Competition in periodic media:Ⅰ-Existence of pulsating fronts

Mathematical Biosciences

2019

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Does a high dispersal rate provide a competitive advantage when risking competitive exclusion? To this day, the theoretical literature cannot answer this question in full generality. The present paper focuses on the simplest mathematical model with two populations differing only in dispersal abilities and whose one-dimensional territories are spatially segregated. Although the motion of the border between the two territories remains elusive in general, all cases investigated in the literature concur: either the border does not move at all because of some environmental heterogeneity or the fast diffuser chases the slow diffuser. Counterintuitively, it is better to randomly explore the hostile enemy territory, even if it means certain death of some individuals, than to "stay united". This directly contradicts a celebrated result on the intermediate competition case, emphasizing the importance of the competition intensity. Overall, the larger picture remains unclear and the optimal strategy regarding dispersal remains ambiguous. Several open problems worthy of a special attention are raised.

Section: 22mentioning

confidence: 99%

“…Extension to more general heterogeneous environments. Extending the spatially periodic "Unity is not strength"-type results of Girardin et al [29,31,32] in two-or three-dimensional environments (periodic tilings) should be possible, to a certain degree at least, but difficult mathematical obstacles will arise.…”

Section: 2mentioning

confidence: 99%

The effect of random dispersal on competitive exclusion – A review

Mathematical Biosciences

2019

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“…This is assumed indeed from now on. Notice also that, although all results of [16,17] are stated for ω = 1, they are readily extended to the case of non-constant ω.…”

Section: Introductionmentioning

confidence: 99%

“…Recall that if (u 1 , u 2 ) is a solution of (1.3), then the system satisfied by (u 1 , 1 − u 2 ) is a monotone system, whence its linearization admits indeed a periodic principal eigenvalue (details can be found in [16]). Hereafter, a solution z ∈ H 2 L-per (R) of (1.4) such that the L-periodic function The definition of linear stability in the sense of (1.2) can be formally understood by plugging perturbations of the form e −λt ϕ(x), with ϕ L-periodic, into the equation (1.2) linearized at an almost everywhere nonzero steady state z.…”

Section: Introductionmentioning

confidence: 99%

Competition in Periodic Media: III—Existence and Stability of Segregated Periodic Coexistence States

Zilio

2019

J Dyn Diff Equat

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In this paper we consider a system of parabolic reaction-diffusion equations with strong competition and two related scalar reaction-diffusion equations. We show that in certain space periodic media with large periods, there exist periodic, non-constant, non-trivial, stable stationary states. We compare our results with already known results about the existence and nonexistence of such solutions. Finally, we provide ecological interpretations for these results.Here L, a 1 , a 2 , α and d are positive constants, µ 1 , µ 2 ∈ L ∞ (R, (0, +∞)) are positive L-periodic functions, z + = max (z, 0) and z − = − min (z, 0) (so that z = z + − z − ).

“…In the first part [23] of this sequel, the first author studied the existence of bistable pulsating front solutions for the following problem:…”

Section: Introductionmentioning

confidence: 99%

Competition in periodic media: II – Segregative limit of pulsating fronts and “Unity is not Strength”-type result

Journal of Differential Equations

Nadin

2018

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This paper is concerned with the limit, as the interspecific competition rate goes to infinity, of pulsating front solutions in space-periodic media for a bistable two-species competition-diffusion Lotka-Volterra system. We distinguish two important cases: null asymptotic speed and non-null asymptotic speed. In the former case, we show the existence of a segregated stationary equilibrium. In the latter case, we are able to uniquely characterize the segregated pulsating front, and thus full convergence is proved. The segregated pulsating front solves an interesting free boundary problem. We also investigate the sign of the speed as a function of the parameters of the competitive system. We are able to determine it in full generality, with explicit conditions depending on the various parameters of the problem. In particular, if one species is sufficiently more motile or competitive than the other, then it is the invader. This is an extension of our previous work in space-homogeneous media.2000 Mathematics Subject Classification. 35B40, 35K57, 35R35, 92D25.