具捕获的三阶段单种群自食模型及其计算机模拟
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摘要
近年来我国的经济快速发展,但是随之也带来了很多问题,特别是在生态环境上,出现了环境的严重破环,资源的过度开发等问题。人们为此付出了惨重的代价,所以我们需要不断地寻求解决这些问题的方法。而种群动力学模型就可以用来帮助我们研究解决这些问题。种群动力学模型主要研究种群的动态变化规律以及平衡稳定状态。这对环境的保护、种群的开发利用具有重要现实意义。
种群的增长是一个从幼年到成年,从成年到老年的过程。而种群在幼年、成年和老年的不同年龄阶段中其生理机能(出生率、死亡率、竞争能力、捕食能力)不同。因此,在建立种群动力学模型时,考虑具有阶段结构的种群更符合现实情况。
怎样才能合理的开发利用种群资源呢?这需要我们对种群资源的开发进行研究。通过建立种群的动力学模型,进行定性和定量分析,预测种群的发展变化以及人们的开发利用对种群的影响,确定合理的捕获策略,使种群资源自身能够持续生存,同时也满足人类需要,这些研究工作具有十分重要实际意义。
本文所做的工作及主要研究结论如下：
1.介绍了种群动力学的研究背景和研究意义,对国内外关于具有阶段结构的单种群及其开发利用问题的研究进行概括总结。
2.对已有的模型进行改进,得到一个具有三个年龄阶段,并且成年种群和老年种群同时捕获幼年种群的自食模型；证明了模型具有正解性和解的有界性；解出了非负平衡点,给出了平凡平衡点处的局部渐进稳定性和全局渐进稳定性的充分条件、正平衡点处的局部渐进稳定性稳定的充分条件。
3.在改进的模型上,分别考虑了对幼年种群,成年种群和老年种群进行捕获的模型,得到其平衡点及平衡点处的稳定性的充分条件,给出了相应的最优收获策略。
4.利用Maple数学软件,对模型中参数和初值给定一定的数值；在无捕获和有捕获的情况下,绘制了系统的向量场图、轨线图、趋势图、收获量图,并求出数值解；通过图像和数值的分析验证了前面得出的结论。
China's economy has developed rapidly in recent years. But it also brought up lots of problems. It especially cause many problems in the ecological environ-ment like the seriously destroyed environment and the over exploited resources and so on. People have to pay a heavy price for it, so we need to seek meth-ods for these issues continually. The population dynamics model can be used to help us to study and solve these problems. The regular of population and the state of stable steady are studied in this model. It is important in protecting the environment and developing and utilizing the population.
There is a process that population's growth from immature to mature, from mature to old. And population's physiological function is different in immature, mature and the old, such as the rate of birth and death, the ability of competi-tion and prey. Therefore, it is more access to the real situation to consider the population with stage structure in the population dynamics model.
How to develop and use the biological resources rationally? It needs us to study the development of biological resources. Through the establishing of the population model, qualitative and quantitative analyzing of the population model, predicting of the population changes and the impact on the population with people's development and making the right optimal harvesting policy to make the biological resources to maintain the survival continually, and meeting human's needs in the same time, these studies are very important in practice.
In this paper, the works and the main conclusions are as follows:
First, we introduce the background and significance of population dynamics, and summarize the model of Stage Structure Single population and its develop-ment and utilization in our country and abroad.
Second, we improve the existed models and get a single cannibalistic popu- lation model with three-stage-structure in which the immature species are preyed by the mature as well as the old ones. We prove that the model has a positive so-lution. Then we give the non-negative equilibrium point. We obtain the sufficient conditions for the existence of a local asymptotical stability and global asymp-totical stability of the trivial equilibrium point as well as the sufficient conditions for the local asymptotical stability of positive equilibrium point.
Third, on the basis of the improved model, we consider the model with capture immature species, mature species and old species respectively. Then we get equilibrium points and the sufficient conditions for the stability of equilibrium points, and make the optimal harvesting policy.
