doi: 10.3934/dcdsb.2020279

Dynamics of the food-chain population in a polluted environment with impulsive input of toxicant

Department of Mathematics, City University of Hong Kong, Hong Kong, China

* Corresponding author: wingclo@cityu.edu.hk

Received  April 2020 Revised  July 2020 Published  September 2020

Some industrial behaviors, such as wasting outputs and inadequately treated and stored hazardous materials, may pollute our environment, so some populations in the polluted habitats are at the edge of extinction. In this work, we develop a mathematical model that validates the dynamics of the food-chain population in a polluted environment with impulsive toxicant input. Based on the model, we obtain a sufficient condition for the extinction of populations. When the concentration of toxicants surpasses the threshold, it will contribute to the extinction of populations in the related environment. Also, sufficient conditions for the permanence of populations are obtained in our analysis. Several numerical simulations validate the theoretical conclusions and further reflect the influence of toxicants.

Citation: Yu Mu, Wing-Cheong Lo. Dynamics of the food-chain population in a polluted environment with impulsive input of toxicant. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020279
References:
[1]

S. Aquino and D. Stuckey, Soluble microbial products formation in anaerobic chemostats in the presence of toxic compounds, Water Res., 38 (2004), 255-266.  doi: 10.1016/j.watres.2003.09.031.  Google Scholar

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F. S. BacelarS. DueriE. Hernández-García and J. Zaldívar, Joint effects of nutrients and contaminants on the dynamics of a food chain in marine ecosystems, Math. Biosci., 218 (2009), 24-32.  doi: 10.1016/j.mbs.2008.12.002.  Google Scholar

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M. P. BoerB. W. Kooi and S. A. L. M. Kooijman, Multiple attractors and boundary crises in a tri-trophic food chain, Math. Biosci., 169 (2001), 109-128.  doi: 10.1016/S0025-5564(00)00058-4.  Google Scholar

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M. M. A. El-Sheikh and S. A. A. Mahrouf, Stability and bifurcation of a simple food chain in a chemostat with removal rates, Chaos Soliton. Fract., 23 (2005), 1475-1489.  doi: 10.1016/j.chaos.2004.06.079.  Google Scholar

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R. Fekih-SalemC. Lobry and T. Sari, A density-dependent model of competition for one resource in the chemostat, Math. Biosci., 286 (2017), 104-122.  doi: 10.1016/j.mbs.2017.02.007.  Google Scholar

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R. Fekih-SalemA. Rapaport and T. Sari, Emergence of coexistence and limit cycles in the chemostat model with flocculation for a general class of functional responses, Appl. Math. Model., 40 (2016), 7656-7677.  doi: 10.1016/j.apm.2016.03.028.  Google Scholar

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J. W. FleegerK. 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

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A. GragnaniO. De Feo and S. Rinaldi, Food chains in the chemostat: Relationships between mean yield and complex dynamics, B. Math. Biol., 60 (1998), 703-719.  doi: 10.1006/bulm.1997.0039.  Google Scholar

[11]

C. R. Gwaltney and M. A. Stadtherr, Reliable computation of equilibrium states and bifurcations in ecological systems analysis, Comput. Chem. Eng., 31 (2007), 993-1005.  doi: 10.1016/j.compchemeng.2006.10.011.  Google Scholar

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A. Hastings and T. Powell, Chaos in a three-species food chain, Ecology, 72 (1991), 896-903.  doi: 10.2307/1940591.  Google Scholar

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S. B. HsuS. Hubbell and P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, SIAM J. Appl. Math., 32 (1977), 366-383.  doi: 10.1137/0132030.  Google Scholar

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B. Li and Y. Kuang, Simple food chain in a chemostat with distinct removal rates, J. Math. Anal. Appl., 242 (2000), 75-92.  doi: 10.1006/jmaa.1999.6655.  Google Scholar

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Y. Li and X. Meng, Dynamics of an impulsive stochastic nonautonomous chemostat model with two different growth rates in a polluted environment, Discrete Dyn. Nat. Soc., 2019 (2019), ID 5498569, 15 pp. doi: 10.1155/2019/5498569.  Google Scholar

