AIMS Mathematics, 2020, 5(6): 6749-6765. doi: 10.3934/math.2020434.

Research article

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Survival analysis of single-species population diffusion models with chemotaxis in polluted environment

1 School of Data Science of Tongren University, Tongren, 554300, PR China
2 Tongren Preschool Education College, Tongren, 554300, PR China

In this paper, single-species population diffusion models with chemotaxis in polluted environment are proposed and studied. For the deterministic single-species population diffusion model, the sufficient conditions for the extinction and strong persistence of the single-species population are established. For the stochastic single-species population diffusion model. First, we show that system has unique global positive solution. And then, the sufficient conditions for extinction and strongly persistent in the mean of the single-species are obtained. Numerical simulations are used to confirm the efficiency of the main results.
  Article Metrics

Keywords chemotaxis; persistence; extinction; stochastic perturbations; polluted environment

Citation: Xiangjun Dai, Suli Wang, Baoping Yan, Zhi Mao, Weizhi Xiong. Survival analysis of single-species population diffusion models with chemotaxis in polluted environment. AIMS Mathematics, 2020, 5(6): 6749-6765. doi: 10.3934/math.2020434


  • 1. E. H. Colombo, C. Anteneodo, Metapopulation dynamics in a complex ecological landscape, Phys. Rev. E, 92 (2015), 022714.
  • 2. E. Beretta, F. Solimano, Y. Takeuchi, Global stability and periodic orbits for two-patch predatorprey diffusion-delay models, Math. Biosci., 85 (1987), 153-183.    
  • 3. G. Z. Zeng, L. S. Chen, J. F. Chen, Persistence and periodic orbits for two-species nonautonomous diffusion lotka-volterra models, Math. Comput. Model., 20 (1994), 69-80.
  • 4. E. Beretta, Y. Takeuchi, Global stability of single-species diffusion Volterra models with continuous time delays, B. Math. Biol., 49 (1987), 431-448.    
  • 5. L. Zhang, Z. Teng, The dynamical behavior of a predator-prey system with Gompertz growth function and impulsive dispersal of prey between two patches, Math. Method. Appl. Sci., 39 (2016), 3623-3639.    
  • 6. L. J. S. Allen, Persistence and extinction in single-species reaction-diffusion models, B. Math. Biol., 45 (1983), 209-227.    
  • 7. H. I. Freedman, Single species migration in two habitats: Persistence and extinction, Math. Model., 8 (1987), 778-780.    
  • 8. M. Bengfort, H. Malchow, F. M. Hilker, The Fokker-Planck law of diffusion and pattern formation in heterogeneous environments, J. Math. Biol., 73 (2016), 683-704.    
  • 9. D. Li, S. J. Guo, Stability and Hopf bifurcation in a reaction-diffusion model with chemotaxis and nonlocal delay effect, Int. J. Bifurcat. Chaos, 28 (2018), 1850046.
  • 10. Y. Tan, C. Huang, B. Sun, et al. Dynamics of a class of delayed reaction-diffusion systems with Neumann boundary condition, J. Math. Anal. Appl., 458 (2018), 1115-1130.    
  • 11. Y. Xie, Q. Li, K. Zhu, Attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity, Nonlinear Anal. Real, 31 (2016), 23-37.    
  • 12. F. Wei, S. A. H. Geritz, J. Cai, A stochastic single-species population model with partial pollution tolerance in a polluted environment, Appl. Math. Lett., 63 (2016), 130-136.
  • 13. F. Y. Wei, L. H. Chen, Psychological effect on single-species population models in a polluted environment, Math. Biosci., 290 (2017), 22-30.    
  • 14. J. J. Jiao, L. S. Chen, The extinction threshold on a single population model with pulse input of environmental toxin in a polluted environment, Math. Appl., 22 (2009), 11-19.
  • 15. Y. Xiao, L. Chen, Effects of toxicants on a stage-structured population growth model, Appl. Math. Comput., 123 (2001), 63-73.
  • 16. J. D. Stark, J. E. Banks, Population-level effects of pesticides and other toxicants on arthropods, Annu. Rev. Entomol., 48 (2003), 505-519.    
  • 17. W. Jing, W. Ke, Analysis of a single species with diffusion in a polluted environment, Electron. J. Differ. Eq., 112 (2006), 285-296.
  • 18. R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, 2001.
  • 19. A. Friedman, Stochastic Differential Equations and Applications, Academic Press, 1976.
  • 20. M. Liu, K. Wang, Persistence and extinction in stochastic non-autonomous logistic systems, J. Math. Anal. Appl., 375 (2011), 443-457.    
  • 21. X. J. Dai, Z. Mao, X. J. Li, A stochastic prey-predator model with time-dependent delays, Adv. Differ. Equ., 2017 (2017), 1-15.    
  • 22. M. Liu, K. Wang, Q. Wu, Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle, B. Math. Biol., 73 (2011), 1969-2012.    
  • 23. X. Zou, D. Fan, K. Wang, Effects of dispersal for a logistic growth population in random environments, Abstr. Appl. Anal., 2013 (2013), 1-9.
  • 24. L. Zu, D. Jiang, ORegan. Donal, Stochastic permanence, stationary distribution and extinction of a single-species nonlinear diffusion system with Rrandom perturbation, Abstr. Appl. Anal., 2014 (2014), 1-14.
  • 25. M. Liu, M. Deng, B. Du, Analysis of a stochastic logistic model with diffusion, Appl. Comput. Model., 266 (2015), 169-182.
  • 26. X. Zou, K. Wang, A robustness analysis of biological population models with protection zone, Appl. Math. Model., 35 (2011), 5553-5563.    
  • 27. X. Zou, K. Wang, M. Liu, Can protection zone potentially strengthen protective effects in random environments?, Appl. Math. Comput., 231 (2014), 26-38.
  • 28. F. Y. Wei, C. J. Wang, Survival analysis of a single-species population model with fluctuations and migrations between patches, Appl. Math. Model., 81 (2020), 113-127.    
  • 29. X. Zou, K. Wang, Dynamical properties of a biological population with a protected area under ecological uncertainty, Appl. Math. Model., 39 (2015), 6273-6284.
  • 30. D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.    


Reader Comments

your name: *   your email: *  

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved