臺灣博碩士論文加值系統

我們用改良過後的算法做生物競爭模型Lotka-Volterra model 強競爭的數值模擬，使得我們可以保留行波解的資訊與觀察到一般解收斂至行波解的行為，甚至可以定義出一般解的廣義傳遞速度。藉由特定例子中，理論解與數值解的誤差分析，我們可以以可控制的誤差用一般解的漸進行為逼近解不出確切形式的行波解。

We use modified numerical scheme to simulate strong competitive diffusive Lotka-Volterra model with 2 species in one dimensional compact domain to preserve the information of traveling wave solutions and the asymptotic behaviours of evolution solutions, and we are able to define generalized propagation speeds of evolution solutions by speed of reference frame. With error analysis in special models, we could approximate unsolvable exact traveling wave solutions by controllable errors.

摘要3
Abstract 5
Contents 7
List of Figures 9
Chapter 1 Introduction 11
Chapter 2 Survey 13
2.1 Diffusive LotkaVolterra Model . . . . . . . . . . . . . . . . . . . . 13
2.2 Traveling wave solution . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Variational method . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Chapter 3 Numerical scheme 19
3.1 Original model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Motion of numerical evolution solution . . . . . . . . . . . . . . . 21
3.3 Time-adjusting reference frame . . . . . . . . . . . . . . . . . . . . 24
Chapter 4 Simulation result and error analysis 27
4.1 Simulation for Huxley wave( gradient type model ) . . . . . . . . . . 28
4.2 Simulation for Lotka-Volterra model . . . . . . . . . . . . . . . . . . 30
4.3 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.4 Approximation of exact traveling wave solutions . . . . . . . . . . . 35
4.5 Local stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Chapter 5 Summary 43
Chapter 6 Future work 45
6.1 Proof of observation results . . . . . . . . . . . . . . . . . . . . . . 45
6.2 Generalization to other reaction diffusion models . . . . . . . . . . . 45
References 47

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