從過往到現在，細菌是處處存在在我們的世界當中。然而世界一直
在改變，想當然的細菌也是會隨著空間環境的不同而適應當下的環境，
並且透過遷移的方式去找尋適合自己的生長環境，如果無法遷移則會
透過突變的方式來達到能夠適應當前生長環境。
首先我們會驗證運用離散突變項的技巧來替代規律繁殖的方法是否
可實施在一個物種的常微分方程模型上，並且我們去比較彼此之間的
差異。再者，我們運用相同方法套用在兩個物種的常微分方程模型下，
並且我們去比較彼此之間的差異。最後我們藉由這種方法來建構偏微
分方程模型，並且透過這個模型來模擬出細菌在微生物進化和生長場
(MEGA) 上的行為。另外我們透過不同的成長率以及不同的擴散率下
去比較其差異。

From the past to the present, bacteria are everywhere in our world. However, the world is always changing. Naturally, bacteria will adapt to the current environment according to different spatial environments, and find a suitable growth environment through migration. If they cannot migrate, they will
adapt to the current environment through mutations.
First, we will verify whether the technique of using discrete mutation
terms instead of regular reproduction can be implemented on the ordinary
differential equation model of a species, and we will compare the differences
between them. Furthermore, we apply the same method to the ordinary differential equation model of the two species, and we compare the differences
between them. Finally, we use this method to construct a partial differential
equation model, and use this model to simulate the behavior of bacteria in the
microbial evolution and growth arena(MEGA). In addition, we compare the
differences through different growth rates and different diffusion rates.

Contents
誌謝
摘要 i
Abstract ii
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Literature Discussion and The Structure of Our Thesis . . . . . . . . . . 1
2 Numerical Experiments of The Model of Ordinary Differential Equation 5
2.1 Validation of Discrete Mutation Term Method on ODE . . . . . . . . . . 8
2.2 The Structure of ODE Model and Numerical Simulation . . . . . . . . . 13
3 Numerical Experiments of The Model of Partial Differential Equation 27
3.1 The Structure of The Model Equation of Three Bacteria . . . . . . . . . . 28
3.2 Analysis of Numerical Simulation with Different Diffusion Rate . . . . . 29
3.3 Analysis of Numerical Simulation with Different Growth Rate . . . . . . 33
3.4 Analysis of Numerical Simulation with Fixed Diffusion Rate and Growth
Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 Conclusion 39
References 41

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