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B. LiuL. Chen and Y. Zhang, The effects of impulsive toxicant input on a population in a polluted environment, J. Biol. Syst., 11 (2003), 265-274.  doi: 10.1142/S0218339003000907.  Google Scholar

[21]

X. MengL. Wang and T. Zhang, Global dynamics analysis of a nonlinear impulsive stochastic chemostat system in a polluted environment, J. Appl. Anal. Comput., 6 (2016), 865-875.  doi: 10.11948/2016055.  Google Scholar

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X. Meng, Z. Li and J. J. Nieto, Dynamic analysis of Michaelis–Menten chemostat-type competition models with time delay and pulse in a polluted environment, J. Math. Chem., 47 (2010), 123. doi: 10.1007/s10910-009-9536-2.  Google Scholar

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M. Newman, Fundamentals of ecotoxicology: The science of pollution, 5$^{th}$ edition, Boca Raton, FL: CRC Press, 2019. [26] A. Novick and L. Szilard, Descriptio doi: 10.1201/9781351133999.  Google Scholar

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A. Novick and L. Szilard, Description of the chemostat, Science, 112 (1950), S715–716. doi: 10.1126/science.112.2920.715.  Google Scholar

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A. Novick and L. Szilard, Experiments with the chemostat on spontaneous mutations of bacteria, Proc. Natl. Acad. Sci. USA, 36 (1950), 708-719.  doi: 10.1073/pnas.36.12.708.  Google Scholar

[28]

D. M. PostM. L. Pace and N. G. Hairston, Ecosystem size determines food-chain length in lakes, Nature, 405 (2000), 1047-1049.  doi: 10.1038/35016565.  Google Scholar

[29]

M. P. Rayman, Food-chain selenium and human health: Emphasis on intake, Brit. J. Nutr., 100 (2008), 254-268.  doi: 10.1017/S0007114508939830.  Google Scholar

[30]

R. Relyea and J. Hoverman, Assessing the ecology in ecotoxicology: A review and synthesis in freshwater systems, Ecol. Lett., 9 (2006), 1157-1171.  doi: 10.1111/j.1461-0248.2006.00966.x.  Google Scholar

[31]

J. R. RohrJ. L. Kerby and A. Sih, Community ecology as a framework for predicting contaminant effects, Trends Ecol. Evol., 21 (2006), 606-613.  doi: 10.1016/j.tree.2006.07.002.  Google Scholar

[32]

M. Scheffer and S. R. Carpenter, Catastrophic regime shifts in ecosystems: Linking theory to observation, Trends Ecol. Evol., 18 (2003), 648-656.  doi: 10.1016/j.tree.2003.09.002.  Google Scholar

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M. SchefferS. CarpenterJ. A. FoleyC. Folke and B. Walker, Catastrophic shifts in ecosystems, Nature, 413 (2001), 591-596.  doi: 10.1038/35098000.  Google Scholar

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H. L. Smith and P. Waltman, The theory of the chemostat: Dynamics of microbial competition, Cambridge University Press, Cambridge, UK, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[35]

G. Suter II, Ecological Risk Assessment, 2$^{nd}$ edition, Boca Raton, FL: CRC Press, 2006. Google Scholar

[36]

A. M. VerschoorM. Vos and I. Van Der Stap, Inducible defences prevent strong population fluctuations in bi-and tritrophic food chains, Ecol. Lett., 7 (2004), 1143-1148.  doi: 10.1111/j.1461-0248.2004.00675.x.  Google Scholar

[37]

F. WangC. Hao and L. Chen, Bifurcation and chaos in a Monod–Haldene type food chain chemostat with pulsed input and washout, Chaos Soliton. Fract., 32 (2007), 181-194.  doi: 10.1016/j.chaos.2005.10.083.  Google Scholar

[38]

F. WangC. Hao and L. Chen, Bifurcation and chaos in a Tessiet type food chain chemostat with pulsed input and washout, Chaos Soliton. Fract., 32 (2007), 1547-1561.  doi: 10.1016/j.chaos.2005.12.006.  Google Scholar

[39]

Y. ZhangZ. Xiu and L. Chen, Chaos in a food chain chemostat with pulsed input and washout, Chaos Soliton. Fract., 26 (2005), 159-166.  doi: 10.1016/j.chaos.2004.12.040.  Google Scholar

[40]

