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

Previous Article
An almost periodic Dengue transmission model with age structure and time-delayed input of vector in a patchy environment
DCDS-B Home
This Issue
Next Article
The Keller-Segel system with logistic growth and signal-dependent motility

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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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
[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.
[35]	M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stab. Sys., 8 (1993), 189-217. doi: 10.1080/02681119308806158.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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
[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.
[35]	M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stab. Sys., 8 (1993), 189-217. doi: 10.1080/02681119308806158.
[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.

Figure 1. The illustration of Theorem 3.7

Figure Options

Download full-size image

Download as PowerPoint slide

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 $

Figure Options

Download full-size image

Download as PowerPoint slide

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

Figure Options

Download full-size image

Download as PowerPoint slide

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

Figure Options

Download full-size image

Download as PowerPoint slide

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 $

Figure Options

Download full-size image

Download as PowerPoint slide

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 $

Figure Options

Download full-size image

Download as PowerPoint slide

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 $

Figure Options

Download full-size image

Download as PowerPoint slide

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 $

Table Options

Download as excel

[1]	Alexandre Caboussat, Allison Leonard. Numerical solution and fast-slow decomposition of a population of weakly coupled systems. Conference Publications, 2009, 2009 (Special) : 123-132. doi: 10.3934/proc.2009.2009.123
[2]	Luca Dieci, Cinzia Elia. Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2935-2950. doi: 10.3934/dcdsb.2018112
[3]	Hao Wang, Katherine Dunning, James J. Elser, Yang Kuang. Daphnia species invasion, competitive exclusion, and chaotic coexistence. Discrete and Continuous Dynamical Systems - B, 2009, 12 (2) : 481-493. doi: 10.3934/dcdsb.2009.12.481
[4]	Andrea Giorgini. On the Swift-Hohenberg equation with slow and fast dynamics: well-posedness and long-time behavior. Communications on Pure and Applied Analysis, 2016, 15 (1) : 219-241. doi: 10.3934/cpaa.2016.15.219
[5]	Chunhua Shan. Slow-fast dynamics and nonlinear oscillations in transmission of mosquito-borne diseases. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1447-1469. doi: 10.3934/dcdsb.2021097
[6]	Azmy S. Ackleh, Youssef M. Dib, S. R.-J. Jang. Competitive exclusion and coexistence in a nonlinear refuge-mediated selection model. Discrete and Continuous Dynamical Systems - B, 2007, 7 (4) : 683-698. doi: 10.3934/dcdsb.2007.7.683
[7]	Yixiang Wu, Necibe Tuncer, Maia Martcheva. Coexistence and competitive exclusion in an SIS model with standard incidence and diffusion. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 1167-1187. doi: 10.3934/dcdsb.2017057
[8]	M. R. S. Kulenović, Orlando Merino. Competitive-exclusion versus competitive-coexistence for systems in the plane. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1141-1156. doi: 10.3934/dcdsb.2006.6.1141
[9]	Azmy S. Ackleh, Keng Deng, Yixiang Wu. Competitive exclusion and coexistence in a two-strain pathogen model with diffusion. Mathematical Biosciences & Engineering, 2016, 13 (1) : 1-18. doi: 10.3934/mbe.2016.13.1
[10]	Dan Li, Hui Wan. Coexistence and exclusion of competitive Kolmogorov systems with semi-Markovian switching. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4145-4183. doi: 10.3934/dcds.2021032
[11]	Hua Nie, Sze-bi Hsu, Jianhua Wu. A competition model with dynamically allocated toxin production in the unstirred chemostat. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1373-1404. doi: 10.3934/cpaa.2017066
[12]	C. Connell Mccluskey. Lyapunov functions for tuberculosis models with fast and slow progression. Mathematical Biosciences & Engineering, 2006, 3 (4) : 603-614. doi: 10.3934/mbe.2006.3.603
[13]	Nahla Abdellatif, Radhouane Fekih-Salem, Tewfik Sari. Competition for a single resource and coexistence of several species in the chemostat. Mathematical Biosciences & Engineering, 2016, 13 (4) : 631-652. doi: 10.3934/mbe.2016012
[14]	Ya Li, Z. Feng. Dynamics of a plant-herbivore model with toxin-induced functional response. Mathematical Biosciences & Engineering, 2010, 7 (1) : 149-169. doi: 10.3934/mbe.2010.7.149
[15]	Jie Xu, Yu Miao, Jicheng Liu. Strong averaging principle for slow-fast SPDEs with Poisson random measures. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2233-2256. doi: 10.3934/dcdsb.2015.20.2233
[16]	Seung-Yeal Ha, Dohyun Kim, Jinyeong Park. Fast and slow velocity alignments in a Cucker-Smale ensemble with adaptive couplings. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4621-4654. doi: 10.3934/cpaa.2020209
[17]	Alexandre Vidal. Periodic orbits of tritrophic slow-fast system and double homoclinic bifurcations. Conference Publications, 2007, 2007 (Special) : 1021-1030. doi: 10.3934/proc.2007.2007.1021
[18]	Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257
[19]	Renato Huzak. Cyclicity of the origin in slow-fast codimension 3 saddle and elliptic bifurcations. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 171-215. doi: 10.3934/dcds.2016.36.171
[20]	Chunhua Jin. Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1675-1688. doi: 10.3934/dcdsb.2018069

2021 Impact Factor: 1.497

Tools

Metrics

Tools

Article outline

Figures and Tables

2021 Impact Factor: 1.497

Tools

Figures and Tables

2021 Impact Factor: 1.497

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors