Dynamical analysis of a pest management Leslie–Gower model with ratio-dependent functional response

Original Paper
  • 25 Downloads

Abstract

Agricultural pests are great threat for agricultural production, and the development of effective pest control methods is becoming an interesting topic and attracts great attentions. In this work, an integrated pest management model with Leslie–Gower type and ratio-dependent functional response is investigated, and the sufficient conditions for the existence and stability of the order-1 periodic solution are obtained by applying successor function method and analogue of Poincaré criterion. Meanwhile, a cost minimization model by means of the order-1 periodic solution is formulated to determine the optimal control level. The theoretical results are verified by computer simulations for two specified models, and it indicates that the proposed control strategy could keep the pest below the economic level. In addition, to verify the complex dynamics of the proposed model, an order-2 periodic solution and an order-3 periodic solution are obtained by adjusting one key control parameter in the simulations.

Keywords

Integrated pest management Leslie–Gower model Periodic solution Ratio-dependent Stability 

Notes

Acknowledgements

This work is supposed by the National Natural Science Foundation of China (Nos.: 11671346, 11371306, 11401068), the Project for Science and Technology Open Cooperation of Henan Province (172106000071) and Nanhu Scholars Program of XYNU.

Compliance with ethical standards

Conflict of interests

The authors declare that there is no conflict of interests.

