Two-dimensional traveling waves arising from planar front interaction in a three-species competition-diffusion system

Contento, Lorenzo; Mimura, Mamoru; Tohma, Makoto

doi:10.1007/s13160-015-0186-4

Cited by 8 publications

(23 citation statements)

References 21 publications

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“…It has been recently shown that, when three or more species are considered, dynamical coexistence is possible even in convex homogeneous environments with only random dispersal [1,11,4,3]. This is due to the effect of indirect competition between the species, under the form of the so-called cyclic competition.…”

Section: Introductionmentioning

confidence: 99%

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Ecological invasion in competition–diffusion systems when the exotic species is either very strong or very weak

2018

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Reaction-diffusion systems with a Lotka-Volterra-type reaction term, also known as competition-diffusion systems, have been used to investigate the dynamics of the competition among m ecological species for a limited resource necessary to their survival and growth. Notwithstanding their rather simple mathematical structure, such systems may display quite interesting behaviours. In particular, while for [Formula: see text] no coexistence of the two species is usually possible, if [Formula: see text] we may observe coexistence of all or a subset of the species, sensitively depending on the parameter values. Such coexistence can take the form of very complex spatio-temporal patterns and oscillations. Unfortunately, at the moment there are no known tools for a complete analytical study of such systems for [Formula: see text]. This means that establishing general criteria for the occurrence of coexistence appears to be very hard. In this paper we will instead give some criteria for the non-coexistence of species, motivated by the ecological problem of the invasion of an ecosystem by an exotic species. We will show that when the environment is very favourable to the invading species the invasion will always be successful and the native species will be driven to extinction. On the other hand, if the environment is not favourable enough, the invasion will always fail.

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Section: Introductionmentioning

confidence: 99%

“…In particular, in [11,4,3] the following ecological situation is considered. An ecosystem which is inhabited by two native species u and v which are usually unable to coexist is invaded by a third, exotic species w from outside.…”

Section: Introductionmentioning

confidence: 99%

Ecological invasion in competition–diffusion systems when the exotic species is either very strong or very weak

2018

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“…If r 3 is sufficiently large, the non-trivial wave is the faster of the two. This case and its consequences for the existence of two-dimensional travelling wave solutions have been discussed in [10]. In this paper we are concerned only with values of r 3 for which the trivial wave is faster than the non-trivial wave.…”

Section: Interaction Of Stable Travelling Waves In One Dimensionmentioning

confidence: 99%

“…and take r 3 as a free parameter [56,10]. It can be shown that, as long as r 3 ∈ (7, 29 + 1/3), the assumptions (A1-5) are satisfied.…”

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confidence: 99%

Complex pattern formation driven by the interaction of stable fronts in a competition-diffusion system

Contento

Mimura

2019

J. Math. Biol.

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The ecological invasion problem in which a weaker exotic species invades an ecosystem inhabited by two strongly competing native species is modelled by a three-species competition-diffusion system. It is known that for a certain range of parameter values competitor-mediated coexistence occurs and complex spatiotemporal patterns are observed in two spatial dimensions. In this paper we uncover the mechanism which generates such patterns. Under some assumptions on the parameters the three-species competition-diffusion system admits two planarly stable travelling waves. Their interaction in one spatial dimension may result in either reflection or merging into a single homoclinic wave, depending on the strength of the invading species. This transition can be understood by studying the bifurcation structure of the homoclinic wave. In particular, a time-periodic homoclinic wave (breathing wave) is born from a Hopf bifurcation and its unstable branch acts as a separator between the reflection and merging regimes. The same transition occurs in two spatial dimensions: the stable regular spiral associated to the homoclinic wave destabilizes, giving rise first to an oscillating breathing spiral and then breaking up producing a dynamic pattern characterized by many spiral cores. We find that these complex patterns are generated by the interaction of two planarly stable travelling waves, in contrast with many other well known cases of pattern formation where planar instability plays a central role.

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“…In particular, this system is often considered as a paradigm for biodiversity modelling. New types of patterns have been recently demonstrated in this system which occur for spatially homogeneous diffusion coefficients [2,31,14]. This includes, for example, patchy invasion (the spread of a species via the formation and propagation of chaotic patches without a smooth population front), which was originally believed to occur only in predator-prey or inhibitor-activator types models [39].…”

Section: Introductionmentioning

confidence: 99%

Revealing new dynamical patterns in a reaction–diffusion model with cyclic competition via a novel computational framework

et al. 2018

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Understanding how patterns and travelling waves form in chemical and biological reaction-diffusion models is an area which has been widely researched, yet is still experiencing fast development. Surprisingly enough, we still do not have a clear understanding about all possible types of dynamical regimes in classical reaction-diffusion models, such as Lotka-Volterra competition models with spatial dependence. In this study, we demonstrate some new types of wave propagation and pattern formation in a classical three species cyclic competition model with spatial diffusion, which have been so far missed in the literature. These new patterns are characterized by a high regularity in space, but are different from patterns previously known to exist in reaction-diffusion models, and may have important applications in improving our understanding of biological pattern formation and invasion theory. Finding these new patterns is made technically possible by using an automatic adaptive finite element method driven by a novel error estimate which is proved to provide a reliable bound for the error of the numerical method. We demonstrate how this numerical framework allows us to easily explore the dynamical patterns in both two and three spatial dimensions.

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Two-dimensional traveling waves arising from planar front interaction in a three-species competition-diffusion system

Cited by 8 publications

References 21 publications

Ecological invasion in competition–diffusion systems when the exotic species is either very strong or very weak

Ecological invasion in competition–diffusion systems when the exotic species is either very strong or very weak

Complex pattern formation driven by the interaction of stable fronts in a competition-diffusion system

Revealing new dynamical patterns in a reaction–diffusion model with cyclic competition via a novel computational framework

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