關於三物種間之高流動性Lotka-Vollterra競爭擴散系統的波形極限行為

Limiting Profiles of Lotka-Volterra Competition-diffusion System with Large Advection in Three Species Dynamics

游天福(Tien-fu Yu)

指導教授 ： 陳建隆

國立中央大學/理學院/數學系/碩士(2011年)

摘要

本篇論文中，我們首先考慮兩個相互競爭的物種在異構環境的Lotka-Volter- ra competition-diffusion-advection model。這兩類物種除了他們的流動策略外，是完全相同的，而不同的流動策略是指：一類是隨機擴散，另一類則是“較聰明”─ 結合的隨機擴散和定向運動增加的環境梯度。在[3]裡面，Prof. Chen和Prof. Lou給了一個猜想，如果環境函數有多個局部極大值，那麼“較聰明”的物種，將會集中在所有該環境函數的局部極大值。然而，在[6]裡，Prof. Ni 和Dr. Lam發現，假如隨機擴散的物種在環境函數的局部極大值高於環境函數，會導致“較聰明”的物種被滅絕。在這篇文章中，我們考慮三類物種的Lotka-Volterra competition-diffusion-advection model，並期望會有與Prof. Ni 和Dr. Lam類似的結論可以被證明。

關鍵字

極限行為 ； 競爭擴散系統

並列摘要

In this thesis, we first consider a Lotka-Volterra competition-diffusion-advection model for two competing species in a heterogeneous environment. The two species are identical except for their dispersal strategies: One is just random diffusion while the other is "smarter"- a combination of random diffusion and a directed movement up the environmental gradient. In [3], Chen and Lou conjectured that if the environment function $m$ has multiple local maxima, then the "smarter" species must concentrate at all local maximum of m. Nevertheless, in [6], Lam and Ni found that the "smarter" species will die out if the local maximum of m is smaller than the density of the other species. In this article, we consider a model of three species and expect that the related results will be similar to those in [6].

並列關鍵字

advection ； diffusion ； competition ； Lotka-Volterra

參考文獻

[2] Robert R. Cantrell, Chris Cosner and Yuan Lou, Advection mediation coexistence of competing species, Proc. Royal Soc. Edinburgh (A), 137 (2007), 497-518.

[3] Xinfu Chen and Yuan Lou, Principal eigenvalue and eigenfunction of elliptic operator with large convection and its application to a competition model, Indiana Univ. Math. J., 59 (2008), 627-658.

[4] Jack Dockery, Vivian Hutson, Konstantin Mischaikow and Mark Pernarowski, The evolution of slow dispersal rates: A reaction-diffusion model, J. Math. Biol., 37 (1998), 61-83.

[5] King-Yeung Lam, Concentration phenomena of a semilinear elliptic equation with large advection in population dynamics, J. Differential Equations, 250 (2011), 161-181.

[6] King-Yeung Lam and Wei-Ming Ni, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, Dis. Cont. Dyn. Syst., 28 (2010), no. 3, 1051-1067.