Abstract
In this paper, we study the global dynamics of a population modelwith age structure. The model is given by a nonlocalreaction-diffusion equation carrying a maturation time delay,together with the homogeneous Dirichlet boundary condition. Thenon-locality arises from spatial movements of the immatureindividuals. We are mainly concerned with the case when the birthrate decays as the mature population size becomes large. The analysis is rathersubtle and it is inadequate to apply the powerful theory of monotonedynamical systems. By using the method of super-sub solutions,combined with the careful analysis of the kernel function in thenonlocal term, we prove nonexistence, existence and uniqueness ofthe positive steady states of the model. By establishing anappropriate comparison principle and applying the theory ofdissipative systems, we obtain some sufficient conditions for theglobal asymptotic stability of the trivial solution and the uniquepositive steady state.
Highlights
There has been a growing interest in integrating spatial diffusion and time delay into the mathematical studies of population dynamics since the 1970s [4]. This endeavor is essential for advancing our understanding on how space and time are interwoven to determine the dynamic behavior of biological systems
This seems implausible because the individuals staying at x a time τ earlier are probably shifted to other locations. This delicate issue was first addressed by Britton [2] in 1990 by using a probabilistic argument, and independently, by Smith and Thieme [14] in 1991 by inducing an age structure. Both approaches led to nonlocal diffusion population models
Each individual is tagged with a finite age a ≥ 0, along with time t ≥ 0 and location x ∈ Ω ⊂ RN
Summary
School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, 510006, China, and School of Mathematics and Computer Sciences, Yichun University, Yichun, 336000, China, and Key Laboratory of Mathematics and Interdisciplinary Science of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou, 510006, China
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