双稳年龄结构种群模型的阈值动力学
Sharp threshold dynamics for a bistable age-structured population model
AI总结：
本文研究了双稳出生函数下年龄结构种群模型的阈值动力学，通过单调动力系统理论证明了参数阈值处的明确转变。
摘要：
本文致力于研究Gurtin-MacCamy种群模型在双稳出生函数下的长期动力学。我们考虑一种一参数单调初始分布族，对于参数的小值，相应种群密度随时间推移而灭绝，而对于大值，解表现出不同的行为。我们关注参数的中间集，称为阈值参数。我们证明了这两种渐近动力学之间存在一个明确的转变；即当年龄依赖出生率具有紧支集时，恰好存在一个阈值值，利用单调动力系统理论。当出生率非紧支时，情况更加复杂，即使非线性出生函数是单稳的。然而，本文所用的方法在处理具有非紧支的特定出生率时被证明是有效的，通过将年龄结构模型的动力学转化为积分微分系统。
英文摘要：
This paper is devoted to the long-term dynamics of solutions to the Gurtin-MacCamy population model with a bistable birth function. We consider a one-parameter monotone family of initial distributions for the population such that for small values of the parameter, the corresponding population density gets extinct as time passes, whereas for large values of them, the solutions exhibit a different behavior. We are interested in the intermediate set of values for the parameters, which are called threshold parameters. We prove the existence of a sharp transition between these two asymptotic dynamics; that is, there exists exactly one threshold value when the age-dependent birth rate of the population has compact support, utilizing the theory of monotone dynamical systems. The case when the birth rate is non-compactly supported is more intricate to deal with, as has been observed in several works, even if the nonlinear birth function is monostable. Nevertheless, the approach used in the present work turns out to be effective to handle a particular birth rate with noncompact support by translating the dynamics of the age-structured model into an integro-differential system.
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