The impact of toxins on competition dynamics of three species in a polluted aquatic environment

Yuyue Zhang; Jicai Huang; Qihua Huang

doi:10.3934/dcdsb.2020219

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June 2021, 26(6): 3043-3068. doi: 10.3934/dcdsb.2020219

The impact of toxins on competition dynamics of three species in a polluted aquatic environment

Yuyue Zhang ^1, , Jicai Huang ^1, and Qihua Huang ^2,3,,

1.	School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei 430079, China
2.	School of Mathematical and Statistical Sciences, Southwest University, Chongqing 400715, China
3.	School of Mathematical and Statistical Sciences, Hubei University of Science and Technology, Xianning 437100, China

^*Corresponding author: Qihua Huang

Received February 2020 Revised May 2020 Published June 2021 Early access July 2020

Fund Project: The second author is supported by the NSF of China (11871235). The third author is supported by the NSF of China (11871060), the Venture and Innovation Support Program for Chongqing Overseas Returnees (7820100158), the Fundamental Research Funds for the Central Universities (XDJK2018B031), and the faculty startup fund from Southwest University (20710948)

Accurately assessing the risks of toxins in polluted ecosystems and finding factors that determine population persistence and extirpation are important from both environmental and conservation perspectives. In this paper, we develop and study a toxin-mediated competition model for three species that live in the same polluted aquatic environment and compete for the same resources. Analytical analysis of positive invariance, existence and stability of equilibria, sensitivity of equilibria to toxin are presented. Bifurcation analysis is used to understand how the environmental toxins, plus distinct vulnerabilities of three species to toxins, affect the competition outcomes. Our results reveal that while high concentrations lead to extirpation of all species, sublethal levels of toxins affect competition outcomes in many counterintuitive ways, which include boosting coexistence of species by reducing the abundance of the predominant species, inducing many different types of bistability and even tristability, generating and reducing population oscillations, and exchanging roles of winner and loser in competition. The findings in this work provide a sound theoretical foundation for understanding and assessing population or community effects of toxicity.

Keywords: Toxin, competition, fast-slow dynamics, coexistence, exclusion, bistability.

Mathematics Subject Classification: Primary: 92Bxx, 37N25; Secondary: 34C23.

Citation: Yuyue Zhang, Jicai Huang, Qihua Huang. The impact of toxins on competition dynamics of three species in a polluted aquatic environment. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3043-3068. doi: 10.3934/dcdsb.2020219

References:

