Dynamics of Microorganism Cultivation and Biological Populations: Stochastic, Delay and Impulsive Effects

微生物培養與生態種群的動力學 : 隨機,時滯以及脈衝作用

Student thesis: Doctoral Thesis

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Award date25 Jun 2021


Solving biological issues with mathematical principles started in the 13th century. Later, many famous publications in the 20th century, such as "Theoretical ecology: principles and applications" published by R. May, boosted mathematical biology development. Since then, the investigation of mathematical biology has been applied to many areas like population dynamics, epidemic dynamics, and biochemical dynamics. Theoretical conclusions based on the related mathematical models help us measure and explain the dynamical alternation of populations in the ecosystem and effectively manage and distribute the ecosystem's natural resources. Based on the differential equations theory, dynamical behaviors of microorganism cultivation and biological species in the chemostat or turbidostat have been analyzed in this thesis.

The first part presents some background knowledge and related research works of mathematical models in the chemostat and turbidostat for microorganism cultivation and population dynamics with delay, stochastic perturbations and impulsive effect.

In the second part, a mathematical model of microorganism cultivation under the effect of delay and random perturbation is considered to understand how the dynamics of microorganisms in the turbidostat can be characterized. Sufficient conditions for microorganism extinction and permanence in the turbidostat are obtained with the theory of stochastic differential equations. The system has a stationary distribution under a low-level intensity of stochastic perturbation from the environment. Here, the microorganisms in the turbidostat are persistent, and the microorganisms' concentration fluctuates around a positive value. On the contrary, microorganisms will be extinct with a strong enough intensity of noise. Several numerical simulations are applied to validate the theoretical results for the dynamics of the system.

In the third part, an impulsively mathematical model has been investigated, which validates the food-chain population dynamics in a polluted environment with impulsive toxicant input. Based on the theory of differential equations, sufficient condition for the extinction of populations has been determined. When the concentration of toxicants surpasses the threshold, it will contribute to the die out of species in the related environment. Also, sufficient conditions for the permanence of populations are obtained in our analysis. Several numerical simulations validate the theoretical conclusions and further reflect the influence of toxicants.

The fourth part analyzed a stochastic and impulsive turbidostat model to understand the population dynamics with the influence of toxicant, random perturbations, and prey refuge. For the stochastic system and related deterministic model, sufficient conditions of extinction and permanence for each population have been determined, which reveals the effect of the three factors above on species' dynamics. Several numerical examples are provided to check the theoretical analysis and simulate the phenomena above to the population dynamics for both deterministic and stochastic cases.