45 research outputs found
Coinvasion-Coexistence Traveling Wave Solutions of an Integro-Difference Competition System
This paper is concerned with the traveling wave solutions of an
integro-difference competition system, of which the purpose is to model the
coinvasion-coexistence process of two competitors with age structure. The
existence of nontrivial traveling wave solutions is obtained by constructing
generalized upper and lower solutions. The asymptotic and nonexistence of
traveling wave solutions are proved by combining the theory of asymptotic
spreading with the idea of contracting rectangle
Spreading Speed, Traveling Waves, and Minimal Domain Size in\ud Impulsive Reaction-diffusion Models
How growth, mortality, and dispersal in a species affect the species’ spread and persistence constitutes a central problem in spatial ecology. We propose impulsive reaction-diffusion equation models for species with distinct reproductive and dispersal stages. These models can describe a seasonal birth pulse plus nonlinear mortality and dispersal throughout the year. Alternatively they can describe seasonal harvesting, plus nonlinear birth and mortality as well as dispersal throughout the year. The population dynamics in the seasonal pulse is described by a discrete map that gives the density of the populationat the end stage as a possibly nonmonotone function of the density of the population at the beginning of the stage. The dynamics in the dispersal stage is governed by a nonlinear reaction-diffusion equation in a bounded or unbounded domain. We develop a spatially explicit theoretical framework that links species vital rates (mortality or fecundity) and dispersal characteristics with species’ spreading speeds, traveling wave speeds, as well as and minimal domain size for species persistence. We provide an explicit formula for the spreading speed in terms of model parameters, and show that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions. We also determine an explicit formula for the minimal domain size using model parameters. Our results show how the diffusion coefficient, and the combination of discrete- and continuous-time growth and mortality determine the spread and persistence dynamics of the population in a wide variety of ecological scenarios. Numerical simulations are presented to demonstrate the theoretical results
A Global Attractivity in a Nonmonotone Age-Structured Model with Age Dependent Diffusion and Death Rates
In this paper, we investigated the global attractivity of the positive
constant steady state solution of the mature population governed by
the age-structured model: \begin{equation*} \left\{\begin{array}{ll}
\frac{\partial u}{\partial t}+\frac{\partial u}{\partial a}=D(a)\frac{\partial
^2 u}{\partial x^2} - d(a)u, & t\geq t_0\geq A_l,\;a\geq 0,\; 0< x< \pi,\\
w(t,x)=\int_r^{A_l}u(t,a,x)da,& t\geq t_0\geq A_l,\; 0<x<\pi,\\
u(t,0,x)=f(w(t,x)), & t\geq t_0\geq A_l,\; 0<x<\pi,\\
u_x(t,a,0)=u_x(t,a,\pi)=0,\;& t\geq t_0\geq A_l,\; a \geq 0, \end{array}
\right. \end{equation*} when the diffusion rate and the death rate
are age dependent, and when the birth function is nonmonotone. We
also presented some illustrative examples.Comment: 11 page
Front-like entire solutions for monostable reaction-diffusion systems
This paper is concerned with front-like entire solutions for monostable
reactiondiffusion systems with cooperative and non-cooperative nonlinearities.
In the cooperative case, the existence and asymptotic behavior of spatially
independent solutions (SIS) are first proved. Combining a SIS and traveling
fronts with different wave speeds and directions, the existence and various
qualitative properties of entire solutions are then established using
comparison principle. In the non-cooperative case, we introduce two auxiliary
cooperative systems and establish some comparison arguments for the three
systems. The existence of entire solutions is then proved via the traveling
fronts and SIS of the auxiliary systems. Our results are applied to some
biological and epidemiological models. To the best of our knowledge, it is the
first work to study the entire solutions of non-cooperative reaction-diffusion
systems