Study On Almost Periodic Solutions And The Harvesting Problem Of Impulsive Biosystems

Currently,Mathematical Ecology,which is a branch of Bio-mathematics, devel-oped rapidly in the international and domestic.It is also a part of ecology.In terms of Mathtools-used,there are theoretical ecology, systems ecology and statistical ecol-ogy.In the theoretical ecology,we established the following Logistic model which is one of the mathematical model described the increasing regularity and resource man-agement of single population. where x(t) represents the density of the population or the population level of the re-source at time t,the constant r> 0 is called the intrinsic growth rate,which reflects the inherent characteristics of the species;k(k> 0) reflects the richness of resources.When x= k,the size of population no longer increased. Thus k represents the environment carrying capacity.Since biological systems are always subjected to obvious periodic fluctuations,such as seasonal effects:weather,temperature,food availability and mating habits,etc.It is as-sumed that human activity occurs continuously throughout the year.However,in prac- tice there is another way to exploit a resource.Some species are often harvested at fixed moments every year or in regular pulses,so that human activity can be mod-elled by impulsive perturbations.The impulsive differential equations provide the nat-ural description for such systems [3,10].Recently,impulsive differential system,being one of the focuses of interest,has been applied in many fields such as population dynamics,population ecology,medical and chemistry,etc. Providing a powerful re-search tools for population dynamics,which are chiefly related to population ecol-ogy [8].Consequently, we also consider the following impulsive periodic Logistic sys-tem where N is the set of all positive integers.Ek(0≤Ek≤1) denotes the impulsive har-vesting effort.△χ(tk)=χ(tk+)-χ(tk); r(t),k(t)∈PC[R,R].We assume that q times of harvesting occur at the time t= tk(k=1,2,…,q), Here,n is a positive integer.In earlier research,r, k of Logistic model (*) are constants, we mainly discussed the stability of positive equilibrium point or periodic solutions,the existence of limit cycles.Later,some ecological mathematician began to study the existence of periodic solution of (*) and (**) with periodic coefficients.In recent years,many authors tried to discuss the existence and stability of positive almost periodic solution (such as,Jiang Dongping[6],Ahmad[13]etc.),and obtained many classical results.In part two of this article,we studied the existence and uniqueness of almost pe-riodic solution of impulsive Logistic system (**).Main results:by the method of con- structing Lyapunov function,we proved the system has a unique almost periodic solu-tion,and obtained the following theorem that isTheorem Suppose that (H1) (H2)hold.Assume further that (H3) the set of sequences is unifomly almost periodic; (H4) (?) is almost periodic; (H5) there exist positive constants c,θ＞0, such that-aLma<-θ; Then system admits a unique almost periodic solution.In part three of this article,based on the single species periodic growth model,suppose that the production function is Ex,so the harvesting model where r(t), k(t), E(t) are all continuous,and We studied the optimal exploitation problem and obtained the optimal harvesting policy.And we generalized the corresponding results in literature[12,14].Biological resources is a renewable resource.How to use the lim-ited renewable resources and achieve the sustainable development.Many scholars are concerned about the issues.The main results in this chapter is:Choosing the maximum annual biomass yield as the management objective,we investigate the optimal harvest-ing policies for periodic Logistic ecosystem with impulsive harvest.When the optimal harvesting effort Ek* maximizes the annual biomass yield,the corresponding optimal populationχ* (t),and the maximum periodic biomass yield (?) are obtained.In par-ticular,it is proved that the maximum biomass yield is in fact the maximum sustainable yield (MSY). Provided a theoretical basis for the actual management of renewable resources.