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ABSTRACT. A food chain consisting of species at three trophic levels is modeled using Beddington‐DeAngelis functional responses as the links between trophic levels. The dispersal of the species is modeled by diffusion, so the resulting... more
ABSTRACT. A food chain consisting of species at three trophic levels is modeled using Beddington‐DeAngelis functional responses as the links between trophic levels. The dispersal of the species is modeled by diffusion, so the resulting model is a three component reaction‐diffusion system. The behavior of the system is described in terms of predictions of extinction or persistence of the species. Persistence is characterized via permanence, i.e., uniform persistence plus dissi‐pativity. The way that the predictions of extinction or persistence depend on domain size is studied by examining how they vary as the size (but not the shape) of the underlying spatial domain is changed.
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It is well known that movement strategies in ecology and in economics can make the difference between extinction and persistence. We present a unifying model for the dynamics of ecological populati...
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It is well known that relocation strategies in ecology can make the difference between extinction and persistence. We consider a reaction-advection-diffusion framework to analyze movement strategies in the context of species which are... more
It is well known that relocation strategies in ecology can make the difference between extinction and persistence. We consider a reaction-advection-diffusion framework to analyze movement strategies in the context of species which are subject to a strong Allee effect. The movement strategies we consider are a combination of random Brownian motion and directed movement through the use of an environmental signal. We prove that a population can overcome the strong Allee effect when the signals are super-harmonic. In other words, an initially small population can survive in the long term if they aggregate sufficiently fast. A sharp result is provided for a specific signal that can be related to the Fokker-Planck equation for the Orstein-Uhlenbeck process. We also explore the case of pure diffusion and pure aggregation and discuss their benefits and drawbacks, making the case for a suitable combination of the two as a better strategy.
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The dynamics of a population inhabiting a strongly heterogeneous environment are modeled by diffusive logistic equations of the form $u_1 = \nabla \cdot (d(x,u) + \nabla u) - {\bf b}(x) \cdot \nabla u + m(x)u - cu^2 $ in $\Omega \times... more
The dynamics of a population inhabiting a strongly heterogeneous environment are modeled by diffusive logistic equations of the form $u_1 = \nabla \cdot (d(x,u) + \nabla u) - {\bf b}(x) \cdot \nabla u + m(x)u - cu^2 $ in $\Omega \times (0,\infty )$, where $u$ represents the ...
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... ED1TOR-IN-CH1EF Simon Levin, Princeton University, USA Spatial Ecology via Reaction-Diffusion Equations Robert Stephen Cantrell and ... of Ecology and Evolutionary Biology, Princeton University, USA Associate Editors Zvia Agur,... more
... ED1TOR-IN-CH1EF Simon Levin, Princeton University, USA Spatial Ecology via Reaction-Diffusion Equations Robert Stephen Cantrell and ... of Ecology and Evolutionary Biology, Princeton University, USA Associate Editors Zvia Agur, Tel-Aviv University, Israel Odo Diekmann. ...
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In this chapter, we review some previous studies on modeling spatial spread of specific communicable diseases involving animal hosts. Reaction-diffusion equations are used to model these diseases due to movement of animal hosts. Selected... more
In this chapter, we review some previous studies on modeling spatial spread of specific communicable diseases involving animal hosts. Reaction-diffusion equations are used to model these diseases due to movement of animal hosts. Selected topics include the transmission of rabies in fox populations (Kallen et al., 1984; Kallen et al., 1985; Murray et al., 1986), dengue (Takahashi et al., 2005), West Nile virus (Lewis et al., 2006; Ou & Wu, 2006), hantavirus spread in mouse populations (Abramson and Kenkre, 2002), Lyme disease (Caraco et al., 2002), and feline immunodeficiency virus (FIV) (Fitzgibbon et al., 1995; Hilker et al., 2007).
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õ1. Let D and D * be subsets of R n with D c D * and let u(x) be a solution to the partial differential equation Lu = 0 in D. The "extension problem " is to extend u to a solution of Lu = 0 into the domain D*. Questions of this... more
õ1. Let D and D * be subsets of R n with D c D * and let u(x) be a solution to the partial differential equation Lu = 0 in D. The "extension problem " is to extend u to a solution of Lu = 0 into the domain D*. Questions of this type have been considered by many authors over the last twenty-five years since the work of F. John [6]. The problems are often ill-posed in the sense that extended solutions may not be possible (we envision imposing boundary conditions on u so as to be able to obtain a unique solution in D) and even if they are, they may not depend on the boundary data in a continuous manner. For example in [6] John showed that continuous dependence on the data indeed failed for second order elliptic equations unless one prescribed a fixed global bound for the solutions. Perhaps the best known problem along these lines is the behavior exhibited by the backwards heat equation. In this paper we shall restrict our viewpoint to partial differential equations over regio...
