SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4284-4313, January 2020.
In this work we investigate some asymptotic properties of an age-structured Lotka--Volterra model, where a specific choice of the functional parameters allows us to formulate it as a delayed problem, for which we prove the existence of a unique coexistence equilibrium and characterize the existence of a periodic solution. We also exhibit a Lyapunov functional that enables us to reduce the attractive set to either the nontrivial equilibrium or to a periodic solution. We then prove the asymptotic stability of the nontrivial equilibrium where, depending on the existence of the periodic trajectory, we make explicit the basin of attraction of the equilibrium. Finally, we prove that these results can be extended to the initial PDE problem.

中文翻译：

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具有年龄结构和时滞的Lotka-Volterra模型的渐近行为

SIAM数学分析期刊，第52卷，第5期，第4284-4313页，2020
年1月。在这项工作中，我们研究了具有年龄结构的Lotka-Volterra模型的一些渐近性质，其中特定的功能参数选择使我们能够将其表述为一个延迟问题，为此我们证明了唯一的共存均衡的存在并描述了周期解的存在。我们还展示了Lyapunov函数，该函数使我们能够将吸引集减小为非平凡平衡或周期解。然后，我们证明了非平凡平衡的渐近稳定性，其中，根据周期轨迹的存在，我们明确表明了平衡吸引的盆地。最后，我们证明了这些结果可以扩展到最初的PDE问题。