带有季节性繁殖的种群模型临界域大小研究

Research on Critical Domain Size for a Population Model with Seasonal Breeding

摘要: 针对季节性繁殖物种在一维有界区域上的持续生存问题，本文采用周期单调动力系统理论及周期抛物方程的主特征值理论，研究了一类具有单调出生函数、齐次Dirichlet边界条件的常微分–偏微分混合模型，给出了种群持久存在的临界域长度及计算公式，确定了种群灭绝与持续生存的阈值动力学结果。研究表明，区域长度小于该临界域时，系统的零平衡态全局渐近稳定，种群灭绝；区域长度大于该临界域时，系统存在唯一全局吸引的正平衡态，种群可持续生存。数值模拟讨论了模型关键参数对临界域的影响，并验证了理论结果的准确性。

Abstract: The persistence of species with seasonally breeding on a one-dimensional bounded domain is investigated. By employing the theory of periodic monotone dynamical systems and the principal eigenvalue theory for periodic parabolic equations, we study a hybrid ordinary differential equation-partial differential equation model with monotone birth functions and homogeneous Dirichlet boundary conditions. The critical domain size for population persistence is established and its explicit formula is derived. The threshold dynamics between population extinction and persistence is determined. The results show that when the domain length is below the critical size, the zero equilibrium is globally asymptotically stable, and the population becomes extinct; when the domain length exceeds the critical size, there exists a unique globally attractive positive equilibrium, and the population is persistent. Numerical simulations are conducted to observe the effects of key model parameters on the critical domain size and also to verify the accuracy of the theoretical results.