Each species is subject to various biotic and abiotic factors during growth. This paper formulates a deterministic model with the consideration of various factors regulating population growth such as age-dependent birth and death rates, spatial movements, seasonal variations, intra-specific competition and time-varying maturation simultaneously. The model takes the form of two coupled reaction–diffusion equations with time-dependent delays, which bring novel challenges to the theoretical analysis. Then, the model is analysed when competition among immatures is neglected, in which situation one equation for the adult population density is decoupled. The basic reproduction number $\mathcal{R}_0$ is defined and shown to determine the global attractivity of either the zero equilibrium (when $\mathcal{R}_0\leq 1$ ) or a positive periodic solution ( $\mathcal{R}_0\gt1$ ) by using the dynamical system approach on an appropriate phase space. When the immature intra-specific competition is included and the immature diffusion rate is neglected, the model is neither cooperative nor reducible to a single equation. In this case, the threshold dynamics about the population extinction and uniform persistence are established by using the newly defined basic reproduction number $\widetilde{\mathcal{R}}_0$ as a threshold index. Furthermore, numerical simulations are implemented on the population growth of two different species for two different cases to validate the analytic results.

中文翻译：

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具有周期性延迟和竞争的非局部模型的空间动力学

每个物种在生长过程中都受到各种生物和非生物因素的影响。本文制定了一个确定性模型，同时考虑了调节人口增长的各种因素，例如与年龄相关的出生率和死亡率、空间运动、季节变化、种内竞争和随时间变化的成熟度。该模型采用两个耦合反应-扩散方程的形式，具有时间相关的延迟，这给理论分析带来了新的挑战。然后，分析了模型在忽略不成熟之间的竞争的情况下，在这种情况下，成年人人口密度的一个方程是解耦的。基本再生数$\mathcal{R}_0$定义并显示以确定零平衡的全局吸引力（当$\mathcal{R}_0\leq 1$) 或正周期解 ($\mathcal{R}_0\gt1$) 通过在适当的相空间上使用动态系统方法。当包括不成熟的种内竞争而忽略不成熟的扩散速率时，该模型既不合作也不可简化为单一方程。在这种情况下，通过使用新定义的基本再生数来建立关于种群灭绝和均匀持续性的阈值动态$\widetilde{\mathcal{R}}_0$作为阈值指标。此外，对两种不同情况下两种不同物种的种群增长进行了数值模拟，以验证分析结果。