王宁 吕月明
摘要: 通过研究周期演化区域上一类三种群互惠模型, 讨论区域的周期演化对种群持续和灭绝的影响. 利用上下解方法、 比较原理、 拟单增系統理论及抛物方程的先验估计理论,研究模型正周期解的存在性及稳定性问题. 记ρ为区域演化速率, ρ-2=1T∫T01ρ2(t)dt. 结果表明: 当ρ-2>1时, 区域周期演化对互惠种群持续性的影响是消极的;当ρ-2<1时, 区域周期演化对互惠种群持续性的影响是积极的; 当ρ-2=1时, 区域周期演化对互惠种群持续性没有影响.
综上,本文研究了周期演化区域上的一类三种群互惠模型. 首先,为克服由区域演化产生的对流项和稀释项引起的分析困难,本文假设区域演化是周期的且各向同性的,并通过Lagrange变换将模型转化为初始区域上的一类反应扩散方程模型,其扩散系数和反应项都依赖于区域演化速率. 其次,将模型在平凡平衡点处线性化,并研究相关的周期抛物特征值问题,获得了主特征值对区域演化速率的依赖性及其他性质. 进一步,用上下解方法讨论了模型正周期解的存在性及稳定性问题.
最后,根据所得结论和数值模拟,给出了区域周期演化对种群持续和灭绝影响的分析. 结果表明: 较大的区域周期演化率有利于互惠种群的持续生存,而较小的区域周期演化率则会促进种群的灭绝.
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