Cross-diffusion induced Turing instability for a three species food chain model

Xie, Zhifu

doi:10.1016/j.jmaa.2011.10.054

Cited by 47 publications

(22 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In developmental biology, recent experimental findings demonstrate that cross-diffusion can be quite significant in generating spatial structure [10]. The effects of cross-diffusion on models for pattern formation (i.e., reaction-diffusion type) have been studied in many theoretical papers [13][14][15][16][17][18][19][20][21][22][23][24]. Recently, in [25] we showed that introducing cross-diffusion to a system of reaction-diffusion equations results in further relaxation of the conditions necessary for the emergency of patterns.…”

Section: Introductionmentioning

confidence: 99%

Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces

Madzvamuse

Barreira

2014

Phys. Rev. E

View full text Add to dashboard Cite

Article (Published Version) http://sro.sussex.ac.uk Madzvamuse, A and Barreira, R (2014) Exhibiting cross-diffusion-induced patterns for reactiondiffusion systems on evolving domains and surfaces. Physical Review E, 90. 043307-1-043307-14. ISSN 1539-3755 This version is available from Sussex Research Online: http://sro.sussex.ac.uk/51334/ This document is made available in accordance with publisher policies and may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the URL above for details on accessing the published version. Copyright and reuse:Sussex Research Online is a digital repository of the research output of the University.Copyright and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable, the material made available in SRO has been checked for eligibility before being made available.Copies of full text items generally can be reproduced, displayed or performed and given to third parties in any format or medium for personal research or study, educational, or not-for-profit purposes without prior permission or charge, provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way.PHYSICAL REVIEW E 90, 043307 (2014) Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces The aim of this manuscript is to present for the first time the application of the finite element method for solving reaction-diffusion systems with cross-diffusion on continuously evolving domains and surfaces. Furthermore we present pattern formation generated by the reaction-diffusion system with cross-diffusion on evolving domains and surfaces. A two-component reaction-diffusion system with linear cross-diffusion in both u and v is presented. The finite element method is based on the approximation of the domain or surface by a triangulated domain or surface consisting of a union of triangles. For surfaces, the vertices of the triangulation lie on the continuous surface. A finite element space of functions is then defined by taking the continuous functions which are linear affine on each simplex of the triangulated domain or surface. To demonstrate the role of cross-diffusion to the theory of pattern formation, we compute patterns with model kinetic parameter values that belong only to the cross-diffusion parameter space; these do not belong to the standard parameter space for classical reaction-diffusion systems. Numerical results exhibited show the robustness, flexibility, versatility, and generality of our methodology; the methodology can deal with complicated evolution laws of the domain and surface, and these include uniform isotropic and anisotropic growth profiles as well as those profiles driven by chemical concentrations residing in...

show abstract

Section: Introductionmentioning

confidence: 99%

Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces

Madzvamuse

Barreira

2014

Phys. Rev. E

View full text Add to dashboard Cite

show abstract

“…Upon introducing the delay (τ = 10.1), in (35) we found that the system does not exhibit blow-up. The chosen parameter set for all of our simulations is ω 4 = 0.210; ω 5 = 1.290; ω 6 = 2.930; ω 7 = 0.540; ω 8 = 0.981; ω 9 = 0.100; ω 10 = 0.445; ω 11 = 0.200; ω 12 = 0.280.…”

Section: Effect Of Time Delaymentioning

confidence: 90%

“…This behavior is called Diffusion driven instability. Models that exhibits this behavior in 2 and 3 species have been extensively studied in [59,60,35], were several different patterns was observed. In order for homogeneous steady state (u * , v * , r * ) to be stable (in the absence of diffusion) we need…”

Section: Turing Instabilitymentioning

confidence: 97%

Predator interference effects on biological control: The “paradox” of the generalist predator revisited

Parshad

Bhowmick

Quansah

et al. 2016

Communications in Nonlinear Science and Numerical Simulation

View full text Add to dashboard Cite

Abstract. An interesting conundrum in biological control questions the efficiency of generalist predators as biological control agents. Theory suggests, generalist predators are poor agents for biological control, primarily due to mutual interference. However field evidence shows they are actually quite effective in regulating pest densities. In this work we provide a plausible answer to this paradox. We analyze a three species model, where a generalist top predator is introduced into an ecosystem as a biological control, to check the population of a middle predator, that in turn is depredating on a prey species. We show that the inclusion of predator interference alone, can cause the solution of the top predator equation to blow-up in finite time, while there is global existence in the no interference case. This result shows that interference could actually cause a population explosion of the top predator, enabling it to control the target species, thus corroborating recent field evidence. Our results might also partially explain the population explosion of certain species, introduced originally for biological control purposes, such as the cane toad (Bufo marinus) in Australia, which now functions as a generalist top predator. We also show both Turing instability and spatio-temporal chaos in the model. Lastly we investigate time delay effects.

show abstract

“…In this paper, motivated by the recent papers [9,[31][32][33][34], we will investigate a class of three-species predator-prey cross-diffusion systems with generalist predator. The aim of this paper is to construct a mathematical model to describe ecosystems with mutualist interaction and generalist predator, and further investigate Turing instability of the predator-prey ecosystem by virtue of mathematical analysis and numerical examples.…”

Section: Introductionmentioning

confidence: 99%

Persistence and Turing instability in a cross-diffusive predator–prey system with generalist predator

Bai-qi

2018

Adv Differ Equ

View full text Add to dashboard Cite

In this paper, we propose and investigate persistence and Turing instability of a cross-diffusion predator-prey system with generalist predator. First, by virtue of the comparison principle, we obtain sufficient conditions of persistence for a corresponding reaction-diffusion system without self-and cross-diffusion. Second, by using the linear stability analysis, we prove that under some conditions the unique positive equilibrium solution is locally asymptotically stable for the corresponding ODE system and the corresponding reaction-diffusion system without cross-diffusion and self-diffusion. Hence it does not belong to the classical Turing instability. Third, under some appropriate sufficient conditions, we obtain that the uniform positive equilibrium solution is linearly unstable for the cross-reaction-diffusion and partial self-diffusion system. The results indicate that cross-diffusion and partial self-diffusion play an important role in the study of Turing instability about reaction-diffusion systems with generalist predator. Fourth, we elaborate on the relations between the theoretical results and the cross-diffusion predator-prey system by relying on some examples. In the end, we conclude our findings and give a brief discussion. MSC: 35B36; 35B51

show abstract

Cross-diffusion induced Turing instability for a three species food chain model

Cited by 47 publications

References 23 publications

Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces

Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces

Predator interference effects on biological control: The “paradox” of the generalist predator revisited

Persistence and Turing instability in a cross-diffusive predator–prey system with generalist predator

Contact Info

Product

Resources

About