DCDS
Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model
Linda J. S. Allen B. M. Bolker Yuan Lou A. L. Nevai
Discrete & Continuous Dynamical Systems - A 2008, 21(1): 1-20 doi: 10.3934/dcds.2008.21.1
To understand the impact of spatial heterogeneity of environment and movement of individuals on the persistence and extinction of a disease, a spatial SIS reaction-diffusion model is studied, with the focus on the existence, uniqueness and particularly the asymptotic profile of the steady-states. First, the basic reproduction number $\R_{0}$ is defined for this SIS PDE model. It is shown that if $\R_{0} < 1$, the unique disease-free equilibrium is globally asymptotic stable and there is no endemic equilibrium. If $\R_{0} > 1$, the disease-free equilibrium is unstable and there is a unique endemic equilibrium. A domain is called high (low) risk if the average of the transmission rates is greater (less) than the average of the recovery rates. It is shown that the disease-free equilibrium is always unstable $(\R_{0} > 1)$ for high-risk domains. For low-risk domains, the disease-free equilibrium is stable $(\R_{0} < 1)$ if and only if infected individuals have mobility above a threshold value. The endemic equilibrium tends to a spatially inhomogeneous disease-free equilibrium as the mobility of susceptible individuals tends to zero. Surprisingly, the density of susceptibles for this limiting disease-free equilibrium, which is always positive on the subdomain where the transmission rate is less than the recovery rate, must also be positive at some (but not all) places where the transmission rates exceed the recovery rates.
keywords: basic reproduction number disease-free equilibrium dispersal endemic equilibrium. Spatial heterogeneity
MBE
Global dynamics for two-species competition in patchy environment
Kuang-Hui Lin Yuan Lou Chih-Wen Shih Tze-Hung Tsai
Mathematical Biosciences & Engineering 2014, 11(4): 947-970 doi: 10.3934/mbe.2014.11.947
An ODE system modeling the competition between two species in a two-patch environment is studied. Both species move between the patches with the same dispersal rate. It is shown that the species with larger birth rates in both patches drives the other species to extinction, regardless of the dispersal rate. The more interesting case is when both species have the same average birth rate but each species has larger birth rate in one patch. It has previously been conjectured by Gourley and Kuang that the species that can concentrate its birth in a single patch wins if the diffusion rate is large enough, and two species will coexist if the diffusion rate is small. We solve these two conjectures by applying the monotone dynamics theory, incorporated with a complete characterization of the positive equilibrium and a thorough analysis on the stability of the semi-trivial equilibria with respect to the dispersal rate. Our result on the winning strategy for sufficiently large dispersal rate might explain the group breeding behavior that is observed in some animals under certain ecological conditions.
keywords: monotone dynamics global dynamics patch model Competition stability.
DCDS-B
Preface
Bo Guan Jong-Shenq Guo Bei Hu Urszula Ledzewicz Yuan Lou Fernando Reitich
Discrete & Continuous Dynamical Systems - B 2012, 17(6): i-iii doi: 10.3934/dcdsb.2012.17.6i
It is our great privilege to serve as Guest Editors for this special issue of Discrete and Continuous Dynamical Systems, Series B honoring Professor Avner Friedman on his 80th birthday.

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DCDS
Dynamics of a three species competition model
Yuan Lou Daniel Munther
Discrete & Continuous Dynamical Systems - A 2012, 32(9): 3099-3131 doi: 10.3934/dcds.2012.32.3099
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.
keywords: competitive exclusion Dispersal permanence reaction-diffusion-advection.
DCDS-B
Preface on the special issue of Discrete and Continuous Dynamical Systems- Series B in honor of Chris Cosner on the occasion of his 60th birthday
Robert Stephen Cantrell Suzanne Lenhart Yuan Lou Shigui Ruan
Discrete & Continuous Dynamical Systems - B 2014, 19(10): i-ii doi: 10.3934/dcdsb.2014.19.1i
Chris Cosner turned 60 on June 3, 2012 and now, at age 62, continues his stellar career at the interface of mathematics and biology. He received his Ph.D. in 1977 at the University of California, Berkeley under the direction of Murray Protter, winning the Bernard Friedman prize for the best dissertation in applied mathematics. From 1977 until 1982 he was on the faculty of Texas A&M University. In 1982 he left A&M to join the faculty of the Department of Mathematics of the University of Miami as Associate Professor, rising to the rank of Professor in 1988. The academic year 2013-2014 marked his 32nd year of distinguished service to the University of Miami and its research and pedagogical missions.

