## Journals

- Advances in Mathematics of Communications
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- Discrete & Continuous Dynamical Systems - A
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### Open Access Journals

DCDS

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.

MBE

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.

DCDS-B

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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keywords:

DCDS

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.

DCDS-B

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

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.

DCDS-B

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

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

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.## Year of publication

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