 = 5 and p = 0.02.…”

mentioning

confidence: 99%

Spatiotemporal Patterns in a General Networked Activator-Substrate Model

Chen

Zheng

et al. 2021

Preprint

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For the purpose of understanding the spatiotemporal pattern formation in the random networked system, a general activator-substrate model with network structure is introduced. Firstly, we investigate the boundedness of the non-constant steady state of the elliptic system of the continuous media system. It is found that the non-constant steady state admits their upper and lower bounds with certain conditions. Then, one investigates some properties and non-existence of the non-constant steady state with the no-flux boundary conditions. The main results show that the diffusion rate of activator should greater than the diffusion rate of substrate. Otherwise, there might be no pattern formation of the system. Afterwards, a general random networked activator-substrate model is made public. The conditions of the stability, the Hopf bifurcation, the Turing instability and a co-dimensional-two Turing-Hopf bifurcation are yield by the method of stability analysis and bifurcation theorem. Finally, we choose a suitable sub-system of the general activator-substrate model to verify the theoretical results, and full numerical simulations are well verified these results. Especially, an interesting finding is that the stability of the positive equilibrium will switch from unstable to stable one with the change of the connection probability of the nodes, this is different from the pattern formation in the continuous media systems.

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Stationary and oscillatory patterns of a food chain model with diffusion and predator‐taxis

Han

Röst

2023

Math Methods in App Sciences

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In this paper, we investigate pattern dynamics in a reaction-diffusionchemotaxis food chain model with predator-taxis, which extends previous studies of reaction-diffusion food chain model. By virtue of diffusion semigroup theory, we first prove global classical solvability and boundedness for the considered model over a bounded domain Ω ⊂ R n (n ≥ 1) with smooth boundary for arbitrary predator-taxis sensitivity coefficient. Then the linear stability analysis for the considered model shows that chemotaxis can induce the losing of stability of the unique positive spatially homogeneous steady state via Turing bifurcation and Turing-spatiotemporal Hopf bifurcation. These bifurcations results in the formation of two kinds of important spatiotemporal patterns: stationary Turing pattern and oscillatory pattern. Simultaneously, the threshold values for Turing bifurcation and Turing-spatiotemporal Hopf bifurcation are given explicitly. Finally, numerical simulations are performed to illustrate and support our theoretical findings, and some interesting non-Turing patterns are found in temporal Hopf parameter space by numerical simulation.

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Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis

Cited by 28 publications

References 40 publications

Dynamics and pattern formation in a cross-diffusion model with stage structure for predators

Dynamics and pattern formation in a cross-diffusion model with stage structure for predators

Spatiotemporal Patterns in a General Networked Activator-Substrate Model

Stationary and oscillatory patterns of a food chain model with diffusion and predator‐taxis

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