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Global dynamics of a nonlocal population model with age structure in a bounded domain: A non-monotone case

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Abstract

We study the global dynamics of a nonlocal population model with age structure in a bounded domain. We mainly concern with the case where the birth rate decreases as the mature population size become large. The analysis is rather subtle and it is inadequate to apply the powerful theory of monotone dynamical systems. By using the method of super-sub solutions, combined with the careful analysis of the kernel function in the nonlocal term, we prove nonexistence, existence and uniqueness of positive steady states of the model. Moreover, due to the mature individuals do not diffuse, the solution semiflow to the model is not compact. To overcome the difficulty of non-compactness in describing the global asymptotic stability of the unique positive steady state, we first establish an appropriate comparison principle. With the help of the comparison principle, we can employ the theory of dissipative systems to obtain the global asymptotic stability of the unique positive steady state. The main results are illustrated with the nonlocal Nicholson’s blowflies equation and the nonlocal Mackey-Glass equation.

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Authors and Affiliations

  1. School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, 510006, China

    YueDing Yuan & ZhiMing Guo

  2. School of Mathematics and Computer Sciences, Yichun University, Yichun, 336000, China

    YueDing Yuan

  3. Key Laboratory of Mathematics and Interdisciplinary Science of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou, 510006, China

    YueDing Yuan & ZhiMing Guo

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  1. YueDing Yuan
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Yuan, Y., Guo, Z. Global dynamics of a nonlocal population model with age structure in a bounded domain: A non-monotone case. Sci. China Math. 58, 1–22 (2015). https://doi.org/10.1007/s11425-015-5004-6

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