Z. ZhaoL. Chen and X. Song, Extinction and permanence of chemostat model with pulsed input in a polluted environment, Commun. Nonlinear Sci., 14 (2009), 1737-1745.  doi: 10.1016/j.cnsns.2008.01.009.  Google Scholar

show all references

References:
[1]

S. Aquino and D. Stuckey, Soluble microbial products formation in anaerobic chemostats in the presence of toxic compounds, Water Res., 38 (2004), 255-266.  doi: 10.1016/j.watres.2003.09.031.  Google Scholar

[2]

F. S. BacelarS. DueriE. Hernández-García and J. Zaldívar, Joint effects of nutrients and contaminants on the dynamics of a food chain in marine ecosystems, Math. Biosci., 218 (2009), 24-32.  doi: 10.1016/j.mbs.2008.12.002.  Google Scholar

[3]

M. P. BoerB. W. Kooi and S. A. L. M. Kooijman, Food chain dynamics in the chemostat, Math. Biosci., 150 (1998), 43-62.  doi: 10.1016/S0025-5564(98)00010-8.  Google Scholar

[4]

M. P. BoerB. W. Kooi and S. A. L. M. Kooijman, Multiple attractors and boundary crises in a tri-trophic food chain, Math. Biosci., 169 (2001), 109-128.  doi: 10.1016/S0025-5564(00)00058-4.  Google Scholar

[5]

G. Cabana and J. B. Rasmussen, Modelling food chain structure and contaminant bioaccumulation using stable nitrogen isotopes, Nature, 372 (1994), 255-257.  doi: 10.1038/372255a0.  Google Scholar

[6]

M. M. A. El-Sheikh and S. A. A. Mahrouf, Stability and bifurcation of a simple food chain in a chemostat with removal rates, Chaos Soliton. Fract., 23 (2005), 1475-1489.  doi: 10.1016/j.chaos.2004.06.079.  Google Scholar

[7]

R. Fekih-SalemC. Lobry and T. Sari, A density-dependent model of competition for one resource in the chemostat, Math. Biosci., 286 (2017), 104-122.  doi: 10.1016/j.mbs.2017.02.007.  Google Scholar

[8]

R. Fekih-SalemA. Rapaport and T. Sari, Emergence of coexistence and limit cycles in the chemostat model with flocculation for a general class of functional responses, Appl. Math. Model., 40 (2016), 7656-7677.  doi: 10.1016/j.apm.2016.03.028.  Google Scholar

[9]

J. W. FleegerK. 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

[10]

A. GragnaniO. De Feo and S. Rinaldi, Food chains in the chemostat: Relationships between mean yield and complex dynamics, B. Math. Biol., 60 (1998), 703-719.  doi: 10.1006/bulm.1997.0039.  Google Scholar

[11]

C. R. Gwaltney and M. A. Stadtherr, Reliable computation of equilibrium states and bifurcations in ecological systems analysis, Comput. Chem. Eng., 31 (2007), 993-1005.  doi: 10.1016/j.compchemeng.2006.10.011.  Google Scholar

[12]

S. J. Hamilton, Review of selenium toxicity in the aquatic food chain, Sci. Total Environ., 326 (2004), 1-31.  doi: 10.1016/j.scitotenv.2004.01.019.  Google Scholar

[13]

A. Hastings and T. Powell, Chaos in a three-species food chain, Ecology, 72 (1991), 896-903.  doi: 10.2307/1940591.  Google Scholar

[14]

S. B. HsuS. Hubbell and P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, SIAM J. Appl. Math., 32 (1977), 366-383.  doi: 10.1137/0132030.  Google Scholar

[15]

J. JiaoK. Ye and L. Chen, Dynamical analysis of a five-dimensioned chemostat model with impulsive diffusion and pulse input environmental toxicant, Chaos Soliton. Fract., 44 (2011), 17-27.  doi: 10.1016/j.chaos.2010.11.001.  Google Scholar

[16]

J. Jiao and L. Chen, Dynamical analysis of a chemostat model with delayed response in growth and pulse input in polluted environment, J. Math. Chem., 46 (2009), 502-513.  doi: 10.1007/s10910-008-9474-4.  Google Scholar

[17]