Fourth, by using Maple mathematical software, we give a certain model parameters and initial values. In case of no-capture and capture, we draw a lot of images including vector fields, trajectories, trends and harvest. Then we calculate the numerical solutions. By analyzing the images and numerical solutions, we verify conclusions of the previously obtained.
种群的增长是一个从幼年到成年,从成年到老年的过程。而种群在幼年、成年和老年的不同年龄阶段中其生理机能(出生率、死亡率、竞争能力、捕食能力)不同。因此,在建立种群动力学模型时,考虑具有阶段结构的种群更符合现实情况。
怎样才能合理的开发利用种群资源呢?这需要我们对种群资源的开发进行研究。通过建立种群的动力学模型,进行定性和定量分析,预测种群的发展变化以及人们的开发利用对种群的影响,确定合理的捕获策略,使种群资源自身能够持续生存,同时也满足人类需要,这些研究工作具有十分重要实际意义。
本文所做的工作及主要研究结论如下：
1.介绍了种群动力学的研究背景和研究意义,对国内外关于具有阶段结构的单种群及其开发利用问题的研究进行概括总结。
2.对已有的模型进行改进,得到一个具有三个年龄阶段,并且成年种群和老年种群同时捕获幼年种群的自食模型；证明了模型具有正解性和解的有界性；解出了非负平衡点,给出了平凡平衡点处的局部渐进稳定性和全局渐进稳定性的充分条件、正平衡点处的局部渐进稳定性稳定的充分条件。
3.在改进的模型上,分别考虑了对幼年种群,成年种群和老年种群进行捕获的模型,得到其平衡点及平衡点处的稳定性的充分条件,给出了相应的最优收获策略。
4.利用Maple数学软件,对模型中参数和初值给定一定的数值；在无捕获和有捕获的情况下,绘制了系统的向量场图、轨线图、趋势图、收获量图,并求出数值解；通过图像和数值的分析验证了前面得出的结论。
China's economy has developed rapidly in recent years. But it also brought up lots of problems. It especially cause many problems in the ecological environ-ment like the seriously destroyed environment and the over exploited resources and so on. People have to pay a heavy price for it, so we need to seek meth-ods for these issues continually. The population dynamics model can be used to help us to study and solve these problems. The regular of population and the state of stable steady are studied in this model. It is important in protecting the environment and developing and utilizing the population.
There is a process that population's growth from immature to mature, from mature to old. And population's physiological function is different in immature, mature and the old, such as the rate of birth and death, the ability of competi-tion and prey. Therefore, it is more access to the real situation to consider the population with stage structure in the population dynamics model.
How to develop and use the biological resources rationally? It needs us to study the development of biological resources. Through the establishing of the population model, qualitative and quantitative analyzing of the population model, predicting of the population changes and the impact on the population with people's development and making the right optimal harvesting policy to make the biological resources to maintain the survival continually, and meeting human's needs in the same time, these studies are very important in practice.
In this paper, the works and the main conclusions are as follows:
First, we introduce the background and significance of population dynamics, and summarize the model of Stage Structure Single population and its develop-ment and utilization in our country and abroad.
Second, we improve the existed models and get a single cannibalistic popu- lation model with three-stage-structure in which the immature species are preyed by the mature as well as the old ones. We prove that the model has a positive so-lution. Then we give the non-negative equilibrium point. We obtain the sufficient conditions for the existence of a local asymptotical stability and global asymp-totical stability of the trivial equilibrium point as well as the sufficient conditions for the local asymptotical stability of positive equilibrium point.
Third, on the basis of the improved model, we consider the model with capture immature species, mature species and old species respectively. Then we get equilibrium points and the sufficient conditions for the stability of equilibrium points, and make the optimal harvesting policy.
Fourth, by using Maple mathematical software, we give a certain model parameters and initial values. In case of no-capture and capture, we draw a lot of images including vector fields, trajectories, trends and harvest. Then we calculate the numerical solutions. By analyzing the images and numerical solutions, we verify conclusions of the previously obtained.
