Skip to main content
SpringerLink
Log in

Qualitative analysis of a predator–prey system with mutual interference and impulsive state feedback control

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper, a predator–prey system with mutual interference and impulsive state feedback control is constructed. Firstly, by using the stability theorem, we obtain the global asymptotically stability of the positive equilibrium for the system without impulse effects. Secondly, by using the geometric theory of differential equations, the method of successor functions and mathematical analysis, we obtain some sufficient conditions for the existence, uniqueness and orbital asymptotically stability of the order-1 periodic solution for the system with impulse effects. Finally, the main conclusions are verified by numerical simulation, and the pest prevention and control strategies on combining light trapping and water-gun spraying are proposed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

References

  1. Fan, H.Q., Qiu, G.L.: Insect pests of Avicennia marina mangroves along the coast of Beibu Gulf in China and the research strategies. Guangxi Zhiwu 24, 558–562 (2004)

    Google Scholar 

  2. Li, H.F.: Analysis of integrated pest management of mangrove in Guangxi Beihai National Coastal Wetland Park. Agric. Research Appl. 148, 59–62 (2013)

    Google Scholar 

  3. Li, Z.G., Dai, J.Q., Ye, J.W., Xu, H.L., Han, S.C.: Species, control status and outbreak causes of main pest insects in mangrove ecosystems in China. Acta Entomol. Sin. 55, 1109–1118 (2012)

    Google Scholar 

  4. Hunter, M.D.: Ecological Causes of Pest Outbreaks. In: Encyclopedia of Pest Management, pp. 214–217. CRC Press, Boca Raton (2002)

  5. Barbosa, P., Schultz, J.C.: Insect Outbreaks. Academic Press, New York (1987)

    Google Scholar 

  6. Singh, T., Satyanarayana, J.: Integrated Pest Management: Innovation-Development Process, In: Insect Outbreaks and Their Management. Springer, Netherlands, pp. 331–350 (2009)

  7. Ikeda, T.: Integrated pest management of Japanese pine wilt disease. For. Path. 14, 398–414 (2015)

    Article  MathSciNet  Google Scholar 

  8. Abrol, D.P.: Integrated Pest Management: Current Concepts and Ecological Perspective. Academic Press, New York (2013)

    Google Scholar 

  9. Abrol, D.P., Shankar, U.: Integrated Pest Management: Principles and Practice. CABI Publishing, Wallingford (2012)

    Book  Google Scholar 

  10. Freedman, H.L.: Graphical stability enrichment and pest control by a natural enemy. Math. Biosci. 31, 207–225 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ludwig, D., Jones, D.D., Holling, C.S.: Qualitative analysis of insect outbreak systems: the Spruce Budworm and Fores. J. Animal Ecol. 47, 315–332 (1978)

    Article  Google Scholar 

  12. Clark, C.W.: Bioeconomic modeling and resource management. In: Applied Mathematical Ecology, pp. 11–57. Springer, Berlin (1989)

  13. Goh, B.S.: Management and analysis of biological populations. Amsterdam Oxford, New York (1980)

    Google Scholar 

  14. Saha, T., Bandyopadhyay, M.: Dynamical analysis of a plant-herbivore model bifurcation and global stability. J. Appl. Math. Comput. 19, 327–344 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Abbott, K.C., Dwyer, G.: Food limitation and insect outbreaks: complex dynamics in plant-herbivore models. J. Anim. Ecol. 76, 1004–1014 (2007)

    Article  Google Scholar 

  16. Kang, Y., Armbruster, D., Kuang, Y.: Dynamics of a plant-herbivore model. J. Biol. Dyn. 2, 89–101 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hughes, J.S., Cobbold, C.A., Haynes, K., Dwyer, G.: Effects of forest spatial structure on insect outbreaks: insights from a simple host-parasitoid model. Am. Nat. 185, 130–152 (2015)

    Article  Google Scholar 

  18. Sun, G.Q., Wang, S.L., Ren, Q., Jin, Z., Wu, Y.P.: Effects of time delay and space on herbivore dynamics: linking inducible defenses of plants to herbivore outbreak. Sci. Rep. 5, 11246 (2015)

    Article  Google Scholar 

  19. Sun, G.Q., Chakraborty, A., Liu, Q.X., Jin, Z., Anderson, K.E., Li, B.L.: Influence of time delay and nonlinear diffusion on herbivore outbreak. Commun. Nonlinear Sci. 19, 1507–1518 (2014)

    Article  MathSciNet  Google Scholar 

  20. Li, L., Jin, Z., Li, J.: Periodic solutions in a herbivore-plant system with time delay and spatial diffusion. Appl. Math. Model. 40, 4765–4777 (2016)

    Article  MathSciNet  Google Scholar 

  21. Sun, G.Q., Jin, Z., Li, L., Haque, M., Li, B.L.: Spatial patterns of a predator-prey model with cross Diffusion. Nonlinear Dyn. 69, 1631–1638 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sun, G.Q., Wu, Z.Y., Wang, Z., Jin, Z.: Influence of isolation degree of spatial patterns on persistence of populations. Nonlinear Dyn. 83, 811–819 (2016)

    Article  MathSciNet  Google Scholar 

  23. Li, L., Jin, Z.: Pattern dynamics of a spatial predator-prey model with noise. Nonlinear Dyn. 67, 1737–1744 (2012)

    Article  MathSciNet  Google Scholar 

  24. Sun, G.Q.: Mathematical modeling of population dynamics with Allee effect. Nonlinear Dyn. 8, 1–12 (2016)

    Article  MathSciNet  Google Scholar 

  25. Grasman, J., Herwaarden, O.A., Hemerik, L., Lenteren, J.C.: A two-component model of host-parasitoid interactions: determination of the size of inundative releases of parasitoids in biological pest control. Math. Biosci. 169, 207–216 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Jana, S., Kar, T.K.: A mathematical study of a prey-predator model in relevance to pest control. Nonlinear Dyn. 74, 667–683 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Barclay, H.J.: Models for pest control using predator release, habitat management and pesticide release in combination. J. Appl. Ecol. 19, 337–348 (1982)

    Article  Google Scholar 

  28. Liu, B., Zhang, Y.J., Chen, L.S.: Dynamic complexities of a Holling I predator-prey model concerning periodic biological and chemical control. Chaos Soliton Fractal 22, 123–134 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kar, T.K., Ghorai, A., Jana, S.: Dynamics of pest and its predator model with disease in the pest and optimal use of pesticide. J. Theor. Biol. 310, 187–198 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wang, X., Tao, Y., Song, X.Y.: Analysis of pest-epidemic model by releasing diseased pest with impulsive transmission. Nonlinear Dyn. 65, 175–185 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Pathak, S., Maiti, A.: Microbial pest control: a mathematical model. J. Biol. Syst. 18, 455–478 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Bhattacharyya, S., Bhattacharya, D.K.: Pest control through viral disease: mathematical modeling and analysis. J. Theor. Biol. 238, 177–197 (2006)

    Article  MathSciNet  Google Scholar 

  33. Pathak, S., Maiti, A.: Pest control using virus as control agent: a mathematical model. Nonlinear Anal.- Model 17, 67–90 (2012)

    MathSciNet  MATH  Google Scholar 

  34. Wang, T.Y., Chen, L.S.: Nonlinear analysis of a microbial pesticide model with impulsive state feedback control. Nonlinear Dyn. 65, 1–10 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wang, T.Y., Chen, L.S.: Dynamic complexity of microbial pesticide model. Nonlinear Dyn. 58, 539–552 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tang, S.Y., Xiao, Y.N., Chen, L.S., Cheke, R.A.: Integrated pest management models and their dynamical behaviour. B. Math. Biol. 67, 115–135 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  37. Tang, S.Y., Xiao, Y.N., Cheke, R.A.: Multiple attractors of host-parasitoid models with integrated pest management strategies: eradication, persistence and outbreak. Theor. Popul. Biol. 73, 181–197 (2008)

    Article  MATH  Google Scholar 

  38. Tang, S.Y., Tang, G., Cheke, R.A.: Optimum timing for integrated pest management: modelling rates of pesticide application and natural enemy releases. J. Theor. Biol. 264, 623–638 (2010)

    Article  MathSciNet  Google Scholar 

  39. Zhang, Y.J., Liu, B., Chen, L.S.: Dynamical behavior of Volterra model with mutual interference concerning IPM. Esaim Math. Model. Num. 38, 143–155 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  40. Georgescu, P., Dimitriu, G., Sinclair, R.: Impulsive control of an integrated pest management models with dispersal between patches. J. Biol. Syst. 18, 535–569 (2010)

    Article  MathSciNet  Google Scholar 

  41. Bhattacharyya, S., Bhattacharya, D.K.: An improved integrated pest management model under 2-control parameters (sterile male and pesticide). Math. Biosci. 209, 256–281 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  42. Liang, J.H., Tang, S.Y., Cheke, R.A.: Optimum timing for integrated pest management: modelling rates of pesticide application and natural enemy releases. Nonlinear Anal. -Real. 13, 2352–2374 (2012)

    Article  MATH  Google Scholar 

  43. Sun, K.B., Zhang, T.H., 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)

    MathSciNet  Google Scholar 

  44. Bhattacharya, D.K., Karan, S.: On bionomic model of integrated pest management of a single Pest population. Differ. Equ. Dyn. Syst. 12, 301–330 (2004)

    MathSciNet  MATH  Google Scholar 

  45. Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)

    Book  MATH  Google Scholar 

  46. Bainov, D., Simeonov, P.: Impulsive Differential Equations: Periodic Solutions and Applications. Longman, England (1993)

