害虫治理SI模型的最优脉冲控制策略
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摘要
考虑到农药的副作用,释放有病害虫作为一种有价值的非化学工具在害虫治理的过程中变得越来越重要.受到Xiang(2009)和Bhattacharyya等人(2006)工作的启发,该文研究了一类害虫管理SI传染病模型.虽然该模型具有多种动力学行为但易感害虫不能灭绝.为此,在该模型中引入多次脉冲干预措施,得到了易感害虫灭绝周期解全局渐近稳定的充分条件.然而,从生态和经济方面来说,让易感害虫灭绝的策略是不可取的,这是因为田间适当数量的害虫对于保护天敌,以及维持农作物的经济补偿是有益的.因此,用最小的成本最小化害虫在终端时刻的数量,基于不同的控制策略,三种最优害虫控制问题被详细研究.通过时间缩放和时间平移变换的方法,计算了目标函数关于脉冲时间间隔,农药致死率和病虫释放量的梯度,这对获得最优害虫控制策略是至关重要的.最后,数值模拟的结果显示,与其他两种策略相比,非固定时刻的交替综合控制策略是最有效的.另外,通过对比发现,该文提出的策略比害虫灭绝策略更可取.
In view of the side effects,the technique relying on diseased pest releases as a valuable non-chemical tool is getting much more essentiality in pest management.Inspired by Xiang(2009)and Bhattacharyya et al(2006),the present thesis firstly focuses on a susceptible and infected pest model for pest management,which possesses multiple dynamic behaviors but does not eradicate susceptible individuals.For eliminating the pests,human impulsive interventions are embroiled in this model.Then the sufficient conditions for the global asymptotic stability of the susceptible pest-eradication periodic solution are established by unlimited pulse interventions.However,the strategy driving susceptible pests to extinction is unadvisable from ecological and economical aspects since the appropriate amount of pests in the field is beneficial for conservation of natural enemies and maintaining the crop overcompensation after pest injury.Hence,three different optimal problems involving different pest control tactics are deliberated in order to diminish the susceptible population at the terminal time and keep this in balance with the cost of the intervention(control).Subsequently,by time scaling and translation transformation techniques,the gradients for the cost functional on durations,fractions of susceptible pests killed due to chemical sprays as well as the number of infected pest released at each impulsive intervention moment are computed,which are vital to capture the optimal control strategy for pest regulation.Finally,on the basis of simulations,the strategy of alternative integrated control at unfixed time is validated to be the most effective compared with the other two policies.In addition,by comparing our optimal strategy with pest-extinction one,it is revealed that our strategy is more desirable.
In view of the side effects,the technique relying on diseased pest releases as a valuable non-chemical tool is getting much more essentiality in pest management.Inspired by Xiang(2009)and Bhattacharyya et al(2006),the present thesis firstly focuses on a susceptible and infected pest model for pest management,which possesses multiple dynamic behaviors but does not eradicate susceptible individuals.For eliminating the pests,human impulsive interventions are embroiled in this model.Then the sufficient conditions for the global asymptotic stability of the susceptible pest-eradication periodic solution are established by unlimited pulse interventions.However,the strategy driving susceptible pests to extinction is unadvisable from ecological and economical aspects since the appropriate amount of pests in the field is beneficial for conservation of natural enemies and maintaining the crop overcompensation after pest injury.Hence,three different optimal problems involving different pest control tactics are deliberated in order to diminish the susceptible population at the terminal time and keep this in balance with the cost of the intervention(control).Subsequently,by time scaling and translation transformation techniques,the gradients for the cost functional on durations,fractions of susceptible pests killed due to chemical sprays as well as the number of infected pest released at each impulsive intervention moment are computed,which are vital to capture the optimal control strategy for pest regulation.Finally,on the basis of simulations,the strategy of alternative integrated control at unfixed time is validated to be the most effective compared with the other two policies.In addition,by comparing our optimal strategy with pest-extinction one,it is revealed that our strategy is more desirable.
