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Mathematical Analysis of a Prey–Predator Model in Presence of Two Controls

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Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 1440))

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Abstract

In this paper, we develop a two-dimensional Lotka–Volterra prey–predator model based on crop and pest. Pontryagin’s maximum principle is deployed to address the control issue of the prey–predator system, with the interaction of the coefficients of the Lotka–Volterra equations as the main input. The optimal control techniques are utilized to enhance agricultural yields while decreasing the predator population, which harms the crops. Control variables are utilized in two ways: one is applied to the predator, which kills the predator, and the other is applied to the crops, which support plant development and soil fertility. The rate of sprayed pesticides is the first control variable and the second control variable is the regularity of usage of organic biomass fertilizers. Furthermore, we establish the existence of an optimal control, the required work out the criteria for optimality, and then carry out numerical computations.

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References

  1. Al Basir, F., Banerjee, A., Ray, S.: Role of farming awareness in crop pest management—a mathematical model. J. Theor. Biol. 461, 59–67 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abraha, T., Basir, F.A., Obsu, L.L., Torres, D.F.M.: Pest control using farming awareness: impact of time delays and optimal use of biopesticides. Chaos, Solitons Fractals 146, 110869 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  3. Anguelov, R., Dufourd, C., Dumont, Y.: Mathematical model for pest-insect control using mating disruption and trapping. Appl. Math. Model. 52, 437–457 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arora, C., Kumar, V.: Dynamics of a high-dimensional stage-structured prey-predator model. Int. J. Appl. Comput. Math. 3(1), 427–445 (2017)

    Article  MathSciNet  Google Scholar 

  5. Birkoff, G., Rota, G.C.: Ordinary Differential Equations. Ginn, Boston (1982)

    Google Scholar 

  6. Chakraborty, K., Chakraborty, M., Kar, T.: Optimal control of Harvest and bifurcation of a prey–predator model with stage structure. Appl. Math. Comput. 217, 8778–8792 (2011)

    MathSciNet  MATH  Google Scholar 

  7. Danca, M., Codreanu, S., Bako, B.: Detailed analysis of a nonlinear prey–predator model. J. Biol. Phys. 23, 11–20 (1997)

    Article  Google Scholar 

  8. Aktar, M.W., Sengupta, D., Chowdhury, A.: Impact of pesticides use in agriculture: their benefits and hazards. Interdisc. toxicol. 2(1), 1–12 (2009)

    Article  Google Scholar 

  9. Fleming, W.H., Rishel, R.W.: Deterministic and Stochastic Optimal Control. Springer-Verlag, New York (1975)

    Book  MATH  Google Scholar 

  10. Kar, T.K., Batabyal, A.: Stability analysis and optimal control of an SIR epidemic model with vaccination. Biosystems 104, 127–135 (2011)

    Article  Google Scholar 

  11. Kant, S., Kumar, V.: Analysis of an eco–epidemiological model with migrating and refuging prey. Math. Anal. Its Appl., pp. 17–36 (2015)

    Google Scholar 

  12. Lenhart, S., Workman, J.T.: Optimal Control Applied to Biological Models. Mathematical and Computational Biology Series. Chapman & Hall/CRC, London (2007)

    Google Scholar 

  13. Mehra, M., Mallik, R.K.: Solutions of differential difference equations arising from mathematical models of granulocytopoiesis. Differ. Eqn. Dyn. Syst. 22(1), 33–49 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Mehra, M., Kumar, V.: Fast wavelet-Taylor Galerkin method for linear and non-linear wave problems. Appl. Math. Comput. 189(2), 1292–1299 (2007)

    MathSciNet  MATH  Google Scholar 

  15. McCalmont, J.P., Hastings, A., McNamara, N.P., Richter, G.M., Robson, P., Donnison, I.S., Clifton, B.J.: Environmental costs and benefits of growing Miscanthus for bioenergy in the UK. GCB Bioenergy 9(3), 489–507 (2017)

