捕食介导的絮凝促进微生物食物链模型一致持久性分析

Uniform Persistence Analysis of a Microbial Food Chain Model with Predator-Mediated Flocculation

摘要: 实际微生物培养体系普遍存在搅拌不充分现象，且原生动物的捕食行为会促进细菌聚集形成絮凝体。基于该生物背景，本文建立非搅拌恒化器内一类四组分反应扩散食物链模型，模型包含营养物、孤立细菌、细菌絮凝体、原生动物四类变量，并引入Holling III型功能反应与饱和絮凝函数，刻画粘合强度对细菌絮凝过程的调控作用。运用主特征值方法推导出两个临界扩散系数作为种群入侵阈值，结合抛物型偏微分方程比较原理与一致持久性理论，分段讨论扩散系数对系统长期动力学的影响：扩散系数较大时平凡解全局渐近稳定，所有微生物趋于灭绝；扩散系数处于两阈值中间时半平凡解全局渐近稳定，仅孤立细菌能够存续；扩散系数足够小时系统满足一致持久性，四种种群可长期共存。最后通过数值模拟验证全部理论结论，同时分析粘合强度、输入营养浓度如何改变种群稳态、周期振荡特征与群落结构，相关结果可为微生物絮凝共存动力学研究提供理论参考。

Abstract: In practical microbial culture systems, mixing is often incomplete, and predation by protozoa can promote the aggregation of bacteria into flocs. Against this biological background, this paper constructs a four-component reaction-diffusion food chain model in an unstirred chemostat, which covers four variables: nutrients, isolated bacteria, bacterial flocs and protozoa. A Holling III functional response and a saturated flocculation function are incorporated to characterize how adhesion strength regulates bacterial flocculation. Two critical diffusion coefficients are derived as invasion thresholds by the principal eigenvalue method. Combined with the comparison principle of parabolic partial differential equations and uniform persistence theory, we analyze the long-term dynamics of the system under different diffusion ranges. When the diffusion coefficient is sufficiently large, the trivial steady state is globally asymptotically stable, and all microorganisms become extinct. If the diffusion coefficient falls between the two thresholds, the semi-trivial steady state is globally asymptotically stable, and only isolated bacteria can survive. When the diffusion coefficient is small enough, the system is uniformly persistent, allowing the long-term coexistence of all four populations. Numerical simulations are carried out to verify all theoretical results, and we further explore how adhesion strength and influent nutrient concentration alter population steady states, periodic oscillations and community structures. The findings offer theoretical references for studying the coexistence dynamics of flocculating microorganisms.