The Minimal Speed of Traveling Fronts for a Diffusive Lotka–Volterra Competition Model

Hosono, Yuzo

doi:10.1006/bulm.1997.0008

Cited by 145 publications

(87 citation statements)

References 13 publications

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“…Unfortunately, this reasoning is, in general, not correct. Hosono (1998) demonstrated that high dispersal of the resident may increase the invader's spread rate above what the linearization predicts. Lewis et al (2002) gave sufficient conditions for the spread rate of an invader to be determined by the single equation obtained from linearizing at the resident-only steady state (linear determinacy).…”

Section: Introductionmentioning

confidence: 94%

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Coexistence and Spread of Competitors in Heterogeneous Landscapes

Samia

Lutscher

2010

Bull. Math. Biol.

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Competition between species is ubiquitous in nature and therefore widely studied in ecology through experiment and theory. One of the central questions is under which conditions a (rare) invader can establish itself in a landscape dominated by a resident species at carrying capacity. Applying the same question with the roles of the invader and resident reversed leads to the principle that "mutual invasibility implies coexistence." A related but different question is how fast a locally introduced invader spreads into a landscape (with or without competing resident), provided it can invade. We explore some aspects of these questions in a deterministic, spatially explicit model for two competing species with discrete non-overlapping generations in a patchy periodic environment. We obtain threshold values for fragmentation levels and dispersal distances that allow for mutual invasion and coexistence even if the non-spatial competition model predicts competitive exclusion. We obtain exact results when dispersal is governed by a Laplace kernel. Using the average dispersal success, we develop a mathematical framework to obtain approximate results that are independent of the exact dispersal patterns, and we show numerically that these approximations are very accurate.

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Section: Introductionmentioning

confidence: 94%

“…When the scale of dispersal of the resident species is much larger than that of the invading species, the spread rate of the invading species is not linearly determined in homogeneous habitats (Hosono, 1998;Lewis et al, 2002). The invading species may spread faster than the linearization predicts (see in particular Fig.…”

Section: Failure Of Linear Determinacymentioning

confidence: 99%

Coexistence and Spread of Competitors in Heterogeneous Landscapes

Samia

Lutscher

2010

Bull. Math. Biol.

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“…These match very closely with speeds calculated from the simulation speeds. Even though the minimum possible speed is often the one selected in multispecies waves [see for example, Murray (1989)], it should be noted that this is not always the case (Hosono, 1998). For further discussion of this see Lewis et al (2000) and Weinberger et al (2000).…”

Section: Linear Analysismentioning

confidence: 99%

How Predation can Slow, Stop or Reverse a Prey Invasion

Owen

Lewis

2001

Bulletin of Mathematical Biology

178

141

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Observations on Mount St Helens indicate that the spread of recolonizing lupin plants has been slowed due to the presence of insect herbivores and it is possible that the spread of lupins could be reversed in the future by intense insect herbivory [Fagan, W. F. and J. Bishop (2000). Trophic interactions during primary sucession: herbivores slow a plant reinvasion at Mount St. Helens. Amer. Nat. 155, [238][239][240][241][242][243][244][245][246][247][248][249][250][251]. In this paper we investigate mechanisms by which herbivory can contain the spatial spread of recolonizing plants. Our approach is to analyse a series of predatorprey reaction-diffusion models and spatially coupled ordinary differential equation models to derive conditions under which predation pressure can slow, stall or reverse a spatial invasion of prey. We focus on models where prey disperse more slowly than predators. We comment on the types of functional response which give such solutions, and the circumstances under which the models are appropriate.

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“…Such waves were rst studied numerically, with analytical solution in a special case, by Okubo et al (1989). Existence of waves was proved by Kan-on (1997), with extension to the case of di¬erent di¬usion coe¯cients considered by Hosono (1998). The work presented in the present paper raises many challenges for the extension of these various results to the new movement term re®ecting contact inhibition.…”

Section: Discussionmentioning

confidence: 87%

Wavefront propagation in a competition equation with a new motility term modelling contact inhibition between cell populations

Sherratt

2000

Proc. R. Soc. Lond. A

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Linear di¬usion is an established model for spatial spread in biological systems, including movement of cell populations. However, for interacting, closely packed cell populations, simple di¬usion is inappropriate, because di¬erent cell populations will not move through one another: rather, a cell will stop moving when it encounters another cell. In this paper, I introduce a nonlinear di¬usion term that re®ects this phenomenon, known as contact inhibition of migration. I study this term in the context of two competing cell populations, one of which has a proliferative advantage over the other; this is motivated by the very early stages of solid tumour growth. I focus in particular on travelling-wave solutions, corresponding to moving interfaces between the two cell populations. Numerical simulations indicate that there are wavefront solutions for wave speeds above a critical minimum value, and I present linear analysis that explains the selection of wave speeds by initial conditions. I obtain an approximation to the shape of these waves for high speeds, and show that the minimum speed arises via quite new behaviour in the travelling-wave equations, with the proportion of cells of each type approaching a step function as the wave speed decreases towards the minimum. Exploiting this structure, I use singular perturbation theory to investigate the wave shape for speeds close to the minimum.

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The Minimal Speed of Traveling Fronts for a Diffusive Lotka–Volterra Competition Model

Cited by 145 publications

References 13 publications

Coexistence and Spread of Competitors in Heterogeneous Landscapes

Coexistence and Spread of Competitors in Heterogeneous Landscapes

How Predation can Slow, Stop or Reverse a Prey Invasion

Wavefront propagation in a competition equation with a new motility term modelling contact inhibition between cell populations

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