Spreading speeds as slowest wave speeds for cooperative systems

Li, Bingtuan; Weinberger, Hans F.; Lewis, Mark A.

doi:10.1016/j.mbs.2005.03.008

Cited by 259 publications

(246 citation statements)

References 10 publications

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“…We note that the existence of a comparison principle for (1.1) can lead to a stronger result than that of Theorem 4. For example, the authors in [26] show that the selected front for this system is always the slowest monotone front. Here, we emphasize the marginal stability criterion, as this criterion applies to a larger set of examples, see the discussion at the end of the paper.…”

Section: Nonlinear Stabilitymentioning

confidence: 99%

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A slow pushed front in a Lotka–Volterra competition model

Holzer

Scheel

2012

Nonlinearity

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We study invasion speeds in the Lotka-Volterra competition model when the rate of diffusion of one species is small. Our main result is the construction of the selected front and a rigorous asymptotic approximation of its propagation speed, valid to second order. We use techniques from geometric singular perturbation theory and geometric desingularization. The main challenge arises from the slow passage through a saddle-node bifurcation. From a perspective of linear versus nonlinear speed selection, this front provides an interesting example as the propagation speed is slower than the linear spreading speed. However, our front shares many characteristics with pushed fronts that arise when the influence of nonlinearity leads to faster than linear speeds of propagation. We show that this is a result of the linear spreading speed arising as a simple pole of the resolvent instead of as a branch pole. Using the pointwise Green's function, we show that this pole poses no a priori obstacle to marginal stability of the nonlinear travelling front, thus explaining how nonlinear systems can exhibit slower spreading that their linearization in a robust fashion.

show abstract

Section: Nonlinear Stabilitymentioning

confidence: 99%

“…In order to appeal to the results of [26], we must show that the front we constructed in Theorem 1 is the slowest monotone front. We sketch the argument.…”

Section: Nonlinear Stabilitymentioning

confidence: 99%

A slow pushed front in a Lotka–Volterra competition model

Holzer

Scheel

2012

Nonlinearity

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“…This result, which generalises Proposition 2. In fact, it can be shown, similarly to [11], that c 0 r is actually the slowest spreading speed for the system (3.1) with non-increasing initial data u(0, x) = u 0 (x) such that u 0 (−∞) = β, u 0 (+∞) = 0, in the sense of slowest spreading speed defined in [11, (2.4 Assume thatμ r is finite,…”

Section: Proof Part (I) Is Immediate From the Definitions Ofcmentioning

confidence: 77%

Lack of symmetry in linear determinacy due to convective effects in reaction-diffusion-convection problems

Alkiffai¹,

Crooks²

2016

Tamkang J. Math.

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Abstract. This paper is concerned with linear determinacy in monostable reactiondiffusion-convection equations and co-operative systems. We present sufficient conditions for minimal travelling-wave speeds (equivalent to spreading speeds) to equal values obtained from linearisations of the travelling-wave problem about the unstable equilibrium. These conditions involve both reaction and convection terms. We present separate conditions for non-increasing and non-decreasing travelling waves, called 'right' and 'left' conditions respectively, because of the asymmetry in propagation caused by the convection terms. We also give a necessary condition on the reaction term for the existence of convection terms such that both the right and left conditions can be satisfied simultaneously. Examples show that our sufficient conditions for linear determinacy are not necessary and compare these conditions in the scalar case with alternative conditions observed in Malaguti-Marcelli [15] and . We also illustrate, for both an equation and a system, the existence of reaction and (non-trivial) convection terms for which the right and left linear determinacy conditions are simultaneously satisfied. An example is given of an equation which is right but not left linearly determinate.

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“…We comment that the estimate (53) shows that the kernel e −2Dt I m (2Dt) behaves like the Gaussian kernel when m is small and the estimate (54) shows that the kernel e −2Dt I m (2Dt) behaves like the exponentially decaying when m is large as the time t is fixed. We have the following corollary for the tail estimates on u(t, x) (Corollary 2.2 in Hu and Li (2015)).…”

Section: Simulations For Two Species Modelmentioning

confidence: 92%

Spreading speeds along shifting resource gradients in reaction-diffusion models and lattice differential equations.

Shang¹

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Spreading speeds as slowest wave speeds for cooperative systems

Cited by 259 publications

References 10 publications

A slow pushed front in a Lotka–Volterra competition model

A slow pushed front in a Lotka–Volterra competition model

Lack of symmetry in linear determinacy due to convective effects in reaction-diffusion-convection problems

Spreading speeds along shifting resource gradients in reaction-diffusion models and lattice differential equations.

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