tailieunhanh - Báo cáo hóa học: " Bifurcation analysis of a diffusive model of pioneer and climax species interaction"

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Bifurcation analysis of a diffusive model of pioneer and climax species interaction | Liu and Wei Advances in Difference Equations 2011 2011 52 http content 2011 1 52 o Advances in Difference Equations a SpringerOpen Journal RESEARCH Open Access Bifurcation analysis of a diffusive model of pioneer and climax species interaction Jianxin Liu and Junjie Wei Correspondence weijj@ Department of Mathematics Harbin Institute of Technology Harbin Heilongjiang 150001 PR China Abstract A diffusive model of pioneer and climax species interaction is considered. We perform a detailed Hopf bifurcation analysis to the model and derive conditions for determining the bifurcation direction and the stability of the bifurcating periodic solutions. Keywords pioneer and climax Hopf bifurcation diffusive model 1 Introduction We consider the following model ut dị u uf cn u v vt d2 Al. vg u c22 v where x e o t 0 and u v represent a measure of a pioneer and a climax species respectively. f z the growth rate of the pioneer population is generally assumed to be smoothly deceasing and has a unique positive root at a value z1 so that the crowding is particularly harmful for pioneer species. But for the climax population it is different from pioneer population. Climax fitness increases at low total density but decreasing at higher densities. So that it has an optimum value of density for growing. Hence g z the growth rate of the climax population is assumed to be non-monotone has a hump and possesses two distinct positive roots at some values z2 and z3 with z2 z3 and g z2 0 g z3 . For the reason above we set f cii u v Z1 Ciiu v g u c22v z2 u c22v z3 u c22v in this article. Equation is often used to describe forestry models. Examples can be found in 1 2 and references therein. The dynamics of pioneer-climax models have been studied widely. Systems described by ordinary differential equations are under the hypothesis of homogeneous environment. The stability of positive equilibrium and bifurcation especial Hopf bifurcation