Dynamics and pattern formation of a diffusive predator–prey model in the presence of toxicity

Abstract

In this paper, we develop a diffusivepredator–prey model with toxins under the homogeneous Neumann boundary condition. First, the persistence property and global asymptotic stability of the constant steady states are established. Then by analyzing the associated characteristic equation, we derive explicit conditions for the existence of nonconstant steady states that emerge through steady-state bifurcation from related constant steady states. Furthermore, the existence and nonexistence of nonconstant positive steady states of this model are studied by considering the effect of large diffusivity. Finally, in order to verify our theoretical results, some numerical simulations are also included. These explicit conditions are numerically verified in detail and further compared to those conditions ensuring Turing instability. It is shown that the numerically observed behaviors are in good agreement with the theoretically proposed results. All theoretical analyses and numerical simulations show that toxic substances have a perceptible effect on the system.

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

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

References

  1. 1.

    Hallam, T.G., Clark, C.E.: Non-autonomous logistic equations as models of populations in a deteriorating environment. J. Theoret. Biol. 93(2), 303–311 (1981)

    Article  Google Scholar 

  2. 2.

    Hallam, T.G., Clark, C.E., Jordan, G.S.: Effects of toxicants on populations: a qualitative approach ii. First order kinetics. J. Math. Biol. 18(1), 25–37 (1983)

    Article  Google Scholar 

  3. 3.

    Hallam, T.G., De Luna, J.T.: Effects of toxicants on populations: a qualitative: approach iii. Environmental and food chain pathways. J. Theoret. Biol. 109(3), 411–429 (1984)

    Article  Google Scholar 

  4. 4.

    Freedman, H.I., Shukla, J.B.: Models for the effect of toxicant in single-species and predator-prey systems. J. Math. Biol. 30(1), 15–30 (1990)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Dubey, B., Hussain, J.: A model for the allelopathic effect on two competing species. Ecol. Modell. 129(2–3), 195–207 (2000)

    Article  Google Scholar 

  6. 6.

    Mukhopadhyay, A., Chattopadhyay, J., Tapaswi, P.K.: A delay differential equations model of plankton allelopathy. Math. Biosci. 149(2), 167–189 (1998)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Moussaoui, A., et al.: Effect of a toxicant on the dynamics of a spatial fishery. Afr. Diaspora J. Math. New Ser. 10(2), 122–134 (2010)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Das, T., Mukherjee, R.N., Chaudhuri, K.S.: Harvesting of a prey–predator fishery in the presence of toxicity. Appl. Math. Modell. 33(5), 2282–2292 (2009)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Jana, D., Dolai, P., Pal, A.K., Samanta, G.P.: On the stability and Hopf-bifurcation of a multi-delayed competitive population system affected by toxic substances with imprecise biological parameters. Model. Earth Syst. Environ. 2(3), 110 (2016)

    Article  Google Scholar 

  10. 10.

    Li, Z., Chen, F., He, M.: Asymptotic behavior of the reaction-diffusion model of plankton allelopathy with nonlocal delays. Nonlinear Anal. Real World Appl. 12(3), 1748–1758 (2011)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Samanta, G.P.: A two-species competitive system under the influence of toxic substances. Appl. Math. Comput. 216(1), 291–299 (2010)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Chattopadhyay, J.: Effect of toxic substances on a two-species competitive system. Ecol. Modell. 84(1–3), 287–289 (1996)

    Article  Google Scholar 

  13. 13.

    Kar, T.K., Chaudhuri, K.S.: On non-selective harvesting of two competing fish species in the presence of toxicity. Ecol. Modell. 161(1–2), 125–137 (2003)

    Article  Google Scholar 

  14. 14.

    Pal, D., Samanta, G.P., Mahapatra, G.S.: Selective harvesting of two competing fish species in the presence of toxicity with time delay. Appl. Math. Comput. 313, 74–93 (2017)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Jianhong, W.: Theory and Applications of Partial Functional Differential Equations. Applied Mathematical Sciences. Springer, New York (1996)

    Google Scholar 

  16. 16.

