Minimum domains for spatial patterns in a class of reaction diffusion equations

Murray, J. D.; Sperb, René P.

doi:10.1007/bf00280665

Cited by 41 publications

(32 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Murray and Sperb [8] extended some of the results of [5] to cases where the domain was two dimensional and where the population was subject to drift due to winds or currents rather than dispersing via pure diffusion. Many of their results were essentially estimates of the first eigenvalues of the elliptic operators describing the dispersal of the populations under consideration.…”

Section: O Introduction Modelling and Interpretationmentioning

confidence: 99%

The effects of spatial heterogeneity in population dynamics

1991

View full text Add to dashboard Cite

The dynamics of a population inhabiting a heterogeneous environment are modelled by a diffusive logistic equation with spatially varying growth rate. The overall suitability of an environment is characterized by the principal eigenvalue of the corresponding linearized equation. The dependence of the eigenvalue on the spatial arrangement of regions of favorable and unfavorable habitat and on boundary conditions is analyzed in a number of cases.One of the major problems in mathematical ecology is that of describing the effects of spatial dispersal and environmental heterogeneity on the dynamics of populations. A particular aspect of the problem which is of practical importance in refuge design, pest control, and environmental planning is that of determining how the spatial arrangement of favorable and unfavorable habitats affects the overall suitability of an environment for a given population. Our objectives in this article are to use mathematical models to examine which arrangements are most or least favorable in a number of specific situations and then to draw some general conclusions about the significance of various environmental factors by comparing or contrasting different concrete situations. Our models for population dynamics are reaction-diffusion equations with spatially varying coefficients. Such models and their discrete analogues form one of the main classes of models for spatial effects in population dynamics; the other main class consists of models based on island biogeography theory. The advantages of using reaction-diffusion equations as our models are that they can be tailored to fit vairous biological hypotheses via relatively minor adjustments of the coefficients or boundary conditions, and that they operate at the species rather than community level. We

show abstract

Section: O Introduction Modelling and Interpretationmentioning

confidence: 99%

The effects of spatial heterogeneity in population dynamics

1991

View full text Add to dashboard Cite

show abstract

“…We can then prove that on some open neighbourhood of 0, III/A;,(rOIII _-< M1 < oo, (11) where M 1 is a positive constant independent of h and Ill" 11t denotes the norm of the operator E~,(q).…”

Section: -4 Firthmentioning

confidence: 99%

“…[43 studied the effect of wind patterns for the two dimensional reactiondiffusion problem and Murray and Sperb [11] considered this problem for a general region.…”

Section: Introductionmentioning

confidence: 99%

Analysis of a non-linear difference scheme in reaction-diffusion

郭本瑜

Mitchell

1986

Numer. Math.

View full text Add to dashboard Cite

Summary. A nonlinear partial difference equation resulting from discretising in space and time the parabolic reaction diffusion equation, which models the spruce budworm problem, is analysed and accuracy estimates obtained for solutions over a finite time range and as t~oo. Although the analysis is restricted to the logistic model in one space dimension, the techniques and comparison principles developed in the paper should prove useful in assessing the merits of numerical solutions of other nonlinear parabolic difference equations.

show abstract

“…Nevertheless, patchy patterns can be achieved with single-species reactiondiffusion models such as the model of Murray and Sperb (1983). This model is based on a reaction-diffusion equation with a convection term which plays a role comparable to the dispersal kernel associated with group centers in the GDM; and a diffusion term which has a role comparable to the Brownian motion of the particles in the GDM.…”

mentioning

confidence: 99%

“…Despite these similarities, the way the foci are generated is very different. Indeed, a key ingredient in the model of Murray and Sperb (1983) is the boundary effect: in their case, complex pattern formation (i.e. with more than one focus) cannot appear with lethal boundaries.…”

mentioning

confidence: 99%

Patchy patterns due to group dispersal

Soubeyrand

Roques

Coville

et al. 2011

Journal of Theoretical Biology

View full text Add to dashboard Cite

The presence of multiple foci in population patterns may be due to various processes arising in the population dynamics. Group dispersal, which has been lightly investigated for airborne species, is one of these processes.We built a stochastic model generating the dispersal of groups of particles.This model may be viewed as an extension of classical dispersal models based on parametric kernels. It has a hierarchical structure: at the first stage group centers are drawn under a classical dispersal kernel; at the second stage the particles are diffused around their group centers. Analytic and simulation results show that group dispersal is a sufficient condition to generate patterns with multiple foci, i.e. patchy patterns, even if the population can remain particularly concentrated.

show abstract

Minimum domains for spatial patterns in a class of reaction diffusion equations

Cited by 41 publications

References 7 publications

The effects of spatial heterogeneity in population dynamics

The effects of spatial heterogeneity in population dynamics

Analysis of a non-linear difference scheme in reaction-diffusion

Patchy patterns due to group dispersal

Contact Info

Product

Resources

About