Yu. A. KuznetsovO. De Feo and S. Rinaldi, Belyakov homoclinic bifurcations in a tritrophic food chain model, SIAM J. Appl. Math., 62 (2001), 462-487.  doi: 10.1137/S0036139900378542.  Google Scholar

[18]

B. Li and Y. Kuang, Simple food chain in a chemostat with distinct removal rates, J. Math. Anal. Appl., 242 (2000), 75-92.  doi: 10.1006/jmaa.1999.6655.  Google Scholar

[19]

Y. Li and X. Meng, Dynamics of an impulsive stochastic nonautonomous chemostat model with two different growth rates in a polluted environment, Discrete Dyn. Nat. Soc., 2019 (2019), ID 5498569, 15 pp. doi: 10.1155/2019/5498569.  Google Scholar

[20]

B. LiuL. Chen and Y. Zhang, The effects of impulsive toxicant input on a population in a polluted environment, J. Biol. Syst., 11 (2003), 265-274.  doi: 10.1142/S0218339003000907.  Google Scholar

[21]

X. MengL. Wang and T. Zhang, Global dynamics analysis of a nonlinear impulsive stochastic chemostat system in a polluted environment, J. Appl. Anal. Comput., 6 (2016), 865-875.  doi: 10.11948/2016055.  Google Scholar

[22]

X. Meng, Z. Li and J. J. Nieto, Dynamic analysis of Michaelis–Menten chemostat-type competition models with time delay and pulse in a polluted environment, J. Math. Chem., 47 (2010), 123. doi: 10.1007/s10910-009-9536-2.  Google Scholar

[23]

J. Monod, La technique de culture continue: Theorie et applications, Selected Papers in Molecular Biology by Jacques Monod, 1978,184–204. doi: 10.1016/B978-0-12-460482-7.50023-3.  Google Scholar

[24]

J. Monod, The growth of bacterial cultures, Annu. Rev. Microbiol., 3 (1949), 371-394.  doi: 10.1146/annurev.mi.03.100149.002103.  Google Scholar

[25]

M. Newman, Fundamentals of ecotoxicology: The science of pollution, 5$^{th}$ edition, Boca Raton, FL: CRC Press, 2019. [26] A. Novick and L. Szilard, Descriptio doi: 10.1201/9781351133999.  Google Scholar

[26]

A. Novick and L. Szilard, Description of the chemostat, Science, 112 (1950), S715–716. doi: 10.1126/science.112.2920.715.  Google Scholar

[27]

A. Novick and L. Szilard, Experiments with the chemostat on spontaneous mutations of bacteria, Proc. Natl. Acad. Sci. USA, 36 (1950), 708-719.  doi: 10.1073/pnas.36.12.708.  Google Scholar

[28]

D. M. PostM. L. Pace and N. G. Hairston, Ecosystem size determines food-chain length in lakes, Nature, 405 (2000), 1047-1049.  doi: 10.1038/35016565.  Google Scholar

[29]

M. P. Rayman, Food-chain selenium and human health: Emphasis on intake, Brit. J. Nutr., 100 (2008), 254-268.  doi: 10.1017/S0007114508939830.  Google Scholar

[30]

R. Relyea and J. Hoverman, Assessing the ecology in ecotoxicology: A review and synthesis in freshwater systems, Ecol. Lett., 9 (2006), 1157-1171.  doi: 10.1111/j.1461-0248.2006.00966.x.  Google Scholar

[31]

J. R. RohrJ. L. Kerby and A. Sih, Community ecology as a framework for predicting contaminant effects, Trends Ecol. Evol., 21 (2006), 606-613.  doi: 10.1016/j.tree.2006.07.002.  Google Scholar

[32]

M. Scheffer and S. R. Carpenter, Catastrophic regime shifts in ecosystems: Linking theory to observation, Trends Ecol. Evol., 18 (2003), 648-656.  doi: 10.1016/j.tree.2003.09.002.  Google Scholar

[33]

M. SchefferS. CarpenterJ. A. FoleyC. Folke and B. Walker, Catastrophic shifts in ecosystems, Nature, 413 (2001), 591-596.  doi: 10.1038/35098000.  Google Scholar

[34]

H. L. Smith and P. Waltman, The theory of the chemostat: Dynamics of microbial competition, Cambridge University Press, Cambridge, UK, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[35]

