Dynamics analysis and control optimization of a pest management predator-prey model with an integrated control strategy

This work presents a new integrated pest management predator-prey model.It has shown the existence of the order-1 periodic orbit for the proposed model.It has verified the stability of the order-1 periodic orbit by geometric method.It has obtained optimal pest control level by an optimization problem. Pest management is a complex issue in real applications, and a practical program in pest control in general involves two pest thresholds, where the biological control and chemical control are activated respectively. Aiming at providing a good balance between the biological control and chemical control, this work presented an integrated pest management predator-prey model, where the yield of releases of predator and the strength of pesticide spraying are linearly dependent on the selected control level. Firstly, to determine the frequency of spraying chemical pesticide and releasing of predator, the existence of the order-1 periodic orbit of the proposed model is discussed by the successor function method. And then, to ensure a certain robustness of adopted control, the stability of the order-1 periodic orbit is verified by a stability criterion extracted for a general semi-continuous dynamical system. In addition, to minimize the total cost (i.e. culturing predators and spraying pesticide) in pest control, an optimization problem is formulated and the optimum pest control level is obtained. At last, to complement the theoretical results, the numerical simulations with a specific model are carried out step by step.

[1]  Frédéric Grognard,et al.  Global stability and optimisation of a general impulsive biological control model. , 2008, Mathematical biosciences.

[2]  P. S. Simeonov,et al.  Orbital stability of periodic solutions of autonomous systems with impulse effect , 1988 .

[3]  Alan J. Terry,et al.  Biocontrol in an impulsive predator-prey model. , 2014, Mathematical biosciences.

[4]  C. S. Holling,et al.  Qualitative Analysis of Insect Outbreak Systems: The Spruce Budworm and Forest , 1978 .

[5]  Chen Lan-sun.  Pest Control and Geometric Theory of Semi-Continuous Dynamical System , 2011 .

[6]  Sanyi Tang,et al.  Modelling and analysis of integrated pest management strategy , 2004 .

[7]  Yongfeng Li,et al.  Complex dynamics of a predator-prey model with impulsive state feedback control , 2014, Appl. Math. Comput..

[8]  F. Grognard,et al.  Two models of interfering predators in impulsive biological control , 2010, Journal of biological dynamics.

[9]  Lansun Chen,et al.  The dynamics of an epidemic model for pest control with impulsive effect , 2010 .

[10]  Guirong Jiang,et al.  Impulsive state feedback control of a predator-prey model , 2007 .

[11]  Sanyi Tang,et al.  Threshold conditions for integrated pest management models with pesticides that have residual effects , 2013, Journal of mathematical biology.

[12]  D. O’Regan,et al.  Variational approach to impulsive differential equations , 2009 .

[13]  Sanyi Tang,et al.  On impulsive pest control using integrated intervention strategies , 2015, Appl. Math. Comput..

[14]  Paul Georgescu,et al.  IMPULSIVE CONTROL STRATEGIES FOR PEST MANAGEMENT , 2007 .

[15]  Tonghua Zhang,et al.  Periodic solution of a prey-predator model with nonlinear state feedback control , 2015, Appl. Math. Comput..

[16]  Qingling Zhang,et al.  The geometrical analysis of a predator-prey model with two state impulses. , 2012, Mathematical biosciences.

[17]  Guirong Jiang,et al.  Complex dynamics of a Holling type II prey–predator system with state feedback control , 2007 .

[18]  Frédéric Grognard,et al.  Optimal Release Policy for Prophylactic Biological Control , 2006 .

[19]  Márcia Federson,et al.  Limit sets and the Poincare-Bendixson Theorem in impulsive semidynamical systems , 2008 .

[20]  Sanyi Tang,et al.  Integrated pest management models and their dynamical behaviour , 2005, Bulletin of mathematical biology.

[21]  Paul Georgescu,et al.  An impulsively controlled predator–pest model with disease in the pest ☆ , 2010 .

[22]  Sanyi Tang,et al.  Global dynamics of a state-dependent feedback control system , 2015, Advances in Difference Equations.

[23]  J. C. van Lenteren,et al.  Integrated pest management in protected crops. , 1995 .

[24]  Yanni Xiao,et al.  The dynamics of an eco-epidemic model with biological control , 2003 .

[25]  Lansun Chen,et al.  On the impulsive controllability and bifurcation of a predator-pest model of IPM , 2008, Biosyst..

[26]  X. Yao,et al.  [Quantitative relationships between leaf total nitrogen concentration and canopy reflectance spectra of rice]. , 1982, Ying yong sheng tai xue bao = The journal of applied ecology.

[27]  Lansun Chen,et al.  Modelling and qualitative analysis of a predator-prey system with state-dependent impulsive effects , 2011, Math. Comput. Simul..

[28]  Sanyi Tang,et al.  Optimum timing for integrated pest management: modelling rates of pesticide application and natural enemy releases. , 2010, Journal of theoretical biology.

[29]  Zhidong Teng,et al.  Existence and stability of periodic solution of a predator-prey model with state-dependent impulsive effects , 2009, Math. Comput. Simul..

[30]  Frédéric Grognard,et al.  Influence of Intrapredatory Interferences on Impulsive Biological Control Efficiency , 2010, Bulletin of mathematical biology.

[31]  Sanyi Tang,et al.  Holling II predator–prey impulsive semi-dynamic model with complex Poincaré map , 2015 .

[32]  Sanyi Tang,et al.  State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences , 2005, Journal of mathematical biology.

[33]  Sanyi Tang,et al.  Sliding Bifurcations of Filippov Two Stage Pest Control Models with Economic Thresholds , 2012, SIAM J. Appl. Math..

[34]  Frédéric Grognard,et al.  Natural enemies deployment in patchy environments for augmentative biological control , 2015, Appl. Math. Comput..

[35]  Rui Liu,et al.  Periodic solution of a pest management Gompertz model with impulsive state feedback control , 2014 .

[36]  Zhidong Teng,et al.  Existence and stability of periodic solution of a Lotka-Volterra predator-prey model with state dependent impulsive effects , 2009 .

[37]  Lansun Chen,et al.  Comment on "Existence and stability of periodic solution of a Lotka-Volterra predator-prey model with state dependent impulsive effects" [J. Computational Applied Mathematics 224 (2009) 544-555] , 2010, J. Comput. Appl. Math..

[38]  Tonghua Zhang,et al.  Theoretical study and control optimization of an integrated pest management predator-prey model with power growth rate. , 2016, Mathematical biosciences.

[39]  Yuan Tian,et al.  Geometric approach to the stability analysis of the periodic solution in a semi-continuous dynamic system , 2014 .

[40]  J. C. van Lenteren,et al.  Environmental manipulation advantageous to natural enemies of pests. , 1987 .

[41]  Ruiqing Shi,et al.  A predator-prey model with disease in the prey and two impulses for integrated pest management , 2009 .