    Google Scholar 

  16. Nassir, S.A.L.: The dynamics and optimal control of a prey-predator system. Glob. J. Pure Appl. Math. 13(9), 5287–5298 (2017)

    Google Scholar 

  17. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Gordon and Breach Science Publishers (1986)

    Google Scholar 

  18. Rafikov, M., Balthazar, J.M., von Bremen, H.F.: Mathematical modeling and control of population systems: applications in biological pest control. Appl. Math. Comput. 200, 557–573 (2008)

    MathSciNet  MATH  Google Scholar 

  19. Ritter, L.: For the ad hoc panel on pesticides and cancer. Report of a panel on the relationship between public exposure to pesticides and cancer. Cancer 80, 2019–2033 (1997)

    Google Scholar 

  20. State of Indian Agriculture, p. 247. Published by Department of Agriculture & Cooperation, Govt. of India, New Delhi (2012–13)

    Google Scholar 

  21. Sasmal, S.K., Mandal, D.S., Chattopadhyay, J.: A predator-pest model with Allee effect and pest culling and additional food provision to the predator-application to pest control. J. Biol. Syst. 25(02), 295–326 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Thind, Singh, T.: Perspectives on crop protection in India. Outlooks Pest Manage. 26(3), 121–127(7) (2015)

    Google Scholar 

  23. Kumar, V., Mehra, M.: Wavelet optimized finite difference method using interpolating wavelets for solving singularly perturbed problems. J. Wavelet Theory Appl. 1(1), 83–96 (2007)

    Google Scholar 

  24. Kumar, V., Srinivasan, B.: A novel adaptive mesh strategy for singularly perturbed parabolic convection diffusion problems. Differ. Eqn. Dyn. Syst. 27(1), 203–220 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  25. Van Driesche, R.G., Bellows, J.T.S.: Biological Control. Chapman and Hall, New York (1996)

    Book  Google Scholar 

  26. Van den Bosh, R., Messenger, P.S., Gutierrez, A.P.: An Introduction to Biological Control. Plenum Press, New York (1982)

    Google Scholar 

  27. Yadav, S., Kumar, V.: Study of a prey-predator model with preventing crop pest using natural enemies and control. AIP Conf. Proc. 2336(1), 020002 (2021)

    Article  Google Scholar 

  28. Yadav, S., Kumar, V.: A prey–predator model and control of a nematodes pest using control in banana: Mathematical modeling and qualitative analysis. Int. J. Biomathematics 2150089 (2021)

    Google Scholar 

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Acknowledgements

The authors are grateful to the reviewers for helpful and relevant suggestions.

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Authors and Affiliations

  1. Department of Applied Mathematics, Delhi Technological University, New Delhi, Delhi, India

    Sudhakar Yadav & Vivek Kumar

Authors
  1. Sudhakar Yadav
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  2. Vivek Kumar
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Corresponding author

Correspondence to Sudhakar Yadav .

Editor information

Editors and Affiliations

  1. Pandit Deendayal Energy University, Gandhinagar, India

    Manoj Sahni

  2. School of Computer Science, University of Technology Sydney, Ultimo, NSW, Australia

    José M. Merigó

  3. Peter Faber Business School, Australian Catholic University, North Sydney, NSW, Australia

    Walayat Hussain

  4. Universidad Católica de la Santísima, Concepcion, Chile

    Ernesto León-Castro

  5. School of Economics and Business, Universidad de Talca, Talca, Chile

    Raj Kumar Verma

  6. Pandit Deendayal Energy University, Gandhinagar, India

    Ritu Sahni

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Yadav, S., Kumar, V. (2023). Mathematical Analysis of a Prey–Predator Model in Presence of Two Controls. In: Sahni, M., Merigó, J.M., Hussain, W., León-Castro, E., Verma, R.K., Sahni, R. (eds) Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy. Advances in Intelligent Systems and Computing, vol 1440. Springer, Singapore. https://doi.org/10.1007/978-981-19-9906-2_15

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