Selected topics on reaction-diffusion-advection models from spatial ecology

We discuss the effects of movement and spatial heterogeneity on population dynamics via reaction-diffusion-advection models, focusing on the persistence, competition, and evolution of organisms in spatially heterogeneous environments. Topics include Lokta-Volterra competition models, river models, evolution of biased movement, phytoplankton growth, and spatial spread of epidemic disease. Open problems and conjectures are presented

Concentration Phenomena of a Semilinear Elliptic Equation with Large Advection in an Ecological Model

We consider a reaction-diffusion-advection equation arising from a biological model of migrating species. The qualitative properties of the globally attracting solution are studied and in some cases the limiting profile is determined. In particular, a conjecture of Cantrell, Cosner and Lou on concentration phenomena is resolved under mild conditions. Applications to a related parabolic competition system is also discussed

On Several Conjectures From Evolution of Dispersal

We address several conjectures raised in Cantrell et al. [Evolution of dispersal and ideal free distribution, Math. Biosci. Eng. 7 (2010), pp. 17–36 [9 Cantrell, R. S., Cosner, C. and Lou, Y. 2010. Evolution of dispersal and ideal free distribution. Math. Biosci. Eng., 7: 17–36. [CrossRef], [PubMed], [Web of Science ®], [Google Scholar]]] concerning the dynamics of a diffusion–advection–competition model for two competing species. A conditional dispersal strategy, which results in the ideal free distribution of a single population at equilibrium, was found in Cantrell et al. [9 Cantrell, R. S., Cosner, C. and Lou, Y. 2010. Evolution of dispersal and ideal free distribution. Math. Biosci. Eng., 7: 17–36. [CrossRef], [PubMed], [Web of Science ®], [Google Scholar]]. It was shown in [9 Cantrell, R. S., Cosner, C. and Lou, Y. 2010. Evolution of dispersal and ideal free distribution. Math. Biosci. Eng., 7: 17–36. [CrossRef], [PubMed], [Web of Science ®], [Google Scholar]] that this special dispersal strategy is a local evolutionarily stable strategy (ESS) when the random diffusion rates of the two species are equal, and here we show that it is a global ESS for arbitrary random diffusion rates. The conditions in [9 Cantrell, R. S., Cosner, C. and Lou, Y. 2010. Evolution of dispersal and ideal free distribution. Math. Biosci. Eng., 7: 17–36. [CrossRef], [PubMed], [Web of Science ®], [Google Scholar]] for the coexistence of two species are substantially improved. Finally, we show that this special dispersal strategy is not globally convergent stable for certain resource functions, in contrast with the result from [9 Cantrell, R. S., Cosner, C. and Lou, Y. 2010. Evolution of dispersal and ideal free distribution. Math. Biosci. Eng., 7: 17–36. [CrossRef], [PubMed], [Web of Science ®], [Google Scholar]], which roughly says that this dispersal strategy is globally convergent stable for any monotone resource function

Dynamics of a Three Species Competition Model

We investigate the dynamics of a three species competition model, in which all species have the same population dynamics but distinct dispersal strategies. Gejji et al. [15] introduced a general dispersal strategy for two species, termed as an ideal free pair in this paper, which can result in the ideal free distributions of two competing species at equilibrium. We show that if one of the three species adopts a dispersal strategy which produces the ideal free distribution, then none of the other two species can persist if they do not form an ideal free pair. We also show that if two species form an ideal free pair, then the third species in general can not invade. When none of the three species is adopting a dispersal strategy which can produce the ideal free distribution, we find some class of resource functions such that three species competing for the same resource can be ecologically permanent by using distinct dispersal strategies

Adventive-diffusive equations of population dynamics

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Departamento de Matemática Aplicada, leída el 17-12-2015Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEunpu

Persistence and Extinction Dynamics in Reaction-Diffusion-Advection Stream Population Model with Allee Effect Growth

The question how aquatic populations persist in rivers when individuals are constantly lost due to downstream drift has been termed the ``drift paradox. Reaction-diffusion-advection models have been used to describe the spatial-temporal dynamics of stream population and they provide some qualitative explanations to the paradox. Here random undirected movement of individuals in the environment is described by passive diffusion, and an advective term is used to describe the directed movement in a river caused by the flow. In this work, the effect of spatially varying Allee effect growth rate on the dynamics of reaction-diffusion-advection models for the stream population is studied. In the first part, a reaction-diffusion-advection equation with strong Allee effect growth rate is proposed to model a single species stream population in a unidirectional flow. Under biologically reasonable boundary conditions, the existence of multiple positive steady states is shown when both the diffusion coefficient and the advection rate are small, which lead to different asymptotic behavior for different initial conditions. On the other hand, when the advection rate is large, the population becomes extinct regardless of initial condition under most boundary conditions. It is shown that the population persistence or extinction depends on Allee threshold, advection rate, diffusion coefficient and initial conditions, and there is also rich transient dynamical behavior before the eventual population persistence or extinction. The dynamical behavior of a reaction-diffusion-advection model of a stream population with weak Allee effect type growth is studied in the second part. Under the open environment, it is shown that the persistence or extinction of population depends on the diffusion coefficient, advection rate and type of boundary condition, and the existence of multiple positive steady states is proved for intermediate advection rate using bifurcation theory. On the other hand, for closed environment, the stream population always persists for all diffusion coefficients and advection rates. In the last part, the dynamics of a reaction-diffusion-advection benthic-drift population model that links changes in the flow regime and habitat availability with population dynamics is studied. In the model, the stream is divided into drift zone and benthic zone, and the population is divided into two interacting compartments, individuals residing in the benthic zone and individuals dispersing in the drift zone. The benthic population growth is assumed to be of strong Allee effect type. The influence of flow speed and individual transfer rates between zones on the population persistence and extinction is considered, and the criteria of population persistence or extinction are formulated and proved. All results are proved rigorously using the theory of partial differential equation, dynamical systems. Various mathematical tools such as bifurcation methods, variational methods, and monotone methods are applied to show the existence of multiple steady state solutions of models