競爭物種在塊片棲息環境的全局動態

Abstract

我們研究競爭物種在塊片棲息環境的全局動態。第一，我們考慮兩個競爭物種在兩個塊片棲息環境，這兩個競爭物種在這兩個塊片棲息環境具有相同的遷徙速率。我們證明當某一物種在這兩個塊片棲息環境都擁有較大的出生率，則該物種會存活，且會導致另一個物種滅亡。較有趣的情況是當兩個物種具有相同的總出生率，但是這兩個物種各有一個較大的出生率在某一個塊片棲息環境。在2005年，史蒂芬．古爾利與況陽兩位教授猜測，當遷徙速率夠大時，物種只要集中出生率在單一的塊片棲息環境，該物種就能夠存活，而另一個物種則滅亡；當遷徙速率夠小時，兩個物種呈現共存的狀態。簡言之，贏的策略是盡可能地集中出生率在單一個塊片棲息環境。我們將證明這兩個猜想，透過單調動態理論，及相對於遷徙速率，分析正平衡點的存在性與邊界平衡點的穩定性。在夠大的遷徙速率下，我們對於贏的策略的結果，可能解釋種群在某些特定的生態環境下的繁殖行為。第二，我們考慮兩個競爭物種在兩個塊片棲息環境，這兩個競爭物種在這兩個塊片棲息環境具有不同的遷徙速率。我們證明一個令人感興趣的現象，就是遷徙速率較慢的物種，總是存活，而遷徙速率較快的物種，卻是滅亡。也就是說，在資源不同的塊片棲息環境下，遷徙速率較慢的物種總是導致遷徙速率較快的物種滅亡。在1998年，傑克．道克瑞等四位教授已經證明這結果透過使用偏微分方程，這裡，我們用的是常微分方程。第三，我們探討m個競爭物種的承載單形，我們證明m個競爭物種，在資源相同的塊片棲息環境下，最終會收斂到它的承載單形。最後，我們提供幾個數值上的例子來模擬我們的結果。

In this thesis, we study several ODE systems modeling the competition among m species in patchy environments. First, we consider the case of two species and two patches, where both species move between the patches with the same dispersal rate. It is shown that the species with larger birth rates in both patches drives the other species to extinction, regardless of the dispersal rate. The more interesting case is when both species have the same average birth rate but each species has a larger birth rate in one patch. It has previously been conjectured by Gourley and Kuang in 2005 that the species that can concentrate its birth in a single patch wins if the diffusion rate is large enough, and two species will coexist if the diffusion rate is small. In short, the winning strategy is simply to focus as much birth in a single patch as possible. We solve these two conjectures by applying the monotone dynamics theory, incorporated with a complete characterization of the positive equilibrium and a thorough analysis on the stability of the semi-trivial equilibria with respect to the dispersal rate. Our result on the winning strategy for a su?ffciently large dispersal rate might explain the group breeding behavior that is observed in some animals under certain ecological conditions. Second, we consider two species and two patches, where both species move between the patches with the different dispersal rates. We establish the interesting phenomenon that the slower diffuser always prevails. That is, in a spatially heterogeneous environment, the slower diffuser always wipes out its faster competitor, regardless of the initial conditions as long as both are nonzero. The ODE model we study herein has its continuous-space counterpart. Such a system of PDEs has been studied by Dockery et al. in 1998. Third, the carrying simplex for m competing species is discussed. We con?rm that all solutions for m competing species in a spatially homogeneous environment ultimately enter the carrying simplex. Finally, we provide several numerical examples to illustrate the dynamical scenarios.