中文摘要：

研究了一个具有阶段结构和时滞的幼年染病单种群模型.通过常微分方程的特征根法,借助几何图形分析了3个平衡点的存在性,得到了它们局部稳定的充要条件.并且在推论中找到了参数τ影响平衡点稳定性的阈值,得到了当参数τ在不同区间取值时对应的平衡点的稳定性,并通过例题验证了定理的结论.最后,对于所得的数学结果给出了生物意义下的解释：若时滞较大,即种群的成熟期较长,则种群走向绝灭;若时滞较小,即种群的成熟期较短,则种群可以持续生存.

英文摘要：

A delayed stage-structured single-species model with disease in the infant is studied. By the method of eigenvalue for ordinary differential equation,and by the geometric figures,the sufficient and necessary conditions for the existence and stability of the three equilibrium points are got. In the inference,some thresholds for the parameter τ are found,which control the stability of the equilibrium points. When the parameter τ has different values,the corresponding stability property of the equilibrium points is obtained,and theoretical result is verified by a simple example. At last,the results from the viewpoint of biology are explained. If the delay is large enough and the maturation period is long enough,then the population will die out. If the delay is small and the maturation period is short,then the population may persist.