    Samanta, G.P.: A two-species competitive system under the influence of toxic substances. Appl. Math. Comput. 216(1), 291–299 (2010)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Banerjee, M., Ghorai, S., Mukherjee, N.: Study of cross-diffusion induced Turing patterns in a ratio-dependent prey-predator model via amplitude equations. Appl. Math. Modell. 55, 383–399 (2018)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Sun, X.K., Huo, H.F., Xiang, H.: Bifurcation and stability analysis in predatorcprey model with a stage-structure for predator. Nonlinear Dyn. 58(3), 497 (2009)

    Article  Google Scholar 

  19. 19.

    Zhang, L., Liu, J., Banerjee, M.: Hopf and steady state bifurcation analysis in a ratio-dependent predator-prey model. Commun. Nonlinear Sci. Numer. Simul. 44, 52–73 (2017)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Yi, F., Wei, J., Shi, J.: Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system. J. Differ. Equ. 246(5), 1944–1977 (2009)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Ni, W., Wang, M.: Dynamics and patterns of a diffusive Leslie-Gower prey-predator model with strong Allee effect in prey. J. Differ. Equ. 261(7), 4244–4274 (2016)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Wang, M.: Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion. Phys. D Nonlinear Phenom. 196(1), 172–192 (2004)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Wang, W., Lin, Y., Rao, F., Zhang, L., Tan, Y.: Pattern selection in a ratio-dependent predator-prey model. J. Stat. Mech. Theory and Exp. 2010(11), P11036 (2010)

    Article  Google Scholar 

  24. 24.

    Yang, R., Wei, J.: The effect of delay on a diffusive predator-prey system with modified Leslie–Gower functional response. Bull. Malays. Math. Sci. Soc. 40, 51–73 (2015)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Ghorai, S., Poria, S.: Pattern formation in a system involving prey-predation, competition and commensalism. Nonlinear Dyn. 89(2), 1309–1326 (2017)

    Article  Google Scholar 

  26. 26.

    Aly, S.: Competition in patchy space with cross-diffusion and toxic substances. Nonlinear Anal. Real World Appl. 10(1), 185–190 (2009)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Maynard Smith, J.: Models in Ecology. Cambridge University Press, Cambridge (1978)

    Google Scholar 

  28. 28.

    Lou, Y., Ming Ni, W.: Diffusion, self-diffusion and cross-diffusion. J. Differ. Equ. 131(1), 79–131 (1996)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Allen, J.C., Schaffer, W.M., Rosko, D.: Chaos reduces species extinction by amplifying local population noise. Nature 364(6434), 229–32 (1993)

    Article  Google Scholar 

  30. 30.

    Heino, M., Kaitala, V., Ranta, E., Lindstrom, J.: Synchronous dynamics and rates of extinction in spatially structured populations. Proc. R. Soc. B Biol. Sci. 264(1381), 481–486 (1997)

    Article  Google Scholar 

  31. 31.

    Schimanskygeier, L., Fiedler, B., Kurths, J., Scholl, E.: Analysis and Control of Complex Nonlinear Processes in Physics. Chemistry and Biology. World Scientific, Singapore (2007)

    Google Scholar 

Download references

Acknowledgements

This study was funded by National Science Foundation of China (Grant Number: 11571170). The study is partially supported by Natural Science Foundation of Jiangsu Province (Grant Number: BK20150420), and also supported by the Startup Foundation for Introducing Talent of NUIST.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Hongyong Zhao.

Ethics declarations

Conflict of interests

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zhang, X., Zhao, H. Dynamics and pattern formation of a diffusive predator–prey model in the presence of toxicity. Nonlinear Dyn 95, 2163–2179 (2019). https://doi.org/10.1007/s11071-018-4683-2

Download citation

Keywords

  • Turing instability
  • Pattern formation
  • Bifurcation
  • Nonconstant steady state