具有表型与空间结构种群的反应扩散模型 I —— 持续性准则
Reaction-diffusion model for a population structured in phenotype and space I -- Criterion for persistence
AI总结：
本文研究异质周期环境中具有表型结构种群的反应扩散模型，证明了适定性并推导出基于广义主特征值的种群持续性准则，揭示了环境异质性与个体移动性增加会加剧物种持续生存的难度，为后续环境优化问题研究奠定数学基础。
摘要：
我们考虑一个具有表型结构的种群反应扩散模型。假设种群生活在异质周期环境中，因此给定的表型性状可能因空间位置不同而具有不同适应性。该模型包含个体的空间移动性与突变。我们首先证明模型的适定性。接着，我们推导出一个种群持续性准则，该准则涉及与线性化椭圆算子相关的广义主特征值。这一概念使我们能够处理算子可能缺乏强制性的问题。随后，我们得到关于广义主特征值的单调性结果，其依据是环境空间波动的频率与空间扩散率。我们推断，环境越异质或个体移动性越高，物种持续生存就越困难。本工作为研究使环境中的种群持续性尽可能困难或容易的其他优化问题奠定了数学基础，这些问题将在后续的姊妹篇中讨论。
英文摘要：
We consider a reaction-diffusion model for a population structured in phenotype. We assume that the population lives in a heterogeneous periodic environment, so that a given phenotypic trait may be more or less fit according to the spatial location. The model features spatial mobility of individuals as well as mutation. We first prove the well-posedness of the model. Next, we derive a criterion for the persistence of the population which involves the generalised principal eigenvalue associated with the linearised elliptic operator. This notion allows us to handle the possible lack of coercivity of the operator. We then obtain a monotonicity result for the generalised principal eigenvalue, in terms of the frequency of spatial fluctuations of the environment and in terms of the spatial diffusivity. We deduce that the more heterogeneous is the environment, or the higher is the mobility of individuals, the harder is the persistence for the species. This work lays the mathematical foundation to investigate some other optimisation problems for the environment to make persistence as hard or as easy as possible, which will be addressed in the forthcoming companion paper.
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