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DCDS-B
On the dependence of population size upon random dispersal rate
Song Liang Yuan Lou
Discrete & Continuous Dynamical Systems - B 2012, 17(8): 2771-2788 doi: 10.3934/dcdsb.2012.17.2771
This paper concerns the dependence of the population size for a single species on its random dispersal rate and its applications on the invasion of species. The population size of a single species often depends on its random dispersal rate in non-trivial manners. Previous results show that the population size is usually not a monotone function of the random dispersal rate. We construct some examples to illustrate that the population size, as a function of the random dispersal rate, can have at least two local maxima. As an application we illustrate that the invasion of exotic species depends upon the random dispersal rate of the resident species in complicated manners. Previous results show that the total population is maximized at some intermediate random dispersal rate for several classes of local intrinsic growth rates. We find one family of local intrinsic growth rates such that the total population is maximized exactly at zero random dispersal rate. We show that the population distribution becomes flatter in average if we increase the random dispersal rate, and the environmental gradient is always steeper than the population distribution, at least in some average sense. We also discuss the dependence of the population size on movement rates in other contexts and propose some open problems.
keywords: Dispersal rate reaction-diffusion. invasion of species population size
DCDS-B
Preface
Robert Stephen Cantrell Suzanne Lenhart Yuan Lou Shigui Ruan
Discrete & Continuous Dynamical Systems - B 2015, 20(6): i-iii doi: 10.3934/dcdsb.2015.20.6i
The movement and dispersal of organisms have long been recognized as key components of ecological interactions and as such, they have figured prominently in mathematical models in ecology. More recently, dispersal has been recognized as an equally important consideration in epidemiology and in environmental science. Recognizing the increasing utility of employing mathematics to understand the role of movement and dispersal in ecology, epidemiology and environmental science, The University of Miami in December 2012 held a workshop entitled ``Everything Disperses to Miami: The Role of Movement and Dispersal in Ecology, Epidemiology and Environmental Science" (EDM).

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DCDS-B
Preface
Chris Cosner Yuan Lou Shigui Ruan Wenxian Shen
Discrete & Continuous Dynamical Systems - B 2017, 22(3): ⅰ-ⅱ doi: 10.3934/dcdsb.201703i
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DCDS
Random dispersal vs. non-local dispersal
Chiu-Yen Kao Yuan Lou Wenxian Shen
Discrete & Continuous Dynamical Systems - A 2010, 26(2): 551-596 doi: 10.3934/dcds.2010.26.551
Random dispersal is essentially a local behavior which describes the movement of organisms between adjacent spatial locations. However, the movements and interactions of some organisms can occur between non-adjacent spatial locations. To address the question about which dispersal strategy can convey some competitive advantage, we consider a mathematical model consisting of one reaction-diffusion equation and one integro-differential equation, in which two competing species have the same population dynamics but different dispersal strategies: the movement of one species is purely by random walk while the other species adopts a non-local dispersal strategy. For spatially periodic and heterogeneous environments we show that (i) for fixed random dispersal rate, if the nonlocal dispersal distance is sufficiently small, then the non-local disperser can invade the random disperser but not vice versa; (ii) for fixed nonlocal dispersal distance, if the random dispersal rate becomes sufficiently small, then the random disperser can invade the nonlocal disperser but not vice versa. These results suggest that for spatially periodic heterogeneous environments, the competitive advantage may belong to the species with much lower effective rate of dispersal. This is in agreement with previous results for the evolution of random dispersal [9, 13] that the slower disperser has an advantage. Nevertheless, if random dispersal strategy with either zero Dirichlet or zero Neumann boundary condition is compared with non-local dispersal strategy with hostile surroundings, the species with much lower effective rate of dispersal may not have the competitive advantage. Numerical results will be presented to shed light on the global dynamics of the system for general values of non-local interaction distance and also to point to future research directions.
keywords: Reaction-diffusion Competition Non-local dispersal Random dispersal Integral kernel.
DCDS
On a limiting system in the Lotka--Volterra competition with cross-diffusion
Yuan Lou Wei-Ming Ni Shoji Yotsutani
Discrete & Continuous Dynamical Systems - A 2004, 10(1&2): 435-458 doi: 10.3934/dcds.2004.10.435
In this paper we investigate a limiting system that arises from the study of steady-states of the Lotka-Volterra competition model with cross-diffusion. The main purpose here is to understand all possible solutions to this limiting system, which consists of a nonlinear elliptic equation and an integral constraint. As far as existence and non-existence in one dimensional domain are concerned, our knowledge of the limiting system is nearly complete. We also consider the qualitative behavior of solutions to this limiting system as the remaining diffusion rate varies. Our basic approach is to convert the problem of solving the limiting system to a problem of solving its "representation" in a different parameter space. This is first done without the integral constraint, and then we use the integral constraint to find the "solution curve" in the new parameter space as the diffusion rate varies. This turns out to be a powerful method as it gives fairly precise information about the solutions.
keywords: existence asymptotic behavior cross-diffusion parameter representation.

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