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How organisms gather and utilize information about their landscapes is central to understanding land-use patterns and population distributions. When such information originates beyond an individual's immediate vicinity, movement... more
How organisms gather and utilize information about their landscapes is central to understanding land-use patterns and population distributions. When such information originates beyond an individual's immediate vicinity, movement decisions require integrating information out to some perceptual range. Such nonlocal information, whether obtained visually, acoustically, or via chemosensation, provides a field of stimuli that guides movement. Classically, however, models have assumed movement based on purely local information (e.g., chemotaxis, step-selection functions). Here we explore how foragers can exploit nonlocal information to improve their success in dynamic landscapes. Using a continuous time/continuous space model in which we vary both random (diffusive) movement and resource-following (advective) movement, we characterize the optimal perceptual ranges for foragers in dynamic landscapes. Nonlocal information can be highly beneficial, increasing the spatiotemporal concentra...
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The systems considered have the form in Ω, on ∂Ω, where is a bounded domain, A is a matrix of second order elliptic operators, and γ is a real parameter. For simplicity the results are stated for a single equation, but the range of... more
The systems considered have the form in Ω, on ∂Ω, where is a bounded domain, A is a matrix of second order elliptic operators, and γ is a real parameter. For simplicity the results are stated for a single equation, but the range of validity for systems is discussed. The first type of a priori estimates give lower bounds for sup in terms of γ and |Ω| when is superlinear, upper bounds for sup when is sublinear, and lower bounds for γ when has linear growth. The second type of estimates generalize to systems results of Brezis and Turner and P. L. Lions for single equations; they give upper bounds for sup in the superlinear case. Those estimates require A to be diagonal. None of the results require a variational structure for the system.
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ABSTRACT
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We consider solutions to the nonlinear eigenvalue problem \[ ( ∗ ) A ( x , u → ) u → + λ f ( x , u → ) = 0 in Ω , u → = 0 on ∂ Ω , u → = 0 , on ∂ Ω , u → = 0 , (*)\quad A(x,\vec u)\vec u + \lambda f(x,\vec u) = 0\:\quad {\text... more
We consider solutions to the nonlinear eigenvalue problem \[ ( ∗ ) A ( x , u → ) u → + λ f ( x , u → ) = 0 in Ω , u → = 0 on ∂ Ω , u → = 0 , on ∂ Ω , u → = 0 , (*)\quad A(x,\vec u)\vec u + \lambda f(x,\vec u) = 0\:\quad {\text {in}}\,\Omega ,\quad \vec u = 0\:\quad {\text {on}}\,\partial \Omega ,\quad \vec u{\text { = }}0,\quad {\text {on}}\partial \Omega ,\quad \vec {u} = 0, \] where (*) is a quasilinear strongly coupled second order elliptic system of partial differential equations and Ω ⊆ R n \Omega \subseteq \mathbf {R}^{n} is a smooth bounded domain. We obtain lower bounds for λ \lambda in the case where f ( x , u → ) f(x,\vec u) has linear growth, and relations between λ , Ω \lambda ,\Omega , and ess sup | u → | |\vec u| when f ( x , u → ) f(x,\vec u) has sub- or superlinear growth. The estimates are based on integration by parts and application of certain Sobolev inequalities. We briefly discuss extensions to higher order systems.
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Research Interests: Genetics, Mathematics, Spatial Ecology, Ecology, Island Biogeography, and 15 moreMedicine, Population, Animals, PARTIAL DIFFERENTIAL EQUATION, Theoretical Models, Analytical Model, Ideal Free Distribution, Boolean Satisfiability, Species Specificity, Ecological Applications, Species Area Relationship, Population dynamic, Ideal Ethics, dynamic model, and discrete model
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Sufficient conditions are given for the pointwise boundedness and decay of solutions to time dependent strongly coupled systems of reaction-diffusion equations on spatially bounded domains. The results are obtained via Lyapunov or energy... more
Sufficient conditions are given for the pointwise boundedness and decay of solutions to time dependent strongly coupled systems of reaction-diffusion equations on spatially bounded domains. The results are obtained via Lyapunov or energy methods.
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The main result is a proof of the existence of solutions which are global in time for a.differential equation arising in the theory of myelinated nerves. The equation differs from the usual reaction-diffusion equations occurring in nerve... more
The main result is a proof of the existence of solutions which are global in time for a.differential equation arising in the theory of myelinated nerves. The equation differs from the usual reaction-diffusion equations occurring in nerve models in that the second derivative operator in the spatial direction is replaced at a sequence of discrete nodes by an operator defined as the jump in the first derivative in the spatial direction across the node. The nonlinear dynamics are assumed to be concentrated at the nodes. Local existence of solutions is obtained via semigroup theory; global bounds and hence global existence via energy or Lyapunov methods.
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ThispaperisasurveyofthecontributionsthatProfessorAlanC.Lazer hasmadetothemathematicaltheoryofpopulationdynamics. Specicar- eas where Professor Lazer has madeimportantcontributions includetime periodic populationmodels with... more
ThispaperisasurveyofthecontributionsthatProfessorAlanC.Lazer hasmadetothemathematicaltheoryofpopulationdynamics. Specicar- eas where Professor Lazer has madeimportantcontributions includetime periodic populationmodels with diusionandnonautonomous modelsfor many competing species.
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ABSTRACT
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