The velocity of spatial population expansion

Bosch, Frank van den; Metz, J.A.J.; Diekmann, Odo

doi:10.1007/bf00164162

Cited by 235 publications

(135 citation statements)

References 46 publications

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“…Since the zero-steady state is unstable, the population projection matrix is bounded by its linearisation at zero (Condition II) and density dependence is local, the dynamics of the invasion are dominated by the wavefront [28,29]. For the mathematical analysis, though not the later simulations, we will assume this Linear Conjecture [29], which has been proved rigorously for a narrower class of problem (see Appendix A).…”

Section: Methods Outlinementioning

confidence: 99%

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Spreading speeds for stage structured plant populations in fragmented landscapes

Gilbert

White

Bullock

et al. 2014

Journal of Theoretical Biology

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Article (refereed) -postprintGilbert, Mark A.; White, Steven M.; Bullock, James M.; Gaffney, Eamonn A. 2014. Spreading speeds for stage structured plant populations in fragmented landscapes.Contact CEH NORA team at noraceh@ceh.ac.ukThe NERC and CEH trademarks and logos ('the Trademarks') are registered trademarks of NERC in the UK and other countries, and may not be used without the prior written consent of the Trademark owner. Spreading Speeds for Stage Structured Plant Populations in Fragmented Landscapes AbstractLandscape fragmentation has huge ecological and economic implications and affects the spatial dynamics of many plant species. Determining the speed of population spread in fragmented/heterogeneous landscapes is therefore of utmost importance to ecologists. Stage-structured Integrodifference Equations (IDEs) are deterministic models which accurately reflect the life cycles and dispersal patterns for numerous species.Existing approximations to wave-speeds consider only particular kernels, or landscapes in which the scale of variation is much smaller than the dispersal scale. We propose an analytical approximation to the wavespeeds of IDE solutions with periodic landscapes of alternating good and bad patches, where the dispersal scale is greater than the extent of each good patch and where the ratio of the demographic rates in the good and bad patches is given by a small parameter, denoted . We formulate this approximation for the Gaussian and Laplace dispersal kernels and for stage structured and non-stage structured populations, and compare the results against numerical simulations. We find that the approximation is accurate for the landscapes considered, and that the type of dispersal kernel affects the relationship between landscape structure, as classified by landscape period and good patch size, and the spreading speed. This indicates that accurately fitting a kernel to data is important in determining the relationship between landscape structure and spreading speed.

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Section: Methods Outlinementioning

confidence: 99%

“…The solution of a spatially homogeneous IDE with no Allee effect is an exponential travelling wave [27], with the spreading behaviour being governed by the behaviour of the solution's wave-front [28]. This is often referred to as the Linear Conjecture [29], and allows us to approximate the IDE by its linearisation.…”

Section: Introductionmentioning

confidence: 99%

Spreading speeds for stage structured plant populations in fragmented landscapes

Gilbert

White

Bullock

et al. 2014

Journal of Theoretical Biology

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“…Furthermore, he proved that the spreading speed of the nonlinear equation is equal to spreading speed of the linearized equation. This concept is known as linear determinacy (van den Bosch et al, 1990), and part of the proof follows from a comparison principle. The idea is as follows.…”

Section: Spreading Speeds and Traveling Waves (ω Unbounded)mentioning

confidence: 99%

Integrodifference equations in patchy landscapes

Musgrave

Lutscher

2013

J. Math. Biol.

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In this dissertation, we study integrodifference equations in patchy landscapes. Specifically, we provide a framework for linking individual dispersal behavior with populationlevel dynamics in patchy landscapes by integrating recent advances in modeling dispersal into an integrodifference equation.First, we formulate a random-walk model in a patchy landscape with patchdependent diffusion, settling, and mortality rates. We incorporate mechanisms for individual behavior at an interface which, in general, results in the probability-density function of the random walker being discontinuous at an interface. We show that the dispersal kernel can be characterized as the Green's function of a second-order differential operator and illustrate the kind of (discontinuous) dispersal kernels that arise from our approach. We examine the dependence of obtained kernels on model parameters.Secondly, we analyze integrodifference equations in patchy landscapes equipped with discontinuous kernels. We obtain explicit formulae for the critical-domain-size problem, as well as explicit formulae for the analogous critical size of good patches on an infinite, periodic, patchy landscape. We examine the dependence of obtained formulae on individual behavior at an interface. Through numerical simulations, we observe that, if the population can persist on an infinite, periodic, patchy landscape, its spatial profile can evolve into a discontinuous traveling periodic wave. We derive a dispersion relation for the speed of the wave and illustrate how interface behavior ii

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“…Characteristics of invading species, such as density dependence (Veit and Lewis 1996) and life history (van den Bosch et al 1990;Neubert and Caswell 2000), and characteristics of the environments through which invasions are occurring, such as landscape heterogeneity (Shigesada et al 1986;Andow et al 1990) and temporal stochasticity , have received attention by ecologists. In addition, theoretical attention to the "dispersal kernels" that characterize the probability of individuals dispersing particular distances has identified the importance of long-distance dispersal to the velocity of spatial spread of an invading species (Kot et al 1996;Clark et al 1998;Caswell et al 2003).…”

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confidence: 99%

When Can Herbivores Slow or Reverse the Spread of an Invading Plant? A Test Case from Mount St. Helens

Fagan¹,

Lewis²,

Neubert³

et al. 2005

The American Naturalist

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Here we study the spatial dynamics of a coinvading consumer-resource pair. We present a theoretical treatment with extensive empirical data from a long-studied field system in which native herbivorous insects attack a population of lupine plants recolonizing a primary successional landscape created by the 1980 volcanic eruption of Mount St. Helens. Using detailed data on the life history and interaction strengths of the lupine and one of its herbivores, we develop a system of integrodifference equations to study plant-herbivore invasion dynamics. Our analyses yield several new insights into the spatial dynamics of coinvasions. In particular, we demonstrate that aspects of plant population growth and the intensity of herbivory under low-density conditions can determine whether the plant population spreads across a landscape or is prevented from doing so by the herbivore. In addition, we characterize the existence of threshold levels of spatial extent and/or temporal advantage for the plant that together define critical values of "invasion momentum," beyond which herbivores are unable to reverse a plant invasion. We conclude by discussing the implications of our findings for successional dynamics and the use of biological control agents to limit the spread of pest species.

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The velocity of spatial population expansion

Cited by 235 publications

References 46 publications

Spreading speeds for stage structured plant populations in fragmented landscapes

Spreading speeds for stage structured plant populations in fragmented landscapes

Integrodifference equations in patchy landscapes

When Can Herbivores Slow or Reverse the Spread of an Invading Plant? A Test Case from Mount St. Helens

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