Gilpin-Ayala种群收获系统的优化控制问题

发布时间：2018-05-11 01:22

本文选题：Gilpin-Ayala种群增长模型 + 周期解　；参考：《陕西师范大学》2015年硕士论文

【摘要】：可再生生物资源(如森林资源、牧业资源、渔业资源等)能够根据自身的特点,借助于自然循环和生物自身生长、发育或者繁殖而不断更新,并保持一定的储量.现代社会由于人类以高投入、高消耗、高污染的粗放型方式谋求经济的快速增长,社会生产对生物资源的摄取消耗能力己远远超过自身其更新循环能力,造成资源枯竭.因此,可再生生物资源优化管理问题的研究和资源的可持续发展密切相关.近年来关于如何利用有限的可再生资源,实现其可持续发展的相关问题,引起了许多学者的关注.某些情形下,人们可能希望在保证物种和生态环境可持续发展的前提下,追求经济净收益的最大化,而在有些情况下,可能需要研究某时间段内在收获量一定的前提下,使得种群在周期末获得最大的剩余量等问题.常见的收获策略包括连续收获、脉冲收获、时间以及时间依赖收获等.为了定量地研究各种关键因素对种群系统可持续发展的影响,确切地刻画各种控制策略并评估其有效性,需要以人类对生物资源的消耗为背景,建立数学模型对种群系统的发展进行描述,通过对数学模型进行系统的理论分析,获得资源自身生长规律与人类收获或者放养等开发行为之间的关系.评估、分析和预测在不同的参数条件下种群系统的变化趋势,为资源管理者能够合理管理可再生资源提供理论指导以及决策依据.本文主要应用脉冲及连续微分系统的极值原理以及种群动力学的基本理论讨论具有脉冲收获或连续收获的种群系统的优化控制问题.研究结果不仅丰富了种群动力学的理论研究,而且为实际生态问题的解决提供决策依据,有一定的实际意义.本文的主要研究内容和成果包括以下几个方面：(1)研究在周期环境下的Gilpin-Ayala种群系统的连续收获问题.对Gilpin-Ayala种群系统进行比例收获,以一个周期内总收获量最大为目标研究获得了最优收获策略.首先研究系统周期解的存在性及稳定性,并利用关于微分系统的极值原理和一些分析技巧,获得了最优控制策略及最优收益的确切表达式.还研究了一类非自治Gilpin-Ayala种群系统在有限时间内的最大收益问题,对种群系统进行连续收获,目的是在给定的时间区间内使得最终的总收益最大.根据优化问题的极值原理及最速逼近原理,对于不同的初值条件,研究获得了由分段函数表达的最优收获函数以及最大收获量.(2)研究一类由周期Gilpin-Ayala模型描述的脉冲收获系统的优化控制问题.在固定时刻对种群实施比例收获,考虑收获成本因素,以最大经济净收益为管理目标,研究不同的收获努力量对经济收益的影响,并获得最优的收获策略.首先研究该系统周期解的存在唯一性和全局渐近稳定性.进一步利用脉冲微分方程的极值原理,得到最优收获策略满足的数值方程组.(3)研究在有限时间区间内,由Gilpin-Ayala模型描述的线性脉冲收获系统的优化控制问题.收获函数包括比例收获和常量收获,在收获量一定的条件下,以种群在周期末的存储量最大为目标函数,对于任意给定的初值条件,研究不同收获时刻对种群的影响以获得最优的收获策略.首先通过脉冲微分方程的极值原理得到最优收获时刻应满足的必要条件,讨论了在时间周期足够长的条件下具有多次脉冲收获的最优收获策略；进一步考虑了在给定时间范围内的最大收获次数及相应的最优收获策略问题.
[Abstract]:Renewable biological resources (such as forest resources, animal husbandry resources, fishery resources, etc.) can be constantly updated with the help of their own characteristics, with the help of natural circulation and biological growth, development or reproduction, and maintain certain reserves. In modern society, the rapid economic growth is sought by human beings with high input, high consumption and high pollution. The ability of social production to consume biological resources is far more than its own regeneration and recycling capacity, resulting in the exhaustion of resources. Therefore, the research on the optimization and management of renewable biological resources is closely related to the sustainable development of resources. In recent years, the problems of how to make use of limited renewable resources to achieve their sustainable development are discussed. In some cases, people may want to maximize the net income of the economy under the premise of ensuring the sustainable development of the species and the ecological environment. In some cases, it may be necessary to study the maximum surplus of the population at the end of the cycle at the end of a certain period of harvest. The common harvesting strategies include continuous harvest, pulse harvest, time and time dependence. In order to quantitatively study the impact of various key factors on the sustainable development of the population system, accurately depict various control strategies and evaluate their effectiveness, a mathematical model is needed to establish a mathematical model to the background of human resource consumption. The development of the group system is described. Through systematic theoretical analysis of the mathematical model, the relationship between the growth law of resources and the development behavior of human harvest or breeding is obtained. The evaluation, analysis and prediction of the change trend of the population system under different parameter conditions can be made for the resource managers to manage the renewable resources reasonably. This paper mainly applies the extremum principle of pulse and continuous differential system and the basic theory of population dynamics to discuss the optimization control problem of the population system with pulsing harvest or continuous harvest. The results not only enrich the theory of population dynamics, but also solve the problem of the actual ecological problem. The main research content and results of this paper include the following aspects: (1) study the continuous harvest problem of the Gilpin-Ayala population system under the periodic environment. The proportion of the Gilpin-Ayala population system is harvested, and the optimization of the maximum total harvest in a cycle is obtained. First, we study the existence and stability of the periodic solution of the system, and obtain the optimal control strategy and the exact expression of the optimal income by using the extreme value principle and some analytical techniques of the differential system. The maximum income problem of a class of Nonautonomous Gilpin-Ayala population systems in the finite time is also studied. The purpose of continuous harvest is to make the ultimate total profit in a given time interval. According to the extreme value principle and the maximum approximation principle of the optimization problem, the optimal harvest function and the maximum yield expressed by piecewise function are obtained for different initial value conditions. (2) a class of pulse described by periodic Gilpin-Ayala model is studied. The optimal control problem of the harvest system is obtained. In the fixed time, the proportion of the population is harvested at the fixed time, the harvest cost factor is considered, and the maximum economic net income is taken as the management goal. The effects of the different harvest efforts on the economic returns are studied and the optimal harvesting strategy is obtained. By using the extreme value principle of impulsive differential equations, we get the numerical equations which are satisfied by the optimal harvesting strategy. (3) the optimization control problem of the linear pulse harvest system, which is described by the Gilpin-Ayala model in the limited time interval, is studied. The harvest function includes the proportional harvest and the constant harvest, and the species under certain harvest conditions. The maximum storage capacity at the end of the period is the objective function. For any given initial value condition, the effect of the different harvest time on the population is studied to obtain the optimal harvest strategy. First, the necessary conditions for the optimal harvest time should be obtained by the extreme value principle of the impulsive differential equation. The optimal harvesting strategy with multiple pulses is considered, and the maximum harvesting times and corresponding optimal harvesting strategies in a given time range are further considered.

【学位授予单位】：陕西师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175

【共引文献】