引文
[1]马知恩.种群生态学的数学模型与研究[M].合肥：安徽教育出版社.1996.
[2]张锦炎.常微分方程几何理论与分支问题[M].北京：科学出版社.1990.
[3]唐三一,肖燕妮.单种群生物动力系统[M].北京：科学出版社.2008,6.
[4]陈兰荪,孟新柱,焦建军.生物动力学[M].北京：科学出版社.2009.7.
[5]刘胜强,陈兰荪.阶段结构种群生物模型与研究[M].北京：科学出版社.2010,7.
[6]王顺庆,王万雄,徐海根.数学生态学稳定性理论与方法[M].北京：科学出版社.2004,10.
[7]smith F E. Population dynamics in Daphnia[J]. Eclogy.1963,44:651-663.
[8]Cui Q, Lawson G J. Study on models of single population:an expansion of the logistic and exponential equations[J]. Thoer.Biol..1982,98:645-659.
[9]Allen L J S. Persistence and extinction in single species reaction diffusion model-s[J]. Bull.Math.Biol..1983,45:209-227.
[10]Cui.J, Chen Lansun. The effect of dispersal on population growth with stage-structure[J]. Computers Math Applic.2000,39:91-102.
[11]Gao Shujing. Optimal Harvesting Policy and Stability In A Stage Structured Single-Species Growth Model With Cannibalism[J]. Journal of Biomathematics. 2002,17(2):194-200.
[12]高淑京.具有三个阶段结构单种群模型的全局渐进稳定性[J].新疆大学学报(理工版).2001,18(2)：154-158.
[13]Gao Shujing. Models for Species with Three Life History Stages and Cannibalis-m[J]. Journal of Biomathematics.2005,20(4):385-391.
[14]邢铁军,杜明银,王生丽等.具三个年龄阶段的自食单种群系统的定性分析[J].兰州交通大学学报.2008,27(6)：157-159.
[15]Clark C W. Mathematical Bioeconomics:The Optimal Management of Renewable Resources[M]. New York:Wiley.1976.
[16]范猛,王克,刘会民,张玉娟,张树文.一类具有退偿增长曲线生物种群的捕获优化问题[J].生物数学学报.1997,4(1)：38-42.
[17]刘会民,张树文,张玉娟等.临界退偿系统的捕获优化问题[J].生物数学学.1998,13(4)：479-483.
[18]李海龙.带扩散的Logistic单种群模型及其最优收获策略[J].生物数学学报.1999,14(3)：293-300.
[19]李海龙.一类具周期系数的单种群模型及其最优收获策略[J].生物数学学报.1999,14(4)：479-483.
[20]王静,王克.具有年龄结构的单种群模型单一捕获的优化问题[J].东北师大学报自然科学版.2003,35(2)：1-6.
[21]彭奇林.具年龄结构的单种群模型捕获策略的优化[J].呼伦贝尔学院学报.2004,12(4)：7-9,20.
[22]梁志清,赵强.一类具三阶段结构的单种群模型的最优收获策略[J].玉林师范学院学报(自然科学).2005,26(3)：1-3.
[23]刘毅婧,雒志学.一类3阶段结构自食系统的最优收获策略[J].重庆工学院学报(自然科学版).2007,21(6)：43-45.
[24]鲁红英,王克.自治单种群模型及其最优捕获策略[J].系统科学与数学.2004,24(2)：200-205.
[25]鲁红英,王克.一类自治单种群模型及其最优捕获策略[J].系统科学与数学.2010,30(1)：33-42.
[26]杨海霞.一种阶段结构种群的最优策略[J].科学技术与工程.2010,10(27)：6715-6718.
[27]程惠东,刘洪霞,张伟伟等.一类非自治单种群阶段结构时滞模型的一致生存与全局渐近稳定性[J].山东科技大学学报(自然科学版).2009,28(5)：96-100,104.
