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Stochastic sensitivity analysis of noise-induced transitions in a predator-prey model with environmental toxins

1 College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China
2 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

Special Issues: Recent Advances in Mathematical Population Dynamics

Huang et al. [1] recently developed a toxin-dependent predator-prey model and analyzed its global dynamics. Their results showed that environmental toxins may influence both predators and prey and induce bistable situation, and intermediate toxin concentrations may affect predators disproportionately through biomagnification. Environmental noises can change the dynamical behaviors of the toxin-based predator-prey model. In this paper, by formulating a stochastically forced predator-prey model with environmental toxins, we study the dynamical phenomenon of noise-induced transitions from coexistence to prey-only extirpation in the bistable zone. Numerical simulations based on the technique of stochastic sensitivity functions are provided for constructing the confidence ellipse and estimating the threshold value of the noise intensity of state switching. Meanwhile, we construct the confidence band and study the configurational arrangement of the stochastic cycle.
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Keywords predator-prey model; environmental toxins; stochastic sensitivity; noise-induced transitions; confidence domain

Citation: Dongmei Wu, Hao Wang, Sanling Yuan. Stochastic sensitivity analysis of noise-induced transitions in a predator-prey model with environmental toxins. Mathematical Biosciences and Engineering, 2019, 16(4): 2141-2153. doi: 10.3934/mbe.2019104

