一类恒化器竞争模型正解存在区域的刻画

计算机工程与应用 ›› 2014, Vol. 50 ›› Issue (17): 68-73.

一类恒化器竞争模型正解存在区域的刻画

刘继远，李艳玲

陕西师范大学数学与信息科学学院，西安 710062

出版日期:2014-09-01 发布日期:2014-09-12

Characterization of existence region of positive solutions for competition model in chemostat

LIU Jiyuan, LI Yanling

College of Mathematics and Information Science, Shaanxi Normal University, Xi’ an 710062, China

Online:2014-09-01 Published:2014-09-12

摘要/Abstract

摘要： 刻画了一类带Ivlev型反应函数的非均匀恒化器竞争模型正解的存在域。利用不动点指数理论和上下解方法证明了在[a≠λ1]且[b≠σ1]的前提下，系统有正解的充要条件是[a>r1(a,b)]且[b>r2(a,b)]。结合单调方法和不动点指数理论，说明存在域[Λ]是[R2+]中的一个无界连通区域，其边界由两条递增的曲线[Γ1:a=F1(b)]和[Γ2:b=F2(a)]构成。证明了系统在存在域[Λ]的某个子区域内至少有两个正解。

关键词: 恒化器, Ivlev型反应函数, 不动点指数, 单调方法

Abstract: The existence region of positive solutions in the unmixed chemostat with the Ivlev response function is portrayed. It is shown that if [a≠λ1] and [b≠σ1] hold, then the necessary and sufficient conditions, where the system possesses positive solutions, are [a>r1(a,b)] and [b>r2(a,b)] by using the fixed point theory and the upper and lower solution method. Combining with the monotone method and the fixed point theory, it is proved that [Λ] is a connected unbounded region in [R2+], whose boundary consists of two monotone nondecreasing curves [Γ1:a=F1(b)] and [Γ2:b=F2(a)]. It is shown that the system has at least two positive solutions in certain subregion of [Λ].

Key words: chemostat, Ivlev response function, fixed point index, monotone method

刘继远，李艳玲. 一类恒化器竞争模型正解存在区域的刻画[J]. 计算机工程与应用, 2014, 50(17): 68-73.

LIU Jiyuan, LI Yanling. Characterization of existence region of positive solutions for competition model in chemostat[J]. Computer Engineering and Applications, 2014, 50(17): 68-73.