Problem Of Free Boundary Of Bamboo System About Total Control

The paper has intention to introduce some one dimension forest models of distribution structure firstly, in the base of that, it set up relative model according to the notable character about bamboo shoots obtain the nature of budding without human cultivation, besides, it contains the free boundary which is important true factor. The existence and uniqueness of Bamboo forest evolution model in a determined condition is proved by characteristic-line and related Fixed-point theorem. The paper is mainly made of following four chapters:1. In the first chapter, firstly,it has brief explanations to some forest development system models of distribution structure what were investigated by forestry researchers and related scholars in the past.Then, we will show some relative theorems that this paper will use.lastly,it roughly introduces major work and investigative thinking in this paper.2. In the second chapter, we will resolve the problem of free boundary with regard to following nonlinear bamboo forest development system about age structure:This model deals with the process of age and density structure for Class A Bamboo forest, and the distinction, compared with other former scholars, resides in that, we think about the variable boundary, it change fixed boundary l into free boundary which is function depends on total quantity, apart from that, the nature of budding spontaneously included as well,it deepen largely the difficulty of discussion.In this chapter, we denote by β(·)β0>0the afforestation retention rate shoots.By using a variety of techniques such as the methods of characteristic-line, Schauder Fixed-point theorem and iteration etc. we will prove:For any given effective retention rate shootsβ(·) and the cut rate μ(x,t),the solution (p(x,t),l(t)) of the forest system exist and is unique.According to the process of proof, we more further conclude the following conclusion:Suppose that p0(x)>0on [0,b), then p0(x)>0.m0is any positive number given in advance. There exist constant m0>0, makes the solution (p>(x,t),l(t)) satisfies:on QT, we have p(x, t)>0, and for any number t∈[0,T], then p(l(t),t)=0, N(t)≥m0.3. In the third chapter, it studies model about free boundary of bamboo forest development system which depends on circumstances.It mainly considers the circumstances impact on the bamboo forest development system in base of the second chapter. The solution of the forest system is proved to exist and unique. The method is still characteristic-line and Schauder Fixed-point theorem as the second chapter.4. This chapter mainly discusses a class nonlinear bamboo forest development system about internal dissipation, gives corresponding model,and deals with the existence and uniqueness of solution.Its corresponding model is following equations:Among it, k(x,p(0,t))p(x,t) is dissipation, k(x,p(0,t)) means reserved bamboo shoots p(0,t) have an influence coefficient on growth of bamboo whose age is x at time t.Of course,we only consider that p(0,t) have an effect on other bamboo only when p(0,t) is very big,so,when p(0,t) is smaller, k(x,p(0,t)) can close to zero.The proof of existence about above-mentioned model is similar with the corresponding proof of the second and third chapter,however,to a large extent,the its proof of uniqueness is quite different from former.