Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation

In this paper, we consider a non-autonomous stochastic Lotka-Volterra competitive system dxi(t) = xi(t)[(bi(t)¡ nPj=1aij (t)xj (t))dt+¾i(t)dBi(t)], where Bi(t) (i = 1; 2; ¢ ¢ ¢ ; n) are independent standard Brownian motions. Some dynamical properties are discussed and the su±cient conditions for the existence of global positive solutions, stochastic permanence, extinction as well as global attractivity are obtained. In addition, the limit of the average in time of the sample paths of solutions is estimated

Population dynamical behavior of Lotka-Volterra system under regime switching

In this paper, we investigate a Lotka-Volterra system under regime switching dx(t) = diag(x1(t); : : : ; xn(t))[(b(r(t)) + A(r(t))x(t))dt + (r(t))dB(t)]; where B(t) is a standard Brownian motion. The aim here is to find out what happens under regime switching. We first obtain the sufficient conditions for the existence of global positive solutions, stochastic permanence and extinction. We find out that both stochastic permanence and extinction have close relationships with the stationary probability distribution of the Markov chain. The limit of the average in time of the sample path of the solution is then estimated by two constants related to the stationary distribution and the coefficients. Finally, the main results are illustrated by several examples

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

Persistence in fluctuating environments

Understanding under what conditions interacting populations, whether they be plants, animals, or viral particles, coexist is a question of theoretical and practical importance in population biology. Both biotic interactions and environmental fluctuations are key factors that can facilitate or disrupt coexistence. To better understand this interplay between these deterministic and stochastic forces, we develop a mathematical theory extending the nonlinear theory of permanence for deterministic systems to stochastic difference and differential equations. Our condition for coexistence requires that there is a fixed set of weights associated with the interacting populations and this weighted combination of populations' invasion rates is positive for any (ergodic) stationary distribution associated with a subcollection of populations. Here, an invasion rate corresponds to an average per-capita growth rate along a stationary distribution. When this condition holds and there is sufficient noise in the system, we show that the populations approach a unique positive stationary distribution. Moreover, we show that our coexistence criterion is robust to small perturbations of the model functions. Using this theory, we illustrate that (i) environmental noise enhances or inhibits coexistence in communities with rock-paper-scissor dynamics depending on correlations between interspecific demographic rates, (ii) stochastic variation in mortality rates has no effect on the coexistence criteria for discrete-time Lotka-Volterra communities, and (iii) random forcing can promote genetic diversity in the presence of exploitative interactions.Comment: 25 page

Hopf bifurcations in a reaction-diffusion population model with delay effect

A reaction-diffusion population model with a general time-delayed growth rate per capita is considered. The growth rate per capita can be logistic or weak Allee effect type. From a careful analysis of the characteristic equation, the stability of the positive steady state solution and the existence of forward Hopf bifurcation from the positive steady state solution are obtained via the implicit function theorem, where the time delay is used as the bifurcation parameter. The general results are applied to a food-limited population model with diffusion and delay effects as well as a weak Allee effect population model. (C) 2009 Elsevier Inc. All rights reserved

Computational and mathematical modelling of plant species interactions in a harsh climate

This thesis will consider the following assumptions which are based on a few insights about the artic climate: (1)the artic climate can be characterised by a growing season called summer and a dormat season called winter (2)in the summer season growing conditions are reasonably favourable and species are more likely to compete for plentiful resources (3)in the winter season there would be no further growth and the plant populations would instead by subjected to fierce weather events such as storms which is more likely to lead to the destruction of some or all of the biomass. Under these assumptions, is it possible to find those change in the environment that might cause mutualism (see section 1.9.2) from competition (see section 1.9.1) to change? The primary aim of this thesis to to provide a prototype simulation of growth of two plant species in the artic that: (1)take account of different models for summer and winter seasons (2)permits the effects of changing climate to be seen on each type of plant species interaction