一类空间异质的微生物生态学模型的动力学分析与模拟

The combined effects of rapid evolution and spatial heterogeneity make the ecosystem to show some complex spatial patterns and spreading mechanism such as multiple steady state and periodic oscillation. Therefore the study on the dynamics of ecological models with spatial heterogeneity and rapid evolution is challenging and has important biological significance, which is also a hot issue of mathematical biology. We establish a spatially heterogeneous predator-prey chemostat model of microorganism ecosystem with rapid evolution based on experimental data. Then, by decoupling techniques, Lyapunov-Schmidt reduction, degree theory and numerical simulation and so on, we study the dynamical behavior of solutions, investigate the interaction mechanism of microorganisms, and analyze the effects on the longtime behavior of solutions of the model such as diffusion, allelopathy and rapid evolution. By flow reactor models with spatial heterogeneity and rapid evolution, we study the growth, competition and predation of plankton, analyze the effects on the dynamics of model such as diffusion and advection, clarify the interaction mechanism of microbial predation, competition and allelopathy, and reveal the natural law of coexistence or extinction of microorganisms. The research results will promote the theoretical research of microbial ecology model with rapid evolution, and provide an important theoretical foundation and basis for improving environment and sustaining ecological balance.

快速演化、空间异性等因素的共同作用使生态系统呈现了多稳态、周期振荡等复杂的空间模式与扩张机制，因此研究具有空间异性和快速演化特征的生态模型的动力学规律是既富有挑战又具有重要生物意义的热点问题。借助于已有实验数据，建立具有快速演化特征的微生物生态系统的空间异质捕食型恒化器模型，利用解耦技术、Lyapunov-Schmidt分解方法、度理论、数值模拟等方法，研究模型解的动力学行为，探索微生物间相互作用机制，深入分析扩散、化感效应、快速演化特征等因素对模型解的长时行为的影响；借助于具有空间异性和快速演化特征的流动反应器模型研究流域中浮游生物的生长、竞争、捕食规律，分析扩散、对流等因素对模型动力学行为的影响，阐明微生物间捕食、竞争、化感等相互作用机制，揭示微生物间共存或灭绝的自然规律。以期所得研究成果能促进具有快速演化特征的生态学模型的理论研究，并为改善环境、保持生态平衡等提供重要理论基础和依据。

具有空间异质和快速演化生态学模型的动力学行为研究是学术界富有挑战且具有重要意义的核心问题之一。本项目主要研究了空间异质的人口动力学模型及微生物生态学模型等几类反应扩散系统解的动力学行为。利用拓扑度理论、分歧理论、摄动理论、解耦技巧、Lyapunov-Schmidt分解及数值模拟等方法，研究了模型正平衡解的存在唯一性及多重性、稳定性、渐近性及分歧结构等定性性质，并进行了大量数值模拟，探索了生物间相互作用机制，分析了扩散、快速演化等因素对模型解的定性性质的影响；研究了生物生长、竞争、捕食规律，分析了扩散、对流、空间异质等因素对模型解的定性性质的影响，揭示了微生物间捕食、竞争、化感等相互作用机制以及微生物间共存或灭绝的自然规律。所得研究成果丰富了这类具有空间异质结构的生态学模型的理论结果，为进一步深入研究该类型模型打下了坚实的基础，也为相关学科的研究提供了理论支撑和重要参考。