A mathematical study of a prey–predator model in relevance to pest control

In this paper, we propose and analyze an ecological system consisting of pest and its natural enemy as predator. Here we also consider the role of infection to the pest population and the presence of some alternative source of food to the predator population. We analyze the dynamics of this system in a systemic manner, study the dependence of the dynamics on some vital parameters and discuss the global behavior and controllability of the proposed system. The investigation also includes the use of pesticide control to the system and finally we use Pontryagin’s maximum principle to derive the optimal pest control strategy. We also illustrate some of the key findings using numerical simulations.

[1]  D. K. Bhattacharya,et al.  The Role of Viral Infection in Pest Control: A Mathematical Study , 2007, Bulletin of mathematical biology.

[2]  P. Debach,et al.  Biological control of insect pests and weeds , 1967 .

[3]  Sanyi Tang,et al.  An integrated pest management model with delayed responses to pesticide applications and its threshold dynamics , 2012 .

[4]  John T. Workman,et al.  Optimal Control Applied to Biological Models , 2007 .

[5]  Soovoojeet Jana,et al.  A theoretical study on mathematical modelling of an infectious disease with application of optimal control , 2013, Biosyst..

[6]  D. K. Bhattacharya,et al.  Optimization in microbial pest control: An integrated approach , 2010 .

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

[8]  Kunal Chakraborty,et al.  Global dynamics and bifurcation in a stage structured prey-predator fishery model with harvesting , 2012, Appl. Math. Comput..

[9]  P D N Srinivasu,et al.  Role of Quantity of Additional Food to Predators as a Control in Predator–Prey Systems with Relevance to Pest Management and Biological Conservation , 2011, Bulletin of mathematical biology.

[10]  T. K. Kar,et al.  Global dynamics and bifurcation in delayed SIR epidemic model , 2011 .

[11]  P. Hartman.  Ordinary Differential Equations , 1965 .

[12]  Yi Song,et al.  The dynamics of a high-dimensional delayed pest management model with impulsive pesticide input and harvesting prey at different fixed moments , 2011 .

[13]  Soovoojeet Jana,et al.  Dynamics of pest and its predator model with disease in the pest and optimal use of pesticide. , 2012, Journal of theoretical biology.

[14]  J. C. Burkill,et al.  Ordinary Differential Equations , 1964 .

[15]  E. Akpanyung,et al.  Endosulfan-Induces Renal Toxicity Independent of the Route of Exposure in Rats , 2011 .

[16]  R. May,et al.  Regulation and Stability of Host-Parasite Population Interactions: I. Regulatory Processes , 1978 .

[17]  Carl J. Walters,et al.  Invulnerable Prey and the Paradox of Enrichment , 1996 .

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

[19]  T. K. Kar,et al.  Sustainability and optimal control of an exploited prey predator system through provision of alternative food to predator , 2012, Biosyst..

[20]  S. Bhattacharyya,et al.  Pest control through viral disease: mathematical modeling and analysis. , 2006, Journal of theoretical biology.

[21]  Ezio Venturino,et al.  Epidemics in predator-prey models: disease in the predators. , 2002, IMA journal of mathematics applied in medicine and biology.

[22]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[23]  Soovoojeet Jana,et al.  Modeling and analysis of a prey–predator system with disease in the prey , 2013 .

[24]  Xia Wang,et al.  Mathematical models for the control of a pest population by infected pest , 2008, Comput. Math. Appl..

[25]  Xinyu Song,et al.  Analysis of pest-epidemic model by releasing diseased pest with impulsive transmission , 2011 .

[26]  N. Apreutesei,et al.  An optimal control problem for a pest, predator, and plant system , 2012 .

[27]  H. D. Burges,et al.  Microbial control of insects and mites , 1971 .

[28]  Colin W. Clark,et al.  Mathematical Bioeconomics: The Optimal Management of Renewable Resources. , 1993 .

[29]  Li Changguo,et al.  Effect of delay on a predator–prey model with parasitic infection , 2011 .

[30]  Chaoyang Wang,et al.  Bifurcation of a predator–prey model with disease in the prey , 2010 .

[31]  Lansun Chen,et al.  A delayed epidemic model with stage-structure and pulses for pest management strategy , 2008 .

[32]  J. Harwood,et al.  The role of alternative prey in sustaining predator populations. , 2005 .

[33]  Suzanne Lenhart,et al.  Optimal control applied to a model for species augmentation. , 2008, Mathematical biosciences and engineering : MBE.

[34]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[35]  Alka Gupta,et al.  Control of the Coconut Pest Oryctes Rhinoceros L. Using the Oryctes Virus , 2001 .

[36]  P.C.J. van Rijn,et al.  When does alternative food promote biological pest control , 2005 .

[37]  Dahlard L. Lukes,et al.  Differential Equations: Classical to Controlled , 2012 .

[38]  T. K. Kar,et al.  Stability analysis and optimal control of an SIR epidemic model with vaccination , 2011, Biosyst..

[39]  P D N Srinivasu,et al.  Biological control through provision of additional food to predators: a theoretical study. , 2007, Theoretical population biology.

[40]  H. I. Freedman,et al.  Predator-prey populations with parasitic infection , 1989, Journal of mathematical biology.

[41]  T. K. Kar,et al.  Conservation of a resource based fishery through optimal taxation , 2013 .

[42]  Hem Raj Joshi,et al.  Optimal control of an HIV immunology model , 2002 .

[43]  Tieying Wang,et al.  Nonlinear analysis of a microbial pesticide model with impulsive state feedback control , 2011 .

[44]  L. Bittner.  L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishechenko, The Mathematical Theory of Optimal Processes. VIII + 360 S. New York/London 1962. John Wiley & Sons. Preis 90/– , 1963 .

[45]  Yongzhen Pei,et al.  Pest regulation by means of continuous and impulsive nonlinear controls , 2010, Math. Comput. Model..

[46]  Lansun Chen,et al.  Time-limited pest control of a Lotka–Volterra model with impulsive harvest , 2009 .

[47]  Roy M. Anderson,et al.  The Population Dynamics of Microparasites and Their Invertebrate Hosts , 1981 .

[48]  Edouard Kurstak,et al.  Microbial and viral pesticides. , 1982 .

[49]  Soovoojeet Jana,et al.  Application of three controls optimally in a vector-borne disease - a mathematical study , 2013, Commun. Nonlinear Sci. Numer. Simul..

[50]  Suzanne Lenhart,et al.  Optimal control of treatments in a two-strain tuberculosis model , 2002 .

[51]  Yuanshun Tan,et al.  Modelling approach for biological control of insect pest by releasing infected pest , 2009 .