Department of Mathematics, University of Miami, Coral Gables, FL, 33146, USA.
School of Mathematical Sciences, CMA-Shanghai, Shanghai Jiao Tong University, Shanghai, 200240, China.
J Math Biol. 2023 Jun 26;87(1):17. doi: 10.1007/s00285-023-01951-3.
A natural biological system under human interventions may exhibit complex dynamical behaviors which could lead to either the collapse or stabilization of the system. The bifurcation theory plays an important role in understanding this evolution process by modeling and analyzing the biological system. In this paper, we examine two types of biological models that Fred Brauer made pioneer contributions: predator-prey models with stocking/harvesting and epidemic models with importation/isolation. First we consider the predator-prey model with Holling type II functional response whose dynamics and bifurcations are well-understood. By considering human interventions such as constant harvesting or stocking of predators, we show that the system under human interventions undergoes imperfect bifurcation and Bogdanov-Takens bifurcation, which induces much richer dynamical behaviors such as the existence of limit cycles or homoclinic loops. Then we consider an epidemic model with constant importation/isolation of infective individuals and observe similar imperfect and Bogdanov-Takens bifurcations when the constant importation/isolation rate varies.
在人类干预下，自然生物系统可能表现出复杂的动态行为，这可能导致系统的崩溃或稳定。分支理论通过对生物系统进行建模和分析，在理解这一演化过程中起着重要作用。在本文中，我们研究了弗雷德·布劳尔（Fred Brauer）做出开创性贡献的两种生物模型：带有放养/捕捞的捕食者-猎物模型和带有输入/隔离的传染病模型。首先，我们考虑具有 Holling 型 II 功能反应的捕食者-猎物模型，其动力学和分支情况已得到很好的理解。通过考虑人类干预措施，如对捕食者进行持续的捕捞或放养，我们表明，在人类干预下的系统经历了不完善分支和 Bogdanov-Takens 分支，这导致了更丰富的动态行为，如极限环或同宿环的存在。然后，我们考虑了一个具有恒定输入/隔离感染个体的传染病模型，并观察到当恒定输入/隔离率变化时，出现类似的不完善和 Bogdanov-Takens 分支。
