Research On The Age-dependent Problems Of One Dimension Pure Forest In Biology Evolution Systems

The paper is intended to discuss the existence,uniqueness and positivity of solutions of of the nonlinear forest evolution model in a determined condition, by characteristic-line and Schauder Fixed-point theorem. Then the existence of the optimal control is proved by the Maximizing Sequence and the Mazur theorem. Finally we get the necessary condition for the optimality by constructing an adjoint equation. The main results include the following four aspects:1. In the first chapter, some common models of quantitative distribution structures are introduced ,and we also illuminate and analyze some researches on these forest models at home and abroad.Besides, we will show some basic knowledge that this paper will use.The difference between this model and other scholar's in the past discussed is that we take into account the issue of forest breeding problem g ( N ( t )) ,that is at time t,forest regenerate out of some saplings spontaneously.It is concerned with the total amount of time and forest .In this chapter,we denote byβ(?)≥β0> 0the afforestation update rate,which depends on the total forest N ( t ).By using a variety of techniques such as the methods of characteristic line and Schauder Fixed-point theorem etc.,we will prove:(OP1):For any given effective proliferation rateβ(?) and the cut rateμ(x,t),the solution p (x,t)of the forest system exist and is unique. And using the results obtained, we further concluded :Suppose that p₀∈O¹([0,l]),β(?)∈O1(R+), Then p₀ (x) > 0.N0is any positive number given in advance. When N(t)≤N₀,we gotβ( N (t )≥1 + k₀,where k₀ > 0is a constant.2. In the second chapter, we will deal with the following linear biology evolution system : If T is big enough, then it must exist a point t 0≥0.3. In the third chapter, the two maximize control problems of system economic benefits have been studied in the case of forests that have been contracted(suppose Contract Time is T , ensure that the total forest N (T ) in the end of the contract time is no less than N 0in the beginning.(OP2): Optimal control problems of afforestation update rate Control CollectionFor finding the ratioμ? of optimal afforestation update and cutting area,makes is the solution of system (2.1) corresponds toμ(x,t).y(x,t) is unit prices of forest of age x at the moment t . y (x,t)∈C(Q),0