[1]	S. M. Bartell, R. A. Pastorok, H. R. Akcakaya, H. Regan, S. Ferson and C. Mackay, Realism and relevance of ecological models used in chemical risk assessment, Hum. Ecol. Risk Assess., 9 (2003), 907-938. doi: 10.1080/713610016. Google Scholar
[2]	J. A. Camargo and Á. Alonso, Ecological and toxicological effects of inorganic nitrogen pollution in aquatic ecosystems: A global assessment, Environ. Int., 32 (2006), 831-849. doi: 10.1016/j.envint.2006.05.002. Google Scholar
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[4]	N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971/72), 193-226. doi: 10.1512/iumj.1972.21.21017. Google Scholar
[5]	N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Diff. Equ., 31 (1979), 53-98. doi: 10.1016/0022-0396(79)90152-9. Google Scholar
[6]	J. W. Fleeger, K. R. Carman and R. M. Nisbet, Indirect effects of contaminants in aquatic ecosystems, Sci. Total Environ., 317 (2003), 207-233. doi: 10.1016/S0048-9697(03)00141-4. Google Scholar
[7]	H. I. Freedman and J. B. Shukla, Models for the effect of toxicant in single-species and predator-prey systems, J. Math. Biol., 30 (1991), 15-30. doi: 10.1007/BF00168004. Google Scholar
[8]	M. Gyllenberg and P. Yan, On the number of limit cycles for three dimensional Lotka-Volterra systems, Discrete Contin. Dynam. Syst. Ser. B, 11 (2009), 347-352. doi: 10.3934/dcdsb.2009.11.347. Google Scholar
[9]	T. G. Hallam, C. E. Clark and G. S. Jordan, Effects of toxicants on populations: A qualitative approach Ⅱ. First order kinetics, J. Math. Biol., 18 (1983), 25-37. doi: 10.1007/BF00275908. Google Scholar
[10]	T. G. Hallam, C. E. Clark and R. R. Lassiter, Effects of toxicants on populations: A qualitative approach Ⅰ. Equilibrium environmental exposure, Ecol. Model., 18 (1983), 291-304. doi: 10.1016/0304-3800(83)90019-4. Google Scholar
[11]	T. G. Hallam and J. T. de Luna, Effects of toxicants on populations: A qualitative approach. III. Environmental and food chain pathways, J. Theor. Biol., 109 (1984), 411-429. doi: 10.1016/S0022-5193(84)80090-9. Google Scholar
[12]	T. Hanazato, Pesticide effects on freshwater zooplankton: An ecological perspective, Environ. Pollut., 112 (2001), 1-10. doi: 10.1016/S0269-7491(00)00110-X. Google Scholar
[13]	M. W. Hirsch, Systems of differential equations which are competitive or cooperative: III. Competing species, Nonlinearity, 1 (1988), 51-71. doi: 10.1088/0951-7715/1/1/003. Google Scholar
[14]	J. Hofbauer and J. W.-H. So, Multiple limit cycles for three dimensional Lotka-Volterra equations, Appl. Math. Lett., 7 (1994), 65-70. doi: 10.1016/0893-9659(94)90095-7. Google Scholar
[15]	Q. Huang, L. Parshotam, H. Wang, C. Bampfylde and M. A. Lewis, A model for the impact of contaminants on fish population dynamics, J. Theor. Biol., 334 (2013), 71-79. doi: 10.1016/j.jtbi.2013.05.018. Google Scholar
[16]	Q. Huang, H. Wang and M. A. Lewis, The impact of environmental toxins on predator-prey dynamics, J. Theor. Biol., 378 (2015), 12-30. doi: 10.1016/j.jtbi.2015.04.019. Google Scholar
[17]	J. Jiang and L. Niu, On the validity of Zeeman's classification for three dimensional competitive differential equations with linearly determined nullclines, J. Diff. Equ., 263 (2017), 7753-7781. doi: 10.1016/j.jde.2017.08.022. Google Scholar
[18]	G. Lan, C. Wei and S. Zhang, Long time behaviors of single-species population models with psychological effect and impulsive toxicant in polluted environments, Physica A: Statistical Mechanics and its Applications, 521 (2019), 828-842. doi: 10.1016/j.physa.2019.01.096. Google Scholar
[19]	Z. Lu and Y. Luo, Two limit cycles in three-dimensional Lotka-Volterra systems, Comput. Math. Appl., 44 (2002), 51-66. doi: 10.1016/S0898-1221(02)00129-3. Google Scholar
