系統偏微分方程式模擬浮游生物競爭與共存之數學分析

Competition and Coexistence for Systems of Partial Differential Equations Modeling Phytoplankton Species with Storage

王埄彬(Feng-Bin Wang)

指導教授 ： 許世壁

國立清華大學/理學院/數學系/博士(2009年)

https://doi.org/10.6843/NTHU.2009.00072

摘要

本篇論文共分成三章：前二章以系統偏微分方程式 (systems of partial differential equations) 描述具內部儲存 (internal storage) 能力之水生浮游生物在非均勻環境中 (partially-mixed environment) 的生長與競爭。第三章探討在流動的棲息地中(如:細長之河流) 再加上一個現代水閘蓄水壩 (a hydraulic storage zone)會如何影響物種 (species) 的生長。
　　 對理論生態學而言，了解物種在變動的棲息地中如何競爭資源 (resources) 是一個具有挑戰性且又重要的問題。具儲存養分能力的物種是如何影響其在變動的環境中競爭呢? 就目前的理論而言，這完全是個未知的問題。最近 Grover 博士利用生態學上的一種手法 (a Lagrangian modeling approach) 針對環境中只有一種養分的情況下做了一些工作。 他做了很多的電腦模擬而給了下面的結論：
在這樣的環境中只能有一個物種生存。然而他並無法寫下一個明確的數學式來描述這個推論。在前二章我們以系統偏微分方程式分別建立了兩個數學模型並對這個重要課題做了深入之探討。
在第三章我們探討了一個非常實際的問題：在河流邊加上一個現代水閘蓄水壩會如何影響物種的生長? 物種在細長的河流中會有擴散 (diffusion) 情形發生；然而蓄水壩可視為均勻的環境 (well-mixed environment) 因此不會有擴散情形發生。我們以偏微分方程與常微分方程之聯立式建立了這個數學模型。
本篇論文用到的數學工具如下：單調的動態系統 (Monotone dynamical system)、分歧理論(Theory of bifurcation)、度理論(Degree theory)、上下解(upper-lower solution)、最大值定理(Maximum principle)。

關鍵字

單調動態系統 ； 內部儲存 ； 上下解 ； 流動棲息地

並列摘要

This dissertation consists of chapter 1-3. Chapter 1-2 treat of applications of the chemostat to model the coexistence and competition for species with internal storage in the partially-mixed environment, while Chapter 3 is concerned with species in flowing habitats with a hydraulic storage zone.
The problem of understanding competition for resources in spatially variable habitats is a challenging and very significant one for theoretical ecology. The specific question of how storage of nutrient resources affects competition in spatially variable habitats is virtually unknown from a theoretical perspective. Recently Grover used a Lagrangian modeling approach to study the competition of phytoplankton for a single nutrient resource. Each competitor population is divided into many subpopulations that move through two model habitats with gradient in nutrient availability: an unstirred chemostat and a partially-mixed water column. By numerical simulations, he concludes that the competitive exclusion holds. However his mathematical model can not be formally formulated and his results are numerical, not analytic. In Chapter 1-2, we construct two systems of reaction-diffusion equations to describe the coexistence and competition for species with internal storage in an unstirred chemostat.
In Chapter 3, we introduce more realistic spatial models – riverine reservoir with hydraulic “storage zones”. The flow reactor model has similar boundary flows as in the unstirred chemostat, but with advective transport in addition to diffusion. Motivated by considering habitats such as broad high-order rivers or riverine reservoirs constructed by damming a river, we introduce a modification of the flow reactor model. Storage zones were originally introduced in hydraulic models to accurately describe transport of nonreactive tracers. Here we introduce a storage zone model for phytoplankton growing both in the flowing zone and the storage zone and derive conditions for persistence of a single species and coexistence of two competing species.
In this dissertation we used the following arguments: Monotone dynamical system, Theory of bifurcation, Degree theory, upper-lower solution, Maximum principle。

並列關鍵字

Monotone dynamical system ； internal storage ； upper-lower solution ； flowing habitats

參考文獻

[1] R. A. Armstorng and R. McGehee, Competitive exclusion, American Naturalist,

115 (1980), pp. 151–170.

[2] J. V. BAXLEY and S. B. ROBINSON, Coexistence in the unstirred chemostat, Appl.

Math. Computation, 89 (1998), pp. 41–65.

[3] K. E. BENCALA and R. A.WALTERS, Simulation of solute transport in a mountain