References

  • 1. Q. Huang, H. Wang and M. A. Lewis, The impact of environmental toxins on predator-prey dynamics, J. Theor. Biol., 378 (2015), 12–30.
  • 2. R. A. Pastorok, S. M. Bartell and S. Ferson, et al., Ecological modeling in risk assessment: chemical effects on populations, ecosystems, and landscapes, Lewis Publishers, Boca Raton, FL, USA, 2001.
  • 3. S. M. Bartell, R. A. Pastorok and H. R. Akcakaya, et al., Realism and relevance of ecological models used in chemical risk assessment, Hum. Ecol. Risk Assess., 9 (2003), 907–938.
  • 4. N. Galic, U. Hommen and J. H. Baveco, et al., Potential application of population models in the European ecological risk assessment of chemicals. II: review of models and their potential to address environmental protection aims, Integr. Environ. Assess. Manag., 6 (2010), 338–360.
  • 5. Y. Zhao, S. Yuan and T. Zhang, The stationary distribution and ergodicity of a stochastic phytoplankton allelopathy model under regime switching, Commun. Nonlinear Sci. Numer. Simtlat., 37 (2016), 131–142.
  • 6. X. Yu, S. Yuan and T. Zhang, The effects of toxin-producing phytoplankton and environmental fluctuations on the planktonic blooms, Nonlinear Dyn., 91 (2018), 1653–1668.
  • 7. H. R. Thieme, Mathematics in population biology, Princeton Series in Theoretical and Computational Biology, 2003.
  • 8. Q. Huang, L. Parshotham and H. Wang, et al., A model for the impact of contaminants on fish population dynamics, J. Theor. Biol., 334 (2013), 71–79.
  • 9. B. C. Kelly, M. G. Ikonomou and J. D. Blair, et al., Food webspecific biomagnification of persistent organic pollutants, Science, 317 (2007), 236–239.
  • 10. R. V. Thomann, Bioaccumulation model of organic chemical distribution in aquatic food chains, Environ. Sci. Technol., 23 (1989), 699–707.
  • 11. V. S. Anishchenko, V. Astakhov and A. Neiman, et al., Nonlinear dynamics of chaotic and stochastic systems, Springer, Berlin, 2007.
  • 12. A. N. Pisarchik and U. Feudel, Control of multistability, Phys. Rep., 540 (2014), 167–218.
  • 13. C. Xu, S. Yuan and T. Zhang, Stochastic sensitivity analysis for a competitive turbidostat model with inhibitory nutrients, Int. J. Bifurcat. Chaos, 8 (2016), 1440020.
  • 14. C. Xu, S. Yuan and T. Zhang, Sensitivity analysis and feedback control of noise-induced extinction for competition chemostat model with mutualism, Physica A, 505 (2018), 891–902.
  • 15. C. Kurrer and K. Schulten, Effect of noise and perturbations on limit cycle systems, Physica D, 50 (1991), 311–320.
  • 16. F. Baras, Lecture notes in physics springer-verlag, New York, 484 (1997), 167–178.
  • 17. L. Gammaitoni, P. H. anggi and P. Jung, et al., Stochastic resonance, Rev. Mod. Phys., 70 (1999), 223.
  • 18. M. D. McDonnell, N. G. Stocks and C. E. M. Pearce, et al., Stochastic resonance: from suprathreshold stochastic resonance to stochastic signal quantization cambridge, University Press, New York, 2008.
  • 19. W. Horsthemke and R. Lefever, Noise-induced transitions, Springer, Berlin, 1984.
  • 20. K. Matsumoto and I. Tsuda, Noise-induced order, J. Stat. Phys., 33 (1983), 757.
  • 21. F. Gassmann, Noise-induced chaos-order transitions, Phys. Rev. E, 55 (1997), 2215–2221.
  • 22. J. Gao, S. Hwang and J. Liu, When can noise induce chaos? Phys. Rev. Lett., 82 (1999), 1132– 1135.
  • 23. M. A. Zaks, X. Sailer and L. Schimansky-Geier, et al., Noise induced complexity: From subthreshold oscillations to spiking in coupled excitable systems, Chaos, 15 (2005), 026117.
  • 24. C. Xu and S. Yuan, An analogue of break-even concentration in a simple stochastic chemostat model, Appl. Math. Lett., 48 (2015), 62–68.
  • 25. Y. Zhao, S. Yuan and J. Ma, Survival and stationary distribution analysis of a stochastic competitive model of three species in a polluted environment, B. Math. Biol., 77 (2015), 1285–1326.
  • 26. Y. Zhao, S. Yuan and T. Zhang, Stochastic periodic solution of a non-autonomous toxic-producing phytoplankton allelopathy model with environmental fluctuation, Commun. Nonlinear Sci. Numer. Simtlat., 44 (2017), 266–276.
  • 27. X. Yu, S. Yuan and T. Zhang, Persistence and ergodicity of a stochastic single species model with Allee effect under regime switching, Commun. Nonlinear Sci. Numer. Simtlat., 59 (2018), 359–374.
  • 28. X. Yu, S. Yuan and T. Zhang, About the optimal harvesting of a fuzzy predator-prey system: A bioeconomic model incorporating a prey refuge and predator mutual interference, Nonlinear Dyn., DOI: 10.1007/s11071-018-4480-y.
  • 29. C. Xu and S. Yuan, Competition in the chemostat: a stochastic multi-species model and its asymptotic behavior, Math. Biosci., 280 (2016), 1–9.
  • 30. S. Kim, S. H. Park and C. S. Ryu, Colored-noise-induced multistability in nonequilibrium phase transitions, Phys. Rev. E, 58 (1998), 7994–7997.
  • 31. S. Kraut and U. Feudel, Multistability, noise, and attractor hopping: the crucial role of chaotic saddles, Phys. Rev. E, 66 (2002), 015207(R).
  • 32. S. L. T. de Souza, A. M. Batista and I. L. Caldas, et al., Noise-induced basin hopping in a vibro-impact system, Chaos Solit. Fract., 32 (2007), 758–767.
  • 33. M. I. Dykman, R. Mannella and P. V. E. McClintock, et al., Fluctuation-induced transitions between periodic attractors: Observation of supernarrow spectral peaks near a kinetic phase transition, Phys. Rev. Lett., 66 (1990), 48.
  • 34. M. I. Dykman, D. G. Luchinsky and R. Mannella, et al., Simulation of critical phenomena in nonlinear optical systems, Phys. Rev. E, 49 (1994), 1198.
  • 35. I. Bashkirtseva, L. Ryashko and I. Tsvetkov, Sensitivity analysis of stochastic equilibria and cycles for the discrete dynamic systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 17 (2010), 501–515.
  • 36. I. Bashkirtseva, T. Ryazanova and L. Ryashko, Confidence domains in the analysis of noiseinduced transition to chaos for Goodwin model of business cycles, Int. J. Bifurc. Chaos., 24 (2014), 1440020.

 

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