    MATH  Google Scholar 

  47. Bainov, D.D., Simeonov, P.S.: Impulsive Differential Equations: Asymptotic Properties of the Solutions. World Scientific, Singapore (1995)

    Book  MATH  Google Scholar 

  48. Chen, L.S.: Pest control and geometric theory of semi–continuous dynamical system. J. Beihua Univ. (Nat. Sci.) 12, 1–9 (2011)

    Google Scholar 

  49. Chen, L.S.: Theory and application of semi–continuous dynamical system. J. Yulin Normal Univ. (Na. Sci.) 34, 1–10 (2013)

  50. Wei, C.J., Chen, L.S.: Periodic solution and heteroclinic bifurcation in a predator-prey system with Allee effect and impulsive harvesting. Nonlinear Dyn. 76, 1109–1117 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  51. Tang, S.Y., Pang, W.H., Cheke, R.A., Wu, J.H.: Global dynamics of a state-dependent feedback control system. Adv. Differ. Equ. 2015, 1–70 (2015)

    Article  MathSciNet  Google Scholar 

  52. Tang, S.Y., Tang, B., Wang, A.L., Xiao, Y.N.: Holling II predator-prey impulsive semi-dynamic model with complex Poincar map. Nonlinear Dyn. 81, 1575–1596 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  53. Pang, G.P., Chen, L.S.: Periodic solution of the system with impulsive state feedback control. Nonlinear Dyn. 78, 743–753 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  54. Pang, G.P., Chen, L.S., Xu, W.J., Fu, G.: A stage structure pest management model with impulsive state feedback control. Commun. Nonlinear Sci. 23, 78–88 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  55. Zhang, T.Q., Meng, X.Z., Liu, R., Zhang, T.H.: Periodic solution of a pest management Gompertz model with impulsive state feedback control. Nonlinear Dyn. 78, 921–938 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  56. Guan, L.P., Lu, H.B., Hu, M.H.: Application of light trapping in controlling forest pests. Prot. For. Sci. Tech. 143, 4–6 (2015)

    Google Scholar 

  57. Khan, H.R., Kumar, S., Prasad, L.: Studies on seasonal activity of some agro-forestry insect pests by light trap. Indian For. 114, 215–229 (1988)

    Google Scholar 

  58. Shimoda, M., Honda, K.I.: Insect reactions to light and its applications to pest management. Appl. Entomol. Zool. 48, 413–421 (2013)

    Article  Google Scholar 

  59. Hassell, M.P.: Mutual interference between searching insect parasites. J. Anim. Ecol. 40, 473–486 (1971)

    Article  Google Scholar 

  60. Rogers, D.J., Hassell, M.P.: General models for insect parasite and predator searching behaviour: interferences. J. Anim. Ecol. 43, 239–253 (1974)

    Article  Google Scholar 

  61. Freedman, H.L.: Stability analysis of a predator-prey system with mutual interference and density-dependent death rates. Bull. Math. Boil. 41, 67–78 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  62. Freedman, H.L.: Stability analysis of a predator-prey system with mutual interference and density-dependent death rates. Bull. Math. Boil. 45, 991–1004 (1983)

    Article  Google Scholar 

  63. Jiao, J.J., Chen, L.S., Cai, S.H.: A delayed stage-structured Holling II predator-prey model with mutual interference and impulsive perturbations on predator. Chaos Soliton Fract. 40, 1946–1955 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  64. Wang, K.: Permanence and global asymptotical stability of a predator–prey model with mutual interference. Nonlinear Anal. Real. 12, 1062–1071 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  65. Ma, Z.Z., Chen, F.D., Wu, C.Q., Chen, W.L.: Dynamic behaviors of a Lotka-Volterra predator-prey model incorporating a prey refuge and predator mutual interference. Appl. Math. Comput. 219, 7945–7953 (2013)

    MathSciNet  MATH  Google Scholar 

  66. Li, Z., Chen, F.D., He, M.X.: Permanence and global attractivity of a periodic predator-prey system with mutual interference and impulses. Commun. Nonlinear Sci. 17, 444–453 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported by the National Natural Science Foundation of P.R. China (61364020, 11161052) and the Major Research Programmes of Yulin Normal University of P.R. China (2015YJZD02).

Author information

Authors and Affiliations

  1. College of Mathematics and Information Science, Guangxi Colleges and Universities Key Lab of Complex System Optimization and Large Data Processing, Yulin Normal University, Yulin, 537000, Guangxi, China

    Zhiqing Liang, Guoping Pang, Xiaping Zeng & Yanhong Liang

Authors
  1. Zhiqing Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guoping Pang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaping Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yanhong Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhiqing Liang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liang, Z., Pang, G., Zeng, X. et al. Qualitative analysis of a predator–prey system with mutual interference and impulsive state feedback control. Nonlinear Dyn 87, 1495–1509 (2017). https://doi.org/10.1007/s11071-016-3129-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-016-3129-y

Keywords

Search

Navigation