引文
[1]Pei Y,Ji X,Li C.Pest regulation by means of continuous and impulsive nonlinear controls.Math Comput Model,2010,51:810-822
[2]Pei Y,Li C,Fan S.A mathematical model of a three species prey-predator system with impulsive control and holling functional response.Appl Math Comput,2013,219:10945-10955
[3]Barclay H.Combining methods of pest control:Minimizing cost during the control program.Theor Popul Biol,1991,40:105-123
[4]Tang S,Cheke R.Models for integrated pest control and their biological implications.Math Biosci,2008,215:115-125
[5]Lenteren J.Success in Biological Control of Arthropods by Augmentation of Natural Enemies.Berlin:Springer,2000
[6]Naranjo S,Ellsworth P,Frisvold G.Economic value of biological control in integrated pest management of managed plant systems.Annu Rev Entomol,2015,60:621-645
[7]Molnar S,LaPez I,Gamez M,Garay J.A two-agent model applied to the biological control of the sugarcane borer(diatraea saccharalis)by the egg parasitoid trichogramma galloi and the larvae parasitoid cotesia flavipes.Biosystems,2016,141:45-54
[8]Song Y,Pei Y,Chen M,Zhu M.Translation,solving scheme,and implementation of a periodic and optimal impulsive state control problem.Adv Differ Equ-Ny,2018,2018:1-20
[9]Jiao J,Chen L.A pest management SI model with periodic biological and chemical control concern.Appl Math Comput,2006,183:1018-1026
[10]Wang L,Chen L,Nieto J.The dynamics of an epidemic model for pest control with impulsive effect.Nonlinear Anal-R;eal,2010,11:1374-1386
[11]Xiang Z.Dynamic analysis of a SI system with periodic biological and chemical control.Applied Mathematical Sciences,2009,3:2327-2334
[12]Shi R,Jiang X,Chen L.A predator-prey model with disease in the prey and two impulses for integrated pest management.Appl Math Model,2009,33:2248-2256
[13]Zhang H,Xu W,Chen L.A impulsive infective transmission SI model for pest control.Math Method Appl Ssi,2007,30:1169-1184
[14]Jiao J,Chen L,Cai S.Impulsive control strategy of a pest management model with nonlinear incidence rate.Appl Math Model,2009,33:555-563
[15]Tan Y,Chen L.Modelling approach for biological control of insect pest by releasing infected pest.Chaos Soliton Fract,2009,39:304-315
[16]Shi R,Chen L.An impulsive predator-prey model with disease in the prey for integrated pest management.Commun Nonlinear Sci,2010,15:421-429
[17]Liu B,Wang Y,Kang B.Dynamics on a pest management SI model with control strategies of different frequencies.Nonlinear Anal-Hybri,2014,12:66-78
[18]Meng X,Song Z,Chen L.A new mathematical model for optimal control strategies of integrated pest management.J Biol Syst,2007,15:219-234
[19]Ding Y,Gao S,Liu Y,Lan Y.A pest management epidemic model with time delay and stage-structure.Applied Mathematics,2010,1:216-221
[20]Zhang T,Meng X,Song Y,Zhang T.A stage-structured predator-prey SI model with disease in the prey and impulsive effects.Math Model Anal,2013,18:505-528
[21]Xiang Z,Tang S,Xiang C,Wu J.On impulsive pest control using integrated intervention strategies.Appl Math Comput,2015,269:930-946
[22]Wang X,Tao Y,Song X.Analysis of pest-epidemic model by releasing diseased pest with impulsive transmission.Nonlinear Dynam,2011,65:175-185
[23]Guan H.Aperiodic solution mathematical model for pest management and optimization chemical control.Adv Mater Res,2014,1079-1080:660-663
[24]Jiao J,Chen L.Nonlinear incidence rate of a pest management SI model with biological and chemical control concern.Appl Math Mech-Engl,2007,28:541-551
[25]Pei Y,Chen M,Liang X,etal.Optimal control problem in an epidemic disease SIS model with stages and delays.Int J Biomath,2016,9:131-152
[26]Tang S,Liang J,Tan Y,Cheke R.Threshold conditions for integrated pest management models with pesticides that have residual effects.J Math Biol,2013,66:1-35
[27]Kamgarpour M,Tomlin C.On optimal control of non-autonomous switched systems with a fixed mode sequence.Automatica,2012,48:1177-1181
[28]Xu X,Antsaklis P.Optimal control of switched systems based on parameterization of the switching instants.IEEE T Automat Contr,2004,49:2-16
[29]Wu C,Teo K.Global impulsive optimal control computation.J Ind Manag Optim,2006,2:435-450
[30]Teo K.Control parametrization enhancing transform to optimal control problems.Nonlinear Analysis,2005,63:e2223-e2236
[31]Liu Y,Teo K,Jennings L,Wang S.On a class of optimal control problems with state jumps.J Optimiz Theory App,1998,98:65-82
[32]Loxton R,Lin Q,Teo K.Switching time optimization for nonlinear switched systems:Direct optimization and the time-scaling transformation.Pac J Optim,2014,10:537-560
[33]Dykhta V.Second order necessary optimality conditions for impulse control problem and multiprocesses.IFAC Proceedings Volumes,1997,30:97-101
[34]Chen M,Pei Y,Liang X,et al.A hybrid optimization problem at characteristic times and its application in agroecological system.Adv Differ Equ-Ny,2016,2016:1-13
[35]Bhattacharyya S,Bhattacharya D.Pest control through viral disease:mathematical modeling and analysis.J Theor Biol,2006,238:177-197
