Accelerating Solutions in Integro-Differential Equations

Garnier, Jimmy

doi:10.1137/10080693x

Cited by 129 publications

(137 citation statements)

References 32 publications

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“…There is no accelerating spread even for the kernels with a very fat tail such as (18) and (19). This apparently contradicts to some previous results (Kot et al 1996;Medlock and Kot 2003;Garnier 2011). However, a closer look reveals that those previous results were obtained for a linearized system (i.e.…”

Section: Rate Of Spreadcontrasting

confidence: 91%

Patchy Invasion of Stage-Structured Alien Species with Short-Distance and Long-Distance Dispersal

et al. 2015

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Abstract. Understanding of spatiotemporal patterns arising in invasive species spread is necessary for successful management and control of harmful species, and mathematical modeling is widely recognized as a powerful research tool to achieve this goal. The conventional view of the typical invasion pattern as a continuous population traveling front has been recently challenged by both empirical and theoretical results revealing more complicated, alternative scenarios. In particular, the so-called patchy invasion has been a focus of considerable interest; however, its theoretical study was restricted to the case where the invasive species spreads by predominantly short-distance dispersal. Meanwhile, there is considerable evidence that the long-distance dispersal is not an exotic phenomenon but a strategy that is used by many species. In this paper, we consider how the patchy invasion can be modified by the effect of the long-distance dispersal and the effect of the fat tails of the dispersal kernels.

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Section: Rate Of Spreadcontrasting

confidence: 91%

Patchy Invasion of Stage-Structured Alien Species with Short-Distance and Long-Distance Dispersal

et al. 2015

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“…This conceptualization of pushed and pulled solutions, whose mathematical definitions will be given in a future work, has the advantage of being intuitive and adaptable to more complex models that do not necessarily admit traveling wave solutions. For instance, we should now be able to determine the pushed-pulled nature of the solutions of (i) integro-differential equations including long-distance dispersal events and resulting in accelerating waves (39,40), (ii) reaction-diffusion equations with spatially heterogeneous coefficients that lead to pulsating or generalized transition waves (41,42), (iii) reaction-diffusion equations with forced speed, which have been used in ref. 43 to study the effects of a shifting climate on the dynamics of a biological species.…”

Section: Numerical Computationsmentioning

confidence: 99%

Allee effect promotes diversity in traveling waves of colonization

Roques

Garnier

Hamel

et al. 2012

Proc. Natl. Acad. Sci. U.S.A.

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168

225

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Most mathematical studies on expanding populations have focused on the rate of range expansion of a population. However, the genetic consequences of population expansion remain an understudied body of theory. Describing an expanding population as a traveling wave solution derived from a classical reaction-diffusion model, we analyze the spatio-temporal evolution of its genetic structure. We show that the presence of an Allee effect (i.e., a lower per capita growth rate at low densities) drastically modifies genetic diversity, both in the colonization front and behind it. With an Allee effect (i.e., pushed colonization waves), all of the genetic diversity of a population is conserved in the colonization front. In the absence of an Allee effect (i.e., pulled waves), only the furthest forward members of the initial population persist in the colonization front, indicating a strong erosion of the diversity in this population. These results counteract commonly held notions that the Allee effect generally has adverse consequences. Our study contributes new knowledge to the surfing phenomenon in continuous models without random genetic drift. It also provides insight into the dynamics of traveling wave solutions and leads to a new interpretation of the mathematical notions of pulled and pushed waves. R apid increases in the number of biological invasions by alien organisms (1) and the movement of species in response to their climatic niches shifting as a result of climate change have caused a growing number of empirical and observational studies to address the phenomenon of range expansion. Numerous mathematical approaches and simulations have been developed to analyze the processes of these expansions (2, 3). Most results focus on the rate of range expansion (4), and the genetic consequences of range expansion have received little attention from mathematicians and modelers (5). However, range expansions are known to have an important effect on genetic diversity (6, 7) and generally lead to a loss of genetic diversity along the expansion axis due to successive founder effects (8). Simulation studies have already investigated the role of the geometry of the invaded territory (9-11), the importance of long-distance dispersal and the shape of the dispersal kernel (12-14), the effects of local demography (15), or existence of a juvenile stage (13). Further research is needed to obtain mathematical results supporting these empirical and simulation studies, as such results could determine the causes of diversity loss and the factors capable of increasing or reducing it.In a simulation study using a stepping-stone model with a lattice structure, Edmonds et. al (16) analyzed the fate of a neutral mutation present in the leading edge of an expanding population. Although in most cases the mutation remains at a low frequency in its original position, in some cases the mutation increases in frequency and propagates among the leading edge. This phenomenon is described as "surfing" (15). Surfing is caused by the strong genetic drift ...

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“…In the case where the integral terms are linear, a sophisticated theory has been developed, which includes existence and uniqueness of solutions, stabilities, and travelling waves [14,19,20,27,40].…”

Section: Discussionmentioning

confidence: 99%

“…For example, for α = 0, it is easily conceivable that β(u) is a decreasing function of the population density, and hence Assumption (A1) might be violated. In that case, we have to include the next higher order approximation, which is a fourth-order term in Equation (14). If we keep this term, then (for α = 0) we obtain instead of Equation (23) the limit equation…”

Section: Discussionmentioning

confidence: 99%

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Birth-jump processes and application to forest fire spotting

Hillen

Greese

Martín

et al. 2014

Journal of Biological Dynamics

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Birth-jump models are designed to describe population models for which growth and spatial spread cannot be decoupled. A birth-jump model is a nonlinear integro-differential equation. We present two different derivations of this equation, one based on a random walk approach and the other based on a two-compartmental reaction-diffusion model. In the case that the redistribution kernels are highly concentrated, we show that the integro-differential equation can be approximated by a reaction-diffusion equation, in which the proliferation rate contributes to both the diffusion term and the reaction term. We completely solve the corresponding critical domain size problem and the minimal wave speed problem. Birth-jump models can be applied in many areas in mathematical biology. We highlight an application of our results in the context of forest fire spread through spotting. We show that spotting increases the invasion speed of a forest fire front.

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Accelerating Solutions in Integro-Differential Equations

Cited by 129 publications

References 32 publications

Patchy Invasion of Stage-Structured Alien Species with Short-Distance and Long-Distance Dispersal

Patchy Invasion of Stage-Structured Alien Species with Short-Distance and Long-Distance Dispersal

Allee effect promotes diversity in traveling waves of colonization

Birth-jump processes and application to forest fire spotting

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