中文摘要：

互惠相互作用关系是生物种群之间相互作用的基本关系之一,是生态学、生物数学的研究热点。2种群互惠系统是指每一种群的存在对另一种群的增长都会起促进作用的系统。由于时滞对一系统所带来的影响,在自然现象中是屡见不鲜的。因此,生态系统中,为了更真实的反应自然,时滞是一种不应忽略的因素。时标理论的提出,整合和统一了连续与离散的分析。因此时标上的动力系统更为一般,包含微分方程与差分方程作为它的特例。在时间测度上研究了具有时滞的两种群互惠系统,利用重合度理论中的延拓定理讨论此系统周期解的存在性问题,从而使这一类系统的连续时间情形与离散时间情形的周期解存在性问题得到了统一研究。并且所获得的周期解存在性定理,推广了文献[15]的主要结果。

英文摘要：

Mutual interaction,which is one of the basic relationships between populations, has long been dominant themes in both ecology and mathematical ecology.Mutualism,an interaction of two-species of organisms that benefits both,is found in many type of communities.Since time delays occur so often in nature,a number of models in ecology can be formulated as systems of differential equations with time delays.So,time delay is a factor that should not be ignored.The theory on time scales unified analysis of continuous process and discrete process.Therefore,the dynamic equations on time scales are more general,including differential equations and difference equations as special cases.In this paper,the existence of periodic solutions for a delayed mutualism system on time scales is considered.By using the continuation theorem of coincidence degree,a set of sufficient conditions which ensure the existence of periodic solutions of the system are obtained.So,the study of existence of periodic solutions for the continuous differential equations and discrete difference equations are unified.In addition,The existence theorem of periodic solutions generalized the main results in .