References

  1. 1.
    Liu, B., Zhang, Y., Chen, L.: Dynamic complexities of a Holling I predator–prey model concerning periodic biological and chemical control. Chaos Solitons Fractals 22, 123–134 (2004)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Liu, B., Teng, Z., Chen, L.: Analysis of a predator–prey model with Holling II functional response concerning impulsive control strategy. J. Comput. Appl. Math. 193, 347–362 (2006)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Li, Z., Wang, W., Wang, H.: The dynamics of a Beddington-type system with impulsive control strategy. Chaos Solitons Fractals 29, 1229–1239 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Wang, X.Q., Wang, W.M., Lin, X.L.: Chaotic behavior of a Watt-type predator–prey system with impulsive control strategy. Chaos Solitons Fractals 37, 706–718 (2008)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Hsu, S.B., Hwang, T.W., Kuang, Y.: Global analysis of the Michaelis–Menten type ratio-dependence predator prey system. J. Math. Biol. 42, 489–506 (2001)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Liang, Z., Pan, H.: Qualitative analysis of a ratio-dependent Holling–Tanner model. J. Math. Anal. Appl. 334(2), 954–964 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Hsu, S.B., Hwang, T.W.: Global stability for a class of predator–prey systems. SIAM J. Appl. Math. 55, 763–783 (1995)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Saez, E., Gonzalez-Olivares, E.: Dynamics of predator–prey model. SIAM J. Appl. Math. 59, 1867–1878 (1999)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Guo, H., Song, X.: An impulsive predator–prey system with modified Leslie–Gower and Holling type II schemes. Chaos Solitons Fractals 36(5), 1320–1331 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Liu, B., Zhang, Y., Chen, L.: The dynamical behaviors of a Lotka–Volterra predator–prey model concerning integrated pest management. Nonlinear Anal. Real World Appl. 6(2), 227–243 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Zhang, T., Meng, X., Song, Y.: The dynamics of a high-dimensional delayed pest management model with impulsive pesticide input and harvesting prey at different fixed moments. Nonlinear Dyn. 64(1), 1–12 (2011)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Bale, J.S., Lenteren, J.C.V., Bigler, F.: Biological control and sustainable food production. Philos. Trans. R. Soc. Lond. 363, 761–776 (2008)CrossRefGoogle Scholar
  13. 13.
    Zhang, Y.J., Liu, B., Chen, L.S.: Extinction and permanence of a two-prey one-predator system with impulsive effect. Math. Med. Biol. 20(4), 309–325 (2003)CrossRefMATHGoogle Scholar
  14. 14.
    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. 33(5), 2248–2256 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Sun, S., Chen, L.: Mathematical modelling to control a pest population by infected pests. Appl. Math. Model. 33(6), 2864–2873 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Wang, X., Tao, Y., Song, X.: Mathematical model for the control of a pest population with impulsive perturbations on diseased pest. Appl. Math. Model. 33(7), 3099–3106 (2009)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Wang, L., Chen, L., Nieto, J.J.: The dynamics of an epidemic model for pest control with impulsive effect. Nonlinear Anal. Real World Appl. 11(3), 1374–1386 (2010)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lu, Z.H., Chi, X.B., Chen, L.S.: Impulsive control strategies in biological control of pesticide. Theor. Popul. Biol. 64(1), 39–47 (2003)CrossRefMATHGoogle Scholar
  19. 19.
    Jiao, J.J., Chen, L.S., Cai, S.H.: Impulsive control strategy of a pest management SI model with nonlinear incidence rate. Appl. Math. Model. 33(1), 555–563 (2009)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Stern, V.M., Smith, R.F., van den Bosch, R., Hagen, K.S.: The integrated control concept. Hilgardia 29, 81–93 (1959)CrossRefGoogle Scholar
  21. 21.
    Van Lenteren, J.C.: Integrated pest management in protected crops. In: Dent, D. (ed.) Integrated Pest Management, pp. 311–320. Chapman & Hall, London (1995)Google Scholar
  22. 22.
    Van Lenteren, J.C., Woets, J.: Biological and integrated pest control in greenhouses. Ann. Rev. Entomol. 33, 239–250 (1988)CrossRefGoogle Scholar
  23. 23.
    Van Lenteren, J.C.: Measures of success in biological control of arthropods by augmentation of natural enemies. In: Wratten, S., Gurr, G. (eds.) Measures of Success in Biological Control, pp. 77–89. Kluwer Academic Publishers, Dordrecht (2000)CrossRefGoogle Scholar
  24. 24.
    Chen, L.: Pest control and geometric of semi continuous dynamical system. J. Beihua Univ. (Nat. Sci.) 12(1), 1–9 (2011)Google Scholar
  25. 25.
