Journal of Mathematical Biology

, Volume 78, Issue 3, pp 739–766 | Cite as

Direct and indirect effects of toxins on competition dynamics of species in an aquatic environment

  • Chunhua Shan
  • Qihua HuangEmail author


When two competing species are simultaneously exposed in a polluted environment, one species may be more vulnerable to toxins than the other. To study the impact of environmental toxins on competition dynamics of two species, we develop a toxin-dependent competition model that incorporates both direct and indirect toxic effects on the species. The direct effects of toxins typically reduce population abundance by increasing mortality and reducing reproduction. However, the indirect effects, which are mediated through competitive interactions, may lead to counterintuitive effects. We investigate the toxin-dependent competition model and explore the impact of the interplay between environmental toxins and distinct toxic tolerance of two species on the competition outcomes. The results of theoretical analysis and numerical studies reveal that while high level of toxins is harmful to both species, possibly leading to extirpation of both species, intermediate level of toxins, plus different vulnerabilities of two species to toxins, affect competition outcomes in many counterintuitive ways. It turns out that sublethal toxins may boost coexistence of two species (hence keep species diversity in ecosystems) by reducing the abundance of the predominant species; sublethal toxins may overturn and exchange roles of winner and loser in competition; sublethal toxins may also induce different types of bistability of the competition dynamics, where the competition outcome is doomed to exclusion or coexistence, depending on initial population densities. The theory developed here provides a sound foundation for understanding competitive interactions between two species in a polluted aquatic environment.


Toxin Competition Fast-slow dynamics Coexistence Exclusion Bistability 

Mathematics Subject Classification

92Bxx 37N25 34C23 



The authors thank Gunog Seo (Colgate University) for fruitful discussions. The authors gratefully acknowledge two anonymous referees for careful reading and insightful comments which greatly improve the manuscript.


  1. CCME (2003) The Canadian Council of Ministers of the Environment, Canadian water quality guidelines for the protection of aquatic life: guidance on the site-specific application of water quality guidelines in Canada: procedures for deriving numerical water quality objectives.
  2. Cody ML, Diamond JM (1975) Ecology and evolution of communities. Belknap Press of Harvard University Press, CambridgeGoogle Scholar
  3. Fenichel N (1971) Persistence and smoothness of invariant manifolds for flows. Indiana Univ Math J 21:193–226MathSciNetCrossRefzbMATHGoogle Scholar
  4. Fenichel N (1979) Geometric singular perturbation theory for ordinary differential equations. J Diff Equ 31:53–98MathSciNetCrossRefzbMATHGoogle Scholar
  5. Forbes V, Hommen U, Thorbek P, Heimbach F, den Brink PV, J Wogram HT, Grimm V (2009) Ecological models in support of regulatory risk assessments of pesticides: developing a strategy for the future. Integr Environ Asses Manag 5:167–172CrossRefGoogle Scholar
  6. Forbes V, Sibly R, Calow P (2001) Toxicant impacts on density-limited populations: a critical review of theory, practice, and results. Ecol Appl 11:1249–1257CrossRefGoogle Scholar
  7. Freedman HI, Shukla JB (1991) Models for the effect of toxicant in single-species and predator–prey systems. J Math Biol 30:15–30MathSciNetCrossRefzbMATHGoogle Scholar
  8. Grover JP (1997) Resource competition. Chapman & Hall, LondonCrossRefGoogle Scholar
  9. Hallam TG, Clark CE (1983) Effect of toxicants on populations: a qualitative approach. I. Equilibrium environmental exposure. Ecol Model 18:291–304CrossRefzbMATHGoogle Scholar
  10. Hallam TG, Clark CE, Jordan GS (1983) Effect of toxicants on populations: a qualitative approach. II. First order kinetics. J Math Biol 18:25–37CrossRefzbMATHGoogle Scholar
  11. Hallam TG, Luna JD (1984) Extinction and persistence in models of population-toxicant interactions. Ecol Model 22:13–20CrossRefGoogle Scholar
  12. Hallam TG, Luna JD (1990) Toxicant-induced mortality in models of daphnia populations. Environ Toxicol Chem 9:597–621CrossRefGoogle Scholar
  13. Huang Q, Parshotam L, Wang H, Bampfylde C, Lewis M (2013) A model of the impact of contaminants on fish population dynamics. J Theor Biol 334:71–79MathSciNetCrossRefzbMATHGoogle Scholar
  14. Huang Q, Wang H, Lewis MA (2015) The impact of environmental toxins on predator–prey dynamics. J Theor Biol 378:12–30MathSciNetCrossRefzbMATHGoogle Scholar
  15. Kot M (2001) Elements of mathematical ecology. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  16. Luna JT, Hallam TG (1987) Effect of toxicants on populations: a qualitative approach. IV. Resource-consumer-toxiocant models. Ecol Model 35:249–273CrossRefGoogle Scholar
  17. McElroy AE, Barron MG, Beckvar N, Driscoll SBK, Meador JP, Parkerton TF, Preuss TG, Steevens JA (2010) A review of the tissue residue approach for organic and organometallic compounds in aquatic organisms. Integr Environ Assess Manag 7:50–74CrossRefGoogle Scholar
  18. Schoener TW (1982) The controversy over interspecific competition: despite spirited criticism, competition continues to occupy a major domain in ecological thought. Am Sci 70:586–595Google Scholar
  19. Smith H (1995) Monotone dynamical systems. An introduction to the theory of competitive and cooperative systems. Mathematical surveys and monographs, vol 41. American Mathematical Society, ProvidenceGoogle Scholar
  20. Thieme HR (2003) Mathematics in population biology. Princeton University Press, PrincetonzbMATHGoogle Scholar
  21. Thomas DM, Snell TW, Jaffar SM (1996) A control problem in a polluted environment. Math Biosci 133:139–163CrossRefzbMATHGoogle Scholar
  22. USNARA (2013) US national archives and records administration, code of federal regulations, title 40-protection of environment, Appendix A to part 423–126 priority pollutantsGoogle Scholar
  23. Waltman P (1983) Competition models in population biology. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsThe University of ToledoToledoU.S.A.
  2. 2.School of Mathematical and Statistical SciencesSouthwest UniversityChongqingChina

Personalised recommendations