[20]	Z. Ma, G. Cui and W. Wang, Persistence and extinction of a population in a polluted environment, Math. Bios., 101 (1990), 75-97. doi: 10.1016/0025-5564(90)90103-6. Google Scholar
[21]	A. E. McElroy, M. G. Barron, N. Beckvar, S. B. K. Driscoll, J. P. Meador, T. F. Parkerton, T. G. Preuss and J. A. Steevens, A review of the tissue residue approach for organic and organometallic compounds in aquatic organisms, Integ. Environ. Assess. Manage., 7 (2011) 50–74. doi: 10.1002/ieam.132. Google Scholar
[22]	K. Murakami, A concrete example with multiple limit cycles for three dimensional Lotka-Volterra systems, J. Math. Anal. Appl., 457 (2018), 1-9. doi: 10.1016/j.jmaa.2017.07.076. Google Scholar
[23]	K. Murakami, A concrete example with three limit cycles in Zeeman's class 29 for three dimensional Lotka-Volterra competitive systems, Math. Bios., 308 (2019), 38-41. doi: 10.1016/j.mbs.2018.12.006. Google Scholar
[24]	R. A. Pastorok, S. M. Bartell, S. Ferson and L. R. Ginzburg, Ecological modeling in risk assessment: chemical effects on populations, ecosystems, and landscapes, CRC, Boca Raton, FL, USA, 2001. doi: 10.1201/9781420032321. Google Scholar
[25]	C. Shan and Q. Huang, Direct and indirect effects of toxins on competition dynamics of species in an aquatic environment, J. Math. Biol., 78 (2019), 739-766. doi: 10.1007/s00285-018-1290-2. Google Scholar
[26]	A. J. Smith and C. P. Tran, A weight-of-evidence approach to define nutrient criteria protective of aquatic life in large rivers, J. North Am. Benthol. Soc., 29 (2010), 875-891. doi: 10.1899/09-076.1. Google Scholar
[27]	D. M. Thomas, T. W. Snell and S. M. Jaffar, A control problem in a polluted environment, Math. Bios., 133 (1996), 139-163. doi: 10.1016/0025-5564(95)00091-7. Google Scholar
[28]	P. van den Driessche and M. L. Zeeman, Three-dimensional competitive Lotka-Volterra systems with no periodic orbits, SIAM J. Appl. Math., 58 (1998), 227-234. doi: 10.1137/S0036139995294767. Google Scholar
[29]	D. Xiao and W. Li, Limit cycles for the competitive three dimensional Lotka-Volterra system, J. Diff. Equ., 164 (2000), 1-15. doi: 10.1006/jdeq.1999.3729. Google Scholar
[30]	S. Yang, F. Xu, F. Wu, S. Wang and B. Zheng, Development of PFOS and PFOA criteria for the protection of freshwater aquatic life in China, Sci. Total Environ., 470-471 (2014), 677-683. doi: 10.1016/j.scitotenv.2013.09.094. Google Scholar
[31]	P. Yu, M. Han and D. Xiao, Four small limit cycles arond a Hopf singular point in 3-dimensional competitive Lotka-Volterra systems, J. Math. Anal. Appl., 436 (2016), 521-555. doi: 10.1016/j.jmaa.2015.12.002. Google Scholar
[32]	T. F. Zabel and S. Cole, The derivation of environmental quality standards for the protection of aquatic life in the UK, J. Chart. Inst. Water Environ. Manag., 13 (1999), 436-440. doi: 10.1111/j.1747-6593.1999.tb01082.x. Google Scholar
[33]	E. C. Zeeman and M. L. Zeeman, On the convexity of carrying simplices in competitive Lotka-Volterra systems., In Differential Equations, Dynamical Systems, and Control Science, Lecture Notes in Pure and Appl. Math., 152, Dekker, New York, (1994), 353–364 Google Scholar
[34]	E. C. Zeeman and M. L. Zeeman, An n-dimensional competitive Lotka-Volterra system is generically determined by the edges of its carrying simplex, Nonlinearity, 15 (2002), 2019-2032. doi: 10.1088/0951-7715/15/6/312. Google Scholar
[35]	M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stab. Sys., 8 (1993), 189-217. doi: 10.1080/02681119308806158. Google Scholar
[36]	T. Zhang, H. Jin and H. Zhu, Quality criteria of acrylonitrile for the protection of aquatic life in China, Chemosphere, 32 (1996), 2083-2093. doi: 10.1016/0045-6535(96)00110-5. Google Scholar

show all references

References:

[1]	S. M. Bartell, R. A. Pastorok, H. R. Akcakaya, H. Regan, S. Ferson and C. Mackay, Realism and relevance of ecological models used in chemical risk assessment, Hum. Ecol. Risk Assess., 9 (2003), 907-938. doi: 10.1080/713610016. Google Scholar
[2]	J. A. Camargo and Á. Alonso, Ecological and toxicological effects of inorganic nitrogen pollution in aquatic ecosystems: A global assessment, Environ. Int., 32 (2006), 831-849. doi: 10.1016/j.envint.2006.05.002. Google Scholar
[3]	W. H. Clements and C. Kotalik, Effects of major ions on natural benthic communities: an experimental assessment of the US Environmental Protection Agency aquatic life benchmark for conductivity, Freshw. Sci., 35 (2016), 126-138. doi: 10.1086/685085. Google Scholar
[4]	N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971/72), 193-226. doi: 10.1512/iumj.1972.21.21017. Google Scholar
[5]	N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Diff. Equ., 31 (1979), 53-98. doi: 10.1016/0022-0396(79)90152-9. Google Scholar
[6]	J. W. Fleeger, K. R. Carman and R. M. Nisbet, Indirect effects of contaminants in aquatic ecosystems, Sci. Total Environ., 317 (2003), 207-233. doi: 10.1016/S0048-9697(03)00141-4. Google Scholar
[7]	H. I. Freedman and J. B. Shukla, Models for the effect of toxicant in single-species and predator-prey systems, J. Math. Biol., 30 (1991), 15-30. doi: 10.1007/BF00168004. Google Scholar
[8]	M. Gyllenberg and P. Yan, On the number of limit cycles for three dimensional Lotka-Volterra systems, Discrete Contin. Dynam. Syst. Ser. B, 11 (2009), 347-352. doi: 10.3934/dcdsb.2009.11.347. Google Scholar
[9]	T. G. Hallam, C. E. Clark and G. S. Jordan, Effects of toxicants on populations: A qualitative approach Ⅱ. First order kinetics, J. Math. Biol., 18 (1983), 25-37. doi: 10.1007/BF00275908. Google Scholar
[10]	T. G. Hallam, C. E. Clark and R. R. Lassiter, Effects of toxicants on populations: A qualitative approach Ⅰ. Equilibrium environmental exposure, Ecol. Model., 18 (1983), 291-304. doi: 10.1016/0304-3800(83)90019-4. Google Scholar
[11]	T. G. Hallam and J. T. de Luna, Effects of toxicants on populations: A qualitative approach. III. Environmental and food chain pathways, J. Theor. Biol., 109 (1984), 411-429. doi: 10.1016/S0022-5193(84)80090-9. Google Scholar
[12]	T. Hanazato, Pesticide effects on freshwater zooplankton: An ecological perspective, Environ. Pollut., 112 (2001), 1-10. doi: 10.1016/S0269-7491(00)00110-X. Google Scholar
[13]	M. W. Hirsch, Systems of differential equations which are competitive or cooperative: III. Competing species, Nonlinearity, 1 (1988), 51-71. doi: 10.1088/0951-7715/1/1/003. Google Scholar
[14]	J. Hofbauer and J. W.-H. So, Multiple limit cycles for three dimensional Lotka-Volterra equations, Appl. Math. Lett., 7 (1994), 65-70. doi: 10.1016/0893-9659(94)90095-7. Google Scholar
[15]	Q. Huang, L. Parshotam, H. Wang, C. Bampfylde and M. A. Lewis, A model for the impact of contaminants on fish population dynamics, J. Theor. Biol., 334 (2013), 71-79. doi: 10.1016/j.jtbi.2013.05.018. Google Scholar
[16]	Q. Huang, H. Wang and M. A. Lewis, The impact of environmental toxins on predator-prey dynamics, J. Theor. Biol., 378 (2015), 12-30. doi: 10.1016/j.jtbi.2015.04.019. Google Scholar
[17]	J. Jiang and L. Niu, On the validity of Zeeman's classification for three dimensional competitive differential equations with linearly determined nullclines, J. Diff. Equ., 263 (2017), 7753-7781. doi: 10.1016/j.jde.2017.08.022. Google Scholar
[18]	G. Lan, C. Wei and S. Zhang, Long time behaviors of single-species population models with psychological effect and impulsive toxicant in polluted environments, Physica A: Statistical Mechanics and its Applications, 521 (2019), 828-842. doi: 10.1016/j.physa.2019.01.096. Google Scholar