    Pei, Y.Z., Zeng, G.Z., Chen, L.S.: Species extinction and permanence in a prey–predator model with two-type functional responses and impulsive biological control. Nonlinear Dyn. 52, 71–81 (2008)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Shi, R., Jiang, X., Chen, L.: A predatorprey model with disease in the prey and two impulses for integrated pest management. Appl. Math. Model. 33(5), 2248–2256 (2009)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Tang, S., Tang, G., Cheke, R.A.: Optimum timing for integrated pest management: modelling rates of pesticide application and natural enemy releases. J. Theor. Biol. 264(2), 623–638 (2010)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Liu, B., Teng, Z., Chen, L.: Analysis of a predator–prey model with Holling II functional response concerning impulsive control strategy. J. Comput. Appl. Math. 193(1), 347–362 (2006)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Jiao, J.J., Chen, L.S.: The genic mutation on dynamics of a predator–prey system with impulsive effect. Nonlinear Dyn. 70(1), 141–153 (2012)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Zhang, H., Chen, L.S., Georgescu, P.: Impulsive control strategies for pest management. J. Biol. Syst. 15, 235–260 (2007)CrossRefMATHGoogle Scholar
  31. 31.
    Hui, J., Zhu, D.: Dynamic complexities for prey-dependent consumption integrated pest management models with impulsive effects. Chaos Solitons Fractals 29(1), 233–251 (2006)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Tang, S., Chen, L.: Modelling and analysis of integrated pest management strategy. Discrete Contin. Dyn. Syst. Ser. B (DCDS-B) 4(3), 759–768 (2012)MathSciNetMATHGoogle Scholar
  33. 33.
    Tang, S.Y., Xiao, Y.N., Chen, L.S., Cheke, R.A.: Integrated pest management models and their dynamical behaviour. Bull. Math. Biol. 67, 115–135 (2005)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Tang, S., Cheke, R.A.: State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences. J. Math. Biol. 50(3), 257–292 (2005)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Nie, L., Teng, Z., Hu, L., et al.: Qualitative analysis of a modified Leslie–Gower and Holling-type II predator–prey model with state dependent impulsive effects. Nonlinear Anal. Real World Appl. 11(3), 1364–1373 (2010)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Zhang, T., Meng, X., Liu, R., et al.: Periodic solution of a pest management Gompertz model with impulsive state feedback control. Nonlinear Dyn. 78(2), 921–938 (2014)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Wei, C., Chen, L.: Periodic solution and heteroclinic bifurcation in a predator–prey system with Allee effect and impulsive harvesting. Nonlinear Dyn. 76(2), 1109–1117 (2014)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Wei, C.J., Liu, J.N., Chen, L.S.: Homoclinic bifurcation of a ratio-dependent predator–prey system with impulsive harvesting. Nonlinear Dyn. 89(3), 2001–2012 (2017)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Huang, M.Z., Chen, L.C., Song, X.Y.: Stability of a convex order one periodic solution of unilateral asymptotic type. Nonlinear Dyn. 90(1), 83–93 (2017)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Pang, G.P., Chen, L.S.: Periodic solution of the system with impulsive state feedback control. Nonlinear Dyn. 78(1), 743–753 (2014)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Sun, K., Zhang, T., Tian, Y.: Theoretical study and control optimization of an integrated pest management predator–prey model with power growth rate. Math. Biosci. 279, 13–26 (2016)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Sun, K., Zhang, T., Tian, Y.: Dynamics analysis and control optimization of a pest management predator–prey model with an integrated control strategy. Appl. Math. Comput. 292, 253–271 (2017)MathSciNetGoogle Scholar
  43. 43.
    Nie, L.F., Teng, Z.D., Hu, L., Peng, J.G.: Existence and stability of periodic solution of a predator–prey model with state-dependent impulsive effects. Math. Comput. Simul. 79, 2122–2134 (2009)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Zhao, L., Chen, L., Zhang, Q.: The geometrical analysis of a predator–prey model with two state impulses. Math. Biosci. 238(2), 55–64 (2012)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Tian, Y., Sun, K.B., Chen, L.S.: Modelling and qualitative analysis of a predator–prey system with state-dependent impulsive effects. Math. Comput. Simul. 82, 318–331 (2011)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Flores, J.D.: A modified Leslie–Gower predator-prey model with ratio-dependent functional response and alternative food for the predator. Math. Methods Appl. Sci. 40(7), 2313–2328 (2017)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Chicone, C.: Ordinary Differential Equations with Applications. Texts in Applied Mathematics, vol. 34, 2nd edn. Springer, New York (2006)MATHGoogle Scholar
  48. 48.
    Perko, L.: Differential Equation and Dynamical System, 3rd edn. Springer, New York (2006)Google Scholar
  49. 49.
    Bainov, D., Simeonov, P.S.: Impulsive Differential Equations: Periodic Solutions and Applications, vol. 66. Longman Scientific & Technical, New York (1993)MATHGoogle Scholar
  50. 50.
    Simeonov, P.S., Bainov, D.D.: Orbital stability of periodic solutions of autonomous systems with impulse effect. Int. J. Syst. Sci. 19(12), 2561–2585 (1989)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Holmes, P., Shea-Brown, E.T.: Stability. Scholarpedia 1(10), 1838 (2006)CrossRefGoogle Scholar
  52. 52.
    Tian, Y., Sun, K.B., Sun, L.S.: Geometric approach to the stability analysis of the periodic solution in a semi-continuous dynamic system. Int. J. Biomath. 7(2), 1450018 (19 pages) (2014)Google Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXinyang Normal UniversityXinyangChina
  2. 2.School of Information EngineeringDalian UniversityDalianChina

Personalised recommendations