本文選題：反應擴散模型 + 自由邊界�。� 參考：《中國科學技術大學》2017年博士論文

【摘要】：隨著全球化進程的不斷加快,特別是對外貿易以及出國旅游的快速增長,外來物種入境變得更加頻繁,這樣往往會形成生物入侵.生物入侵不但對當?shù)厣锒鄻有詷嫵闪司薮笸{,破壞了生態(tài)平衡,造成了難以估量的經濟損失,還直接威脅到人類的健康.同時,由于全球氣候的變化引起的動植物分布范圍發(fā)生改變,可能也會影響到生物入侵的進程.為了能夠很好地理解由環(huán)境遷移導致的自然資源分布不均對外來物種擴散的影響,生物數(shù)學家嘗試應用簡單的反應擴散模型來模擬外來物種在新棲息地的傳播情況.生物數(shù)學家通過數(shù)學模型分析物種在新環(huán)境中的擴散速度和傳播形式,這是他們研究生物入侵進程的兩大方向.本文我們將考慮用具有自由邊界條件的反應擴散方程來分析預測單個物種在非均勻環(huán)境中的入侵過程.我們將根據所建立的數(shù)學模型利用零點數(shù)理論、比較原理等基本方法分析入侵物種在非均勻環(huán)境中的傳播速度和擴散形式.此博士論文中我們將緊緊圍繞非均勻環(huán)境討論兩類具有自由邊界條件的反應擴散問題.對于第一類自由邊界問題,在高維徑向對稱情形下,我們將非均勻環(huán)境分為好環(huán)境和壞環(huán)境,進而討論環(huán)境的好壞這種外因及物種的擴散率這種內因對外來物種入侵過程的影響,并且分析探討入侵物種長時間的漸近行為.為了更好地考察非均勻環(huán)境對物種漸近傳播行為的影響,我們首先給出所研究自由邊界問題解的適定性結果;其次,我們給出所研究自由邊界問題相應的閾值定義及其性質;之后,我們根據所定義的閾值給出判斷物種成功入侵或者消亡的充分條件.之后,再利用物種的擴散率、初始占有區(qū)域面積及初始密度來考察物種最終的傳播情況并給出"擴張-消亡"二擇一定理.最后,我們通過構造合適的上下解并結合比較原理得到物種入侵前沿的漸近擴張速度.對于第二類自由邊界問題,在一維情形下,當氣候變化速度小于某臨界速度時,在物種成功入侵的情況下,我們考察外來物種精細的長時間動力學行為.即對于氣候變化速度小于某常值速度的情形,當物種成功入侵時我們可以得到物種的最終擴散形式及擴張前沿的最終傳播形態(tài)和漸近擴張速度.因此,我們將緊緊圍繞物種成功入侵的情形,首先通過構造合適的上下解來說明物種的擴張前沿與某條直線關于時間一致有界,再利用零點數(shù)方法說明擴張前沿最終將以線性形式向外擴張.也就是說,當入侵成功時,物種最終將以某一直線函數(shù)的形式向外傳播并且漸進擴張速度是一個固定的常數(shù).進一步地,我們利用比較原理構造上下解給出入侵物種最終詳細的傳播形式.
[Abstract]:With the accelerating process of globalization, especially the rapid growth of foreign trade and travel abroad, the entry of alien species becomes more frequent, which often forms biological invasion. Biological invasion not only poses a great threat to local biodiversity, destroys the ecological balance, causes incalculable economic losses, but also directly threatens human health. At the same time, changes in the distribution of animals and plants due to global climate change may also affect the process of biological invasion. In order to understand the influence of the uneven distribution of natural resources caused by environmental migration on the diffusion of alien species, biological mathematicians try to simulate the spread of alien species in new habitats by using a simple reactive diffusion model. Biological mathematicians use mathematical models to analyze the diffusion speed and propagation form of species in the new environment, which are their two main directions in the study of biological invasion process. In this paper, we will consider using the reaction diffusion equation with free boundary conditions to analyze and predict the invasion process of a single species in a non-uniform environment. According to the established mathematical model, we will analyze the propagation velocity and diffusion form of invasive species in non-uniform environment by using zero number theory, comparison principle and other basic methods. In this dissertation, we will discuss two kinds of reaction-diffusion problems with free boundary conditions. For the first kind of free boundary problem, in the case of high dimensional radial symmetry, we divide the non-uniform environment into good environment and bad environment. The influence of the external cause of environment and the diffusion rate of species on the invasive process of alien species is discussed, and the asymptotic behavior of invasive species over a long period of time is analyzed. In order to better investigate the effect of non-uniform environment on the asymptotic propagation behavior of species, we first give the suitable qualitative results of the solution of the free boundary problem studied, secondly, we give the corresponding threshold definition and properties of the free boundary problem studied. Then we give sufficient conditions for the species to succeed in invading or disappearing according to the defined threshold. After that, the diffusion rate of species, the area of initial occupied area and the initial density of species are used to investigate the final propagation of species and to give the "expansion-extinction" alternative theory. Finally, by constructing appropriate upper and lower solutions and combining the principle of comparison, we obtain the asymptotic expansion rate of the species invasion front. For the second kind of free boundary problem, in one dimensional case, when the climate change velocity is less than a certain critical velocity, and in the case of successful species invasion, we investigate the fine long-term dynamic behavior of alien species. In other words, when the climate change speed is less than a constant velocity, we can obtain the final diffusion form of species and the final propagation form and asymptotic expansion velocity of the expansion frontier when species invades successfully. Therefore, we will focus on the successful invasion of species, first of all, by constructing appropriate upper and lower solutions, we will show that the expansion front of the species is consistent with the time bound of a straight line. Then the zero point method is used to show that the extension front will eventually expand outwards in a linear form. That is to say when the invasion is successful the species will eventually propagate outwards in the form of a linear function and the rate of gradual expansion is a fixed constant. Furthermore, we use the comparison principle to construct the upper and lower solutions for the final detailed propagation of invasive species.
【學位授予單位】：中國科學技術大學
【學位級別】：博士
【學位授予年份】：2017
【分類號】：O175

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