含有保护区的两斑块鱼类生物经济模型
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摘要
研究了同一鱼类种群同时分布于保护区斑块和非保护区斑块,且鱼群密度服从Gompertz增长的生物经济系统.首先,证明了系统内存在唯一的非平凡奇点并且局部稳定,运用Bendixson-Dulac定理得到该奇点的全局稳定性;然后,通过Pontryagin最大值原理获得该鱼类种群的最优捕捞路径;最后,通过数值模拟来说明部分结论.
In order to protect the population on the verge of extinction and sustain the biodiversity of marine ecosystem,marine protected area(MPA)is established to provide a refuge for fishery resource.However,the influence of current harvesting policy on the MPA is still poorly understood.We consider a bioeconomic Gompertz population model in a two-patch environment:a free fishing zone and a protected zone where fishing is strictly prohibited.First,we have proved the existence of the equilibrium in which it is also local stable.In virtue of Bendixson-dulac theorem,the global stability of the nontrivial steady state is obtained.Then,the optimal harvesting policy is established by use of Pontryagin's maximum principle.In the final,the results are illustrated with the help of some numerical examples.
In order to protect the population on the verge of extinction and sustain the biodiversity of marine ecosystem,marine protected area(MPA)is established to provide a refuge for fishery resource.However,the influence of current harvesting policy on the MPA is still poorly understood.We consider a bioeconomic Gompertz population model in a two-patch environment:a free fishing zone and a protected zone where fishing is strictly prohibited.First,we have proved the existence of the equilibrium in which it is also local stable.In virtue of Bendixson-dulac theorem,the global stability of the nontrivial steady state is obtained.Then,the optimal harvesting policy is established by use of Pontryagin's maximum principle.In the final,the results are illustrated with the help of some numerical examples.
引文
[1]谢丛波,范猛.种群服从Gompertz增长的数学生物经济模型的定性分析[J].东北师大学报自然科学版,2003,35(1):10-17.
[2]Hart D R,Jacobson L D,and Tang et al.To split or not to split:Assessment of Georges Bank sea scallops in the presence of marine protected areas[J].Fisheries Research,2013,144:74-83.
[3]Eide A.A bioeconomic MPA study based on cellular automata population growth and distribution[J].Fisheries Research,2012,113:118-132.
[4]Srinvasu P D N,Gayitri I L.Influence of prey reserve capacity on predator-prey dynamics[J].Ecological Modelling,2005,181:192-202.
[5]Pulliam H R.Sources,sinks and population regulation[J].The American Naturalist,1988,132:652-661.
[6]Hilborn R,Micheli et al.Integrating marine protected areas with catch regulation[J].Canadian Journal of Fisheries and Aquatic Sciences,2006,63:642-649.
[7]Kar,T.K.,Matsuda et al.A bioeconomic model of a single-species fishery with a marine reserve[J].Journal of environmental Management,2008,86:171-180.
[8]Dubey B,Chandra et al.A model for fishery resource with reserve area,Nonlinear AnaIysis[J].Real World Applications,2003,4:625-637.
[9]Zou X,Wang K.The protection zone of biological population,Nonlinear Analysis[J].Real World Applications,2011,12:956-964.
[10]Kax T K,Misra et al.Influence of prey reserve in prey-predator fishery[J].Nonlinear Analysis,2006,65:1725-1735.
[11]Olivares E G,Arcos J H.A two-patch model for the optimal management of a fishing resource considering a marine protected area,Nonlinear Analysis[J].Real World Applications,2011,12:2489-2499.
[12]Purohit D,Chaudhuri et al.A bioeconomic model of nonselective harvesting of two competing fish species[J].ANZIAN Journal,2004,46:299-308.12.
[13]Kar T K.Influence of environmental noises on the Gompertz model of two species fishery[J].Ecological Modelling,2004,173:283-293.
[14]Purohit D,Chaudhuri et al.Nonselective harvesting of a prey-predator fishery with Gompertz law of growth[J].International Journal of Mathematical Education in Science and Technology,2002,33(5):671-678.
[15]Pradhan T,Chaudhuri et al.Bioeconomic modelling of a single species fishery with Gompertz law of growth[J].Journal of Biological Systems,1998,6(4):393-409.
[16]Zhang Y J,Xiu Z Let al.Optimal impulsive harvesting of a single-species with Gompertz law of growth(J).Journal of Biological Systems,2006,14(2):303-314.
[17]叶彦谦,丁同仁,黄文灶.极限环论(第2版)[M].上海:上海科学技术出版社,1984:375-387.
[2]Hart D R,Jacobson L D,and Tang et al.To split or not to split:Assessment of Georges Bank sea scallops in the presence of marine protected areas[J].Fisheries Research,2013,144:74-83.
[3]Eide A.A bioeconomic MPA study based on cellular automata population growth and distribution[J].Fisheries Research,2012,113:118-132.
[4]Srinvasu P D N,Gayitri I L.Influence of prey reserve capacity on predator-prey dynamics[J].Ecological Modelling,2005,181:192-202.
[5]Pulliam H R.Sources,sinks and population regulation[J].The American Naturalist,1988,132:652-661.
[6]Hilborn R,Micheli et al.Integrating marine protected areas with catch regulation[J].Canadian Journal of Fisheries and Aquatic Sciences,2006,63:642-649.
[7]Kar,T.K.,Matsuda et al.A bioeconomic model of a single-species fishery with a marine reserve[J].Journal of environmental Management,2008,86:171-180.
[8]Dubey B,Chandra et al.A model for fishery resource with reserve area,Nonlinear AnaIysis[J].Real World Applications,2003,4:625-637.
[9]Zou X,Wang K.The protection zone of biological population,Nonlinear Analysis[J].Real World Applications,2011,12:956-964.
[10]Kax T K,Misra et al.Influence of prey reserve in prey-predator fishery[J].Nonlinear Analysis,2006,65:1725-1735.
[11]Olivares E G,Arcos J H.A two-patch model for the optimal management of a fishing resource considering a marine protected area,Nonlinear Analysis[J].Real World Applications,2011,12:2489-2499.
[12]Purohit D,Chaudhuri et al.A bioeconomic model of nonselective harvesting of two competing fish species[J].ANZIAN Journal,2004,46:299-308.12.
[13]Kar T K.Influence of environmental noises on the Gompertz model of two species fishery[J].Ecological Modelling,2004,173:283-293.
[14]Purohit D,Chaudhuri et al.Nonselective harvesting of a prey-predator fishery with Gompertz law of growth[J].International Journal of Mathematical Education in Science and Technology,2002,33(5):671-678.
[15]Pradhan T,Chaudhuri et al.Bioeconomic modelling of a single species fishery with Gompertz law of growth[J].Journal of Biological Systems,1998,6(4):393-409.
[16]Zhang Y J,Xiu Z Let al.Optimal impulsive harvesting of a single-species with Gompertz law of growth(J).Journal of Biological Systems,2006,14(2):303-314.
[17]叶彦谦,丁同仁,黄文灶.极限环论(第2版)[M].上海:上海科学技术出版社,1984:375-387.