在本論文中，我們要討論在一維度網格中兩物種的競爭模型。此模型是用網格動態系統(Lattice dynamical system)來描述。此模型起源於研究當物種的居住環境是區塊片狀時，有遷移(migration)能力的兩物種之間的競爭。
第一部份，我們考慮單一穩定型(monostable)的情形。我們首先證明存在一個最小的波速(minimal wave speed)使得行進波(traveling wave)存在的充要條件為其波速大於或等於此最小波速。接著，在適當的條件下我們能夠利用系統的參數來刻劃出最小波速。然後，我們證明任何行進波的波形(wave profile)都是嚴格單調的。更近一步，在某些條件下，給定波速後，行進波的波形是唯一的(在不考慮平移的情況下)。最後，在數值的觀點下，我們推論當網孔大小(mesh size)趨近零時，離散型最小波速的收斂性。
第二部份，由於此系統有行進波的存在且此系統滿足比較原理(comparison principle)，我們可以造出某種全域解(entire solution)，其解的行為像兩個行進波從x軸兩側隨時間向彼此方向移動。
最後一個部份，我們將考慮在雙穩定型(bistable)的情形下，波的傳遞(wave propagation)。我們證明只要系統的兩個遷移係數(migration coefficient)夠小時，其(非單調)穩定解(stationary solution)將會存在。而且，波的傳遞失敗 (propagation failure) 現象會產生。在單一的方程系統中，這樣的結果J.P. Keener, SIAM J. Appl. Math. (1987) 已經提出。接著，我們也證明非零波速的行進波的波形必定是嚴格單調的。更進一步，我們也完整的給出行進波的波形在兩端的漸近行為。於是我們可以給出波速的先驗估計。

In this thesis, we study a two-component competition system in one dimensional lattice in the homogenous framework. This model arises in the study of the competition between two species with diffusion (or migration) when the habitat is of one-dimensional and is divided into niches or regions.
In the first part, we focus on the monostable case. We rst show that there is a minimal wave speed such that a traveling wave front exists if and only if its speed is above this minimal wave speed. Next, we characterize the minimal wave speed using the parameters in the system. Then we show that any wave prole is strictly monotone. Moreover, under some conditions, we show that the wave prole is unique (up to translations) for a given wave speed. Finally, for the numerical aspect, we derive the convergence of discretized minimal wave speeds as the mesh size tends to zero.
In the second part, since this system has traveling wave front solutions and enjoys comparison principle, we construct some entire solutions which behave as two traveling wave fronts moving towards each other from both sides of x-axis.
In the nal part, we study the wave propagation in the bistable case. We show that there exist stationary solutions when the migration coecients are suffciently small. Also, the propagation failure phenomenon occurs. These results have been shown by J.P. Keener, SIAM J. Appl. Math. (1987) in the scale equation. Finally, we show that any wave profile of traveling wave with nonzero wave speed is monotone. Moreover, we also give the precise asymptotic behavior of wave tails of waves profiles. We then provide an a priori estimate for the traveling wave speed.

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