[28]李典谟,马祖飞,王正军等.生态模型当前的热点与发展方向[J].生态学报.2002,22(10)：1788-1791.
[29]戴勇,邢铁军,雒志学等.具有三个年龄阶段的自食单种群时滞模型的周期解[J].数学的实践与认识.2011,41(8)：85-91.
[30]张安梅,于书敏.具有年龄结构的单种群模型的脉冲控制及其捕获策略[J].数学的实践与认识.2007,37(16)：151-156.
[31]林琳,雒志学,刘彦平等.具阶段结构两种群捕食模型的渐近行为及最优捕获[J].北华大学学报：自然科学版.2011,12(5)：511-514.
[32]梁志清,周泽文.阶段结构单种群模型的全局渐近稳定性[J].玉林师范学院学报.2004,25(3)：6-8.
[33]于书敏,张安梅.阶段结构单种群捕获的优化策略[J].数学的实践与认识.2008,38(15)：167-173.
[34]李清,王克,范猛等.广义Logistic模型的捕获优化问题[J].生物数学学报.2000,15(4)：408-412.
[35]高淑京.具有阶段结构的自食单种群模型的稳定性[J].鞍山师范学院学报.2002,4(1)：41-43.
[36]梁志清,陈兰荪.具三阶段结构的非自治单种群模型正周期解的存在性[J].华中师范大学学报(自然科学版).2005,39(2)：159-153.
[37]陈兰荪,王东达,杨启昌.阶段结构种群动力学模型[J].北华大学学报(自然科学版).2000,1(3)：185-191.
[38]高淑京,陈兰荪.具有三个成长阶段的单种群时滞模型的永久持续生存和全局稳定性[J].数学物理学报.2006,26(4)：527-533.
[39]Tarzan Legovic, Suncana Gecek. Impact of maximum sustainable yield on inde-pendent populations[J]. Ecological Modelling.2010,221:2108-2111.
[40]Song Xinyu, Chen Lansun. Modelling and Analysis of a Single-Species System with Stage Structure and Harvesting[J]. Mathematical and Computer Modelling. 2002,36:67-82.
[41]Li Wenxue, Wang Ke. Optimal harvesting policy for general stochastic Logistic population model[J]. J.Math.Anal.Appl.2010,368:420-428.
[42]Wang Jing, Wang Ke. Optimal Harvesting Policy for Single Population with Stage Structure[J]. Computers and Mathematics with Applications.2004,48:943-950.
[43]Luis H.R.Alvarez, Erkki Koskela. Optimal harvesting under resource stock and price uncertainty[J]. Journal of Economic Dynamics and Control.2007,31:2461-2485.
[44]Liu Shengqing, Chen Lansun, R.Agarwal. Recent Progress on Stage-Structured Population Dynamics[J]. Mathematical and Computer Modelling.2002,36:1319-1360.
[45]Yasuhisa Saito, Changdo Jung, Yongdo Lim. Effects of cannibalism on a basic stage structure[J]. Applied Mathematics and Computation.2010,217:2133-2141.
[2]张锦炎.常微分方程几何理论与分支问题[M].北京：科学出版社.1990.
[3]唐三一,肖燕妮.单种群生物动力系统[M].北京：科学出版社.2008,6.
[4]陈兰荪,孟新柱,焦建军.生物动力学[M].北京：科学出版社.2009.7.
[5]刘胜强,陈兰荪.阶段结构种群生物模型与研究[M].北京：科学出版社.2010,7.
[6]王顺庆,王万雄,徐海根.数学生态学稳定性理论与方法[M].北京：科学出版社.2004,10.
[7]smith F E. Population dynamics in Daphnia[J]. Eclogy.1963,44:651-663.
[8]Cui Q, Lawson G J. Study on models of single population:an expansion of the logistic and exponential equations[J]. Thoer.Biol..1982,98:645-659.
[9]Allen L J S. Persistence and extinction in single species reaction diffusion model-s[J]. Bull.Math.Biol..1983,45:209-227.
[10]Cui.J, Chen Lansun. The effect of dispersal on population growth with stage-structure[J]. Computers Math Applic.2000,39:91-102.
[11]Gao Shujing. Optimal Harvesting Policy and Stability In A Stage Structured Single-Species Growth Model With Cannibalism[J]. Journal of Biomathematics. 2002,17(2):194-200.
[12]高淑京.具有三个阶段结构单种群模型的全局渐进稳定性[J].新疆大学学报(理工版).2001,18(2)：154-158.
[13]Gao Shujing. Models for Species with Three Life History Stages and Cannibalis-m[J]. Journal of Biomathematics.2005,20(4):385-391.
[14]邢铁军,杜明银,王生丽等.具三个年龄阶段的自食单种群系统的定性分析[J].兰州交通大学学报.2008,27(6)：157-159.
[15]Clark C W. Mathematical Bioeconomics:The Optimal Management of Renewable Resources[M]. New York:Wiley.1976.
[16]范猛,王克,刘会民,张玉娟,张树文.一类具有退偿增长曲线生物种群的捕获优化问题[J].生物数学学报.1997,4(1)：38-42.
[17]刘会民,张树文,张玉娟等.临界退偿系统的捕获优化问题[J].生物数学学.1998,13(4)：479-483.
[18]李海龙.带扩散的Logistic单种群模型及其最优收获策略[J].生物数学学报.1999,14(3)：293-300.
[19]李海龙.一类具周期系数的单种群模型及其最优收获策略[J].生物数学学报.1999,14(4)：479-483.
[20]王静,王克.具有年龄结构的单种群模型单一捕获的优化问题[J].东北师大学报自然科学版.2003,35(2)：1-6.
[21]彭奇林.具年龄结构的单种群模型捕获策略的优化[J].呼伦贝尔学院学报.2004,12(4)：7-9,20.
[22]梁志清,赵强.一类具三阶段结构的单种群模型的最优收获策略[J].玉林师范学院学报(自然科学).2005,26(3)：1-3.
[23]刘毅婧,雒志学.一类3阶段结构自食系统的最优收获策略[J].重庆工学院学报(自然科学版).2007,21(6)：43-45.
[24]鲁红英,王克.自治单种群模型及其最优捕获策略[J].系统科学与数学.2004,24(2)：200-205.
[25]鲁红英,王克.一类自治单种群模型及其最优捕获策略[J].系统科学与数学.2010,30(1)：33-42.
[26]杨海霞.一种阶段结构种群的最优策略[J].科学技术与工程.2010,10(27)：6715-6718.
[27]程惠东,刘洪霞,张伟伟等.一类非自治单种群阶段结构时滞模型的一致生存与全局渐近稳定性[J].山东科技大学学报(自然科学版).2009,28(5)：96-100,104.
[28]李典谟,马祖飞,王正军等.生态模型当前的热点与发展方向[J].生态学报.2002,22(10)：1788-1791.
[29]戴勇,邢铁军,雒志学等.具有三个年龄阶段的自食单种群时滞模型的周期解[J].数学的实践与认识.2011,41(8)：85-91.
[30]张安梅,于书敏.具有年龄结构的单种群模型的脉冲控制及其捕获策略[J].数学的实践与认识.2007,37(16)：151-156.
[31]林琳,雒志学,刘彦平等.具阶段结构两种群捕食模型的渐近行为及最优捕获[J].北华大学学报：自然科学版.2011,12(5)：511-514.
[32]梁志清,周泽文.阶段结构单种群模型的全局渐近稳定性[J].玉林师范学院学报.2004,25(3)：6-8.
[33]于书敏,张安梅.阶段结构单种群捕获的优化策略[J].数学的实践与认识.2008,38(15)：167-173.
[34]李清,王克,范猛等.广义Logistic模型的捕获优化问题[J].生物数学学报.2000,15(4)：408-412.
[35]高淑京.具有阶段结构的自食单种群模型的稳定性[J].鞍山师范学院学报.2002,4(1)：41-43.
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