Systems of ordinary differential equations with some monotonicity property.

摘要

In the first part of this work, a mathematical model formulated by G. Powell in 1986 in considered. This model describes a synthrophic chain of species {dollar}Xsb{lcub}i{rcub}{dollar}, 1 {dollar}leq i leq n{dollar} in a continuous chemostat culture. Species {dollar}Xsb{lcub}i{rcub}{dollar} utilizes substrate {dollar}Ssb{lcub}i{rcub}{dollar} and forms a product {dollar}Ssb{lcub}i+1{rcub}{dollar}. The substrate {dollar}Ssb{lcub}i+1{rcub}{dollar} inhibits the growth of {dollar}Xsb{lcub}i{rcub}{dollar} and constitutes a growth limiting substrate of {dollar}Xsb{lcub}i+1{rcub}{dollar}. The model is generalized and the conditions for coexistence of all species are improved. The local stability results will be extended to global stability. Some interesting cases also will be discussed.; In the second part, a system of ordinary differential equations is considered. The system is of the form {dollar}dot xsb1{dollar} = {dollar}fsb1(xsb1, xsb{lcub}n{rcub}), dot xsb{lcub}i{rcub}{dollar} = {dollar}fsb{lcub}i{rcub}(xsb{lcub}i-1{rcub}, xsb{lcub}i{rcub}, xsb{lcub}i+1{rcub}), i{dollar} = 2, ..., {dollar}n{dollar} {dollar}-{dollar} 1, and {dollar}dot xsb{lcub}n{rcub}{dollar} = {dollar}fsb{lcub}n{rcub}(xsb{lcub}n-1{rcub}, xsb{lcub}n{rcub}){dollar}, where f = {dollar}(fsb1, ..., fsb{lcub}n{rcub}){dollar} is defined on a nonempty, open, convex set {dollar}Omegasubseteq{dollar} R{dollar}sp{lcub}n{rcub}{dollar}, {dollar}fsb{lcub}i{rcub}in Csp{lcub}n-1{rcub}(Omega){dollar}, and {dollar}partial{lcub}fsb{lcub}i{rcub}{rcub}overpartial{lcub}xsb{lcub}j{rcub}{rcub}{dollar} {dollar}>{dollar} 0 for {dollar}i ne j{dollar} except when i = 1 and j = n. It is shown that the omega limit set of any bounded orbit of this system contains an equilibrium point or a periodic orbit. As an application to this result a four dimensional system of Lotka-Volterra type is presented. It is shown that when the interior equilibrium is unstable, the omega limit set of solutions with initial values in R{dollar}sbsp{lcub}+{rcub}{lcub}4{rcub}{dollar} contains periodic orbits. Also, a special case where a Hopf bifurcation appears.