[2]Pei Y,Li C,Fan S.A mathematical model of a three species prey-predator system with impulsive control and holling functional response.Appl Math Comput,2013,219:10945-10955
[3]Barclay H.Combining methods of pest control:Minimizing cost during the control program.Theor Popul Biol,1991,40:105-123
[4]Tang S,Cheke R.Models for integrated pest control and their biological implications.Math Biosci,2008,215:115-125
[5]Lenteren J.Success in Biological Control of Arthropods by Augmentation of Natural Enemies.Berlin:Springer,2000
[6]Naranjo S,Ellsworth P,Frisvold G.Economic value of biological control in integrated pest management of managed plant systems.Annu Rev Entomol,2015,60:621-645
[7]Molnar S,LaPez I,Gamez M,Garay J.A two-agent model applied to the biological control of the sugarcane borer(diatraea saccharalis)by the egg parasitoid trichogramma galloi and the larvae parasitoid cotesia flavipes.Biosystems,2016,141:45-54
[8]Song Y,Pei Y,Chen M,Zhu M.Translation,solving scheme,and implementation of a periodic and optimal impulsive state control problem.Adv Differ Equ-Ny,2018,2018:1-20
[9]Jiao J,Chen L.A pest management SI model with periodic biological and chemical control concern.Appl Math Comput,2006,183:1018-1026
[10]Wang L,Chen L,Nieto J.The dynamics of an epidemic model for pest control with impulsive effect.Nonlinear Anal-R;eal,2010,11:1374-1386
[11]Xiang Z.Dynamic analysis of a SI system with periodic biological and chemical control.Applied Mathematical Sciences,2009,3:2327-2334
[12]Shi R,Jiang X,Chen L.A predator-prey model with disease in the prey and two impulses for integrated pest management.Appl Math Model,2009,33:2248-2256
[13]Zhang H,Xu W,Chen L.A impulsive infective transmission SI model for pest control.Math Method Appl Ssi,2007,30:1169-1184
[14]Jiao J,Chen L,Cai S.Impulsive control strategy of a pest management model with nonlinear incidence rate.Appl Math Model,2009,33:555-563
[15]Tan Y,Chen L.Modelling approach for biological control of insect pest by releasing infected pest.Chaos Soliton Fract,2009,39:304-315
[16]Shi R,Chen L.An impulsive predator-prey model with disease in the prey for integrated pest management.Commun Nonlinear Sci,2010,15:421-429
[17]Liu B,Wang Y,Kang B.Dynamics on a pest management SI model with control strategies of different frequencies.Nonlinear Anal-Hybri,2014,12:66-78
[18]Meng X,Song Z,Chen L.A new mathematical model for optimal control strategies of integrated pest management.J Biol Syst,2007,15:219-234
[19]Ding Y,Gao S,Liu Y,Lan Y.A pest management epidemic model with time delay and stage-structure.Applied Mathematics,2010,1:216-221
[20]Zhang T,Meng X,Song Y,Zhang T.A stage-structured predator-prey SI model with disease in the prey and impulsive effects.Math Model Anal,2013,18:505-528
[21]Xiang Z,Tang S,Xiang C,Wu J.On impulsive pest control using integrated intervention strategies.Appl Math Comput,2015,269:930-946
[22]Wang X,Tao Y,Song X.Analysis of pest-epidemic model by releasing diseased pest with impulsive transmission.Nonlinear Dynam,2011,65:175-185
[23]Guan H.Aperiodic solution mathematical model for pest management and optimization chemical control.Adv Mater Res,2014,1079-1080:660-663
[24]Jiao J,Chen L.Nonlinear incidence rate of a pest management SI model with biological and chemical control concern.Appl Math Mech-Engl,2007,28:541-551
[25]Pei Y,Chen M,Liang X,etal.Optimal control problem in an epidemic disease SIS model with stages and delays.Int J Biomath,2016,9:131-152
[26]Tang S,Liang J,Tan Y,Cheke R.Threshold conditions for integrated pest management models with pesticides that have residual effects.J Math Biol,2013,66:1-35
[27]Kamgarpour M,Tomlin C.On optimal control of non-autonomous switched systems with a fixed mode sequence.Automatica,2012,48:1177-1181
[28]Xu X,Antsaklis P.Optimal control of switched systems based on parameterization of the switching instants.IEEE T Automat Contr,2004,49:2-16
[29]Wu C,Teo K.Global impulsive optimal control computation.J Ind Manag Optim,2006,2:435-450
[30]Teo K.Control parametrization enhancing transform to optimal control problems.Nonlinear Analysis,2005,63:e2223-e2236
[31]Liu Y,Teo K,Jennings L,Wang S.On a class of optimal control problems with state jumps.J Optimiz Theory App,1998,98:65-82
[32]Loxton R,Lin Q,Teo K.Switching time optimization for nonlinear switched systems:Direct optimization and the time-scaling transformation.Pac J Optim,2014,10:537-560
[33]Dykhta V.Second order necessary optimality conditions for impulse control problem and multiprocesses.IFAC Proceedings Volumes,1997,30:97-101
[34]Chen M,Pei Y,Liang X,et al.A hybrid optimization problem at characteristic times and its application in agroecological system.Adv Differ Equ-Ny,2016,2016:1-13
[35]Bhattacharyya S,Bhattacharya D.Pest control through viral disease:mathematical modeling and analysis.J Theor Biol,2006,238:177-197
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