Reciprocal Lotka-volterra Ecosystems Capture Optimization Problem

The paper was written to give complete qualitative analysis to the optimal harvesting control problem of multi-species Lotka-Volterra mutually beneficial system and find available catching strategies which could make the populations sustainably developing. From the view of economical advantages the optimal harvesting tactics that maximize the profit on the basis of persisting existence of the populations was also considered.Since there was hardly special papers or books about this problem we decomposed the question into two parts to be solved stepwisely.Firstly under the actions of certain harvesting efforts given we analysed the asymptotical properties of the system and investigated the positive limit set of any positive initial-valued solution. Because the linear harvestings to these species could be seemed as diminishing the intrinsic rates of themselves, the model could be formally transformed into a cooperative Lotka-Volterra ecological model and we could get some idea about it. By searching some previous related theses on the qualitative analysis of n-species mutual Lotka-Volterra system, some important conclusions were obtained. Thus with citing these results we separatedly and thoroughly examined all possible dynamical manners of the changing of the; populations when the species were developing under the function of catching efforts in three different areas departed by the size relation between the intrisic rates of the system and the values of the harvesting efforts. So we could give out-righter qualitative analysis to this system and the scope of catching efforts which make the species sustainable and the asymptotical properties of the corresponding system were also clear. Furthermore several harvesting policies that could keep ecological balance of the system were given and this could lay foundations for the research of optimal catching efforts.Following that we came to seek the optimal harvesting tactics. By the analysis of the first part, the domain of catching efforts which keep the system enduring and having a unique positive equilibrium that was globally asymptotically stable had been given. So here the issue that we were interested in was to find the very harvesting efforts that maximize the profit function which was closely related with the catching efforts, i.e. the optimal values of harvesting efforts. For the positive equilibrium of the system existed and was unique then we could draw an expression of the positive equilibrium and its corresponding harvesting effortsfrom the model of the system. Therefore the profit expression with these two variablesharvesting efforts E and its mapping equilibrium XE could be translated into an expression which included only one variantthe positive equilibrium XE so that the problem is converted into the one of seeking stationary points in the range we requied by calculus essential principal. This equaled to solve a n-dimension linear equation group and we could easily analyse every possible situation of the solutions. Since that we could get the extreme value XE and the optimal catching efforts E. For the difficulties of giving specific expressions and computational steps for the multi-species system we took a two-dimension model as an example to interpret the steps so that the readers can get a intuitive and peculiar understanding of the process.Finally companied with deep practical backgrounds and applicant values a special case of harvesting-proportionally catching and the question of improving catching tackles which could be transformed into this type of problem were also be discussed. The concrete process of calculation to a two-dimension system was also displayed.In a word this paper had given a comparatively satisfying reply to the problem of how to keep the treasure resources enduring under the actions of harvesting and how to get maximum profit on the basis of persistance of the resources. Some useful results were gained.