[19]	Z. Lu and Y. Luo, Two limit cycles in three-dimensional Lotka-Volterra systems, Comput. Math. Appl., 44 (2002), 51-66. doi: 10.1016/S0898-1221(02)00129-3. Google Scholar
[20]	Z. Ma, G. Cui and W. Wang, Persistence and extinction of a population in a polluted environment, Math. Bios., 101 (1990), 75-97. doi: 10.1016/0025-5564(90)90103-6. Google Scholar
[21]	A. E. McElroy, M. G. Barron, N. Beckvar, S. B. K. Driscoll, J. P. Meador, T. F. Parkerton, T. G. Preuss and J. A. Steevens, A review of the tissue residue approach for organic and organometallic compounds in aquatic organisms, Integ. Environ. Assess. Manage., 7 (2011) 50–74. doi: 10.1002/ieam.132. Google Scholar
[22]	K. Murakami, A concrete example with multiple limit cycles for three dimensional Lotka-Volterra systems, J. Math. Anal. Appl., 457 (2018), 1-9. doi: 10.1016/j.jmaa.2017.07.076. Google Scholar
[23]	K. Murakami, A concrete example with three limit cycles in Zeeman's class 29 for three dimensional Lotka-Volterra competitive systems, Math. Bios., 308 (2019), 38-41. doi: 10.1016/j.mbs.2018.12.006. Google Scholar
[24]	R. A. Pastorok, S. M. Bartell, S. Ferson and L. R. Ginzburg, Ecological modeling in risk assessment: chemical effects on populations, ecosystems, and landscapes, CRC, Boca Raton, FL, USA, 2001. doi: 10.1201/9781420032321. Google Scholar
[25]	C. Shan and Q. Huang, Direct and indirect effects of toxins on competition dynamics of species in an aquatic environment, J. Math. Biol., 78 (2019), 739-766. doi: 10.1007/s00285-018-1290-2. Google Scholar
[26]	A. J. Smith and C. P. Tran, A weight-of-evidence approach to define nutrient criteria protective of aquatic life in large rivers, J. North Am. Benthol. Soc., 29 (2010), 875-891. doi: 10.1899/09-076.1. Google Scholar
[27]	D. M. Thomas, T. W. Snell and S. M. Jaffar, A control problem in a polluted environment, Math. Bios., 133 (1996), 139-163. doi: 10.1016/0025-5564(95)00091-7. Google Scholar
[28]	P. van den Driessche and M. L. Zeeman, Three-dimensional competitive Lotka-Volterra systems with no periodic orbits, SIAM J. Appl. Math., 58 (1998), 227-234. doi: 10.1137/S0036139995294767. Google Scholar
[29]	D. Xiao and W. Li, Limit cycles for the competitive three dimensional Lotka-Volterra system, J. Diff. Equ., 164 (2000), 1-15. doi: 10.1006/jdeq.1999.3729. Google Scholar
[30]	S. Yang, F. Xu, F. Wu, S. Wang and B. Zheng, Development of PFOS and PFOA criteria for the protection of freshwater aquatic life in China, Sci. Total Environ., 470-471 (2014), 677-683. doi: 10.1016/j.scitotenv.2013.09.094. Google Scholar
[31]	P. Yu, M. Han and D. Xiao, Four small limit cycles arond a Hopf singular point in 3-dimensional competitive Lotka-Volterra systems, J. Math. Anal. Appl., 436 (2016), 521-555. doi: 10.1016/j.jmaa.2015.12.002. Google Scholar
[32]	T. F. Zabel and S. Cole, The derivation of environmental quality standards for the protection of aquatic life in the UK, J. Chart. Inst. Water Environ. Manag., 13 (1999), 436-440. doi: 10.1111/j.1747-6593.1999.tb01082.x. Google Scholar
[33]	E. C. Zeeman and M. L. Zeeman, On the convexity of carrying simplices in competitive Lotka-Volterra systems., In Differential Equations, Dynamical Systems, and Control Science, Lecture Notes in Pure and Appl. Math., 152, Dekker, New York, (1994), 353–364 Google Scholar
[34]	E. C. Zeeman and M. L. Zeeman, An n-dimensional competitive Lotka-Volterra system is generically determined by the edges of its carrying simplex, Nonlinearity, 15 (2002), 2019-2032. doi: 10.1088/0951-7715/15/6/312. Google Scholar
[35]	M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stab. Sys., 8 (1993), 189-217. doi: 10.1080/02681119308806158. Google Scholar
[36]	T. Zhang, H. Jin and H. Zhu, Quality criteria of acrylonitrile for the protection of aquatic life in China, Chemosphere, 32 (1996), 2083-2093. doi: 10.1016/0045-6535(96)00110-5. Google Scholar

Figure 1. The illustration of Theorem 3.7

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Figure 2. Bifurcation diagrams with respect to toxin level $ T $ for the case where species-2 only equilibrium $ E_2 $ is globally asymptotically stable when $ T = 0 $. Parameters: $ b_1 = 1, \ b_2 = 1.06, \ b_3 = 1.13, \ p_1 = p_2 = p_3 = k_1 = k_2 = k_3 = 1, \ m_1 = 0.44, \ m_2 = 0.5, \ m_3 = 0.56, \ c_{12} = 1.13, \ c_{13} = 1.19, \ c_{21} = 1.06, \ c_{23} = 0.88, \ c_{31} = 0.75, \ c_{32} = 1.24 $. Here $ T_1^* = 0.28 $, $ T_2^*\approx0.27 $, $ T_3^*\approx0.27 $

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Fig. 2; (b) Limit cycle for $ T = 0.18069 $ in (a); (c) The time series diagram corresponding to (b); (d) The time series diagram of heteroclinic loop for $ T = 0.1801 $; (e) The phase portrait corresponding to (d). The other parameters are the same as those in Fig. 2">Figure 3. (a) The Hopf bifurcation diagram corresponding to Fig. 2; (b) Limit cycle for $ T = 0.18069 $ in (a); (c) The time series diagram corresponding to (b); (d) The time series diagram of heteroclinic loop for $ T = 0.1801 $; (e) The phase portrait corresponding to (d). The other parameters are the same as those in Fig. 2

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Fig. 2">Figure 4. Bifurcation diagrams with respect to toxin level $ T $ for the case where both species 2-only equilibrium $ E_2 $ and species 3-only equilibrium $ E_3 $ are locally asymptotically stable when $ T = 0 $. Parameters: $ c_{13} = 1.15, \ c_{23} = 1.19, \ c_{32} = 1.13 $, the other parameters and the values of $ T_i^*\ (i = 1, \ 2, \ 3) $ are the same as those in Fig. 2

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Figure 5. Bifurcation diagrams with respect to toxin level $ T $ for the case where the coexistence equilibrium $ E^* $ is globally asymptotically stable when $ T = 0 $. Part of panel (a) is enlarged by panel (b) and part of panel (c) is enlarged by panel (d). Parameters: $ b_1 = 1, \ b_2 = 0.97, \ b_3 = 0.96, \ p_1 = p_2 = p_3 = k_1 = k_2 = k_3 = 1, \ m_1 = 0.52, \ m_2 = 0.47, \ m_3 = 0.53, \ c_{12} = 0.11, \ c_{13} = 0.16, \ c_{21} = 0.16, \ c_{23} = 0.21, \ c_{31} = 0.15, \ c_{32} = 0.22 $. Here $ T_1^* = 0.24 $, $ T_2^*\approx0.25 $, $ T_3^*\approx0.22 $

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Fig. 2 except that $ p_2 = 0.5, \ p_3 = 0.2 $. Here, $ T_1^* = 0.28 $, $ T_2^*\approx 0.54 $, $ T_3^*\approx 1.34 $">Figure 6. Bifurcation diagrams with respect to toxin level $ T $, where the parameters are the same to those in Fig. 2 except that $ p_2 = 0.5, \ p_3 = 0.2 $. Here, $ T_1^* = 0.28 $, $ T_2^*\approx 0.54 $, $ T_3^*\approx 1.34 $

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Fig. 2 except that $ p_2 = 2, \ p_3 = 2.5 $. Here $ T_1^* = 0.28 $, $ T_2^*\approx0.14 $, $ T_3^*\approx 0.11 $">Figure 7. Bifurcation diagrams with respect to toxin level $ T $, where the parameters are similar to those in Fig. 2 except that $ p_2 = 2, \ p_3 = 2.5 $. Here $ T_1^* = 0.28 $, $ T_2^*\approx0.14 $, $ T_3^*\approx 0.11 $

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Table 1. The asymptotic dynamics on $ \Sigma $ of system (8). $ \bullet $ signifies an attractive equilibrium on $ \Sigma $, $ \circ $ signifies a repellent equilibrium on $ \Sigma $, the intersection of its hyperbolic manifolds signifies a saddle on $ \Sigma $

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2020 Impact Factor: 1.327

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2020 Impact Factor: 1.327

American Institute of Mathematical Sciences

The impact of toxins on competition dynamics of three species in a polluted aquatic environment

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American Institute of Mathematical Sciences

The impact of toxins on competition dynamics of three species in a polluted aquatic environment

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