G. Suter II, Ecological Risk Assessment, 2$^{nd}$ edition, Boca Raton, FL: CRC Press, 2006. Google Scholar

[36]

A. M. VerschoorM. Vos and I. Van Der Stap, Inducible defences prevent strong population fluctuations in bi-and tritrophic food chains, Ecol. Lett., 7 (2004), 1143-1148.  doi: 10.1111/j.1461-0248.2004.00675.x.  Google Scholar

[37]

F. WangC. Hao and L. Chen, Bifurcation and chaos in a Monod–Haldene type food chain chemostat with pulsed input and washout, Chaos Soliton. Fract., 32 (2007), 181-194.  doi: 10.1016/j.chaos.2005.10.083.  Google Scholar

[38]

F. WangC. Hao and L. Chen, Bifurcation and chaos in a Tessiet type food chain chemostat with pulsed input and washout, Chaos Soliton. Fract., 32 (2007), 1547-1561.  doi: 10.1016/j.chaos.2005.12.006.  Google Scholar

[39]

Y. ZhangZ. Xiu and L. Chen, Chaos in a food chain chemostat with pulsed input and washout, Chaos Soliton. Fract., 26 (2005), 159-166.  doi: 10.1016/j.chaos.2004.12.040.  Google Scholar

[40]

Z. ZhaoL. Chen and X. Song, Extinction and permanence of chemostat model with pulsed input in a polluted environment, Commun. Nonlinear Sci., 14 (2009), 1737-1745.  doi: 10.1016/j.cnsns.2008.01.009.  Google Scholar

Figure 1.  Schematic diagram of system. Fresh nutrient $ S(t) $ is supplied in the chemostat, which represents the supplement of natural resources in the ecosystem. Prey population $ x_1(t) $ consumes the nutrient $ S(t) $ in the system and further provides food to the predator population $ x_2(t) $. Finally, the super predator population $ x_3(t) $ exploits the predator population $ x_2(t) $. The toxicant effect from the ecosystem with impulsive input may disturb the population dynamics and reduce the populations. Besides the dilution process of the chemostat for the nutrient $ S(t) $ and populations $ x_i(t)\ (i = 1, 2, 3) $, the mortality process of the populations $ x_i(t) $ will also alter the dynamics of populations
Figure 2.  Numerical simulation of the solution for the case that $ \frac{m_1S^0}{(n_1+S^0)(d_1+r_1\beta_0)} = 0.747 < 1 $. (a) Temporal dynamic of the solution $ S(t) $. (b) Temporal dynamics of the solutions $ x_i(t)\ (i = 1, 2, 3) $. (c) Temporal dynamic of the solution $ T_0(t) $. (d) Temporal dynamic of the solution $ T_e(t) $
Figure 3.  Numerical simulation of the solution for the case that $ \frac{m_1S^0}{n_1+S^0}-d_1-\frac{m_2S^0}{n_2}-r_1\beta_0 = 0.2311 > 0 $ and $ \frac{m_2S^0}{n_2+S^0}-d_2-r_2\beta_0 = -0.0558 < 0 $. (a) Temporal dynamic of the solution $ S(t) $. (b) Temporal dynamics of the solutions $ x_i(t)\ (i = 1, 2, 3) $
Figure 4.  Numerical simulation of the solution for the case that $ A > 0 $, $ B > 0 $ and $ C<0 $. (a) Temporal dynamic of the solution $ S(t) $. (b) Temporal dynamic of the solution $ x_1(t) $. (c) Temporal dynamic of the solution $ x_2(t) $. (d) Temporal dynamic of the solution $ x_3(t) $
Figure 5.  Numerical simulation of the solution for the case that $ A > 0 $, $ B > 0 $ and $ C<0 $. (a) Portrait phase of $ x_1(t) $ and $ x_2(t) $. (b) Portrait phase of $ S(t) $ and $ x_i(t)\ (i = 1, 2) $
Figure 6.  Numerical simulation of the solution for the case that $ D> 0 $ and $ E> 0 $. (a) Temporal dynamic of the solution $ S(t) $. (b) Temporal dynamic of the solution $ x_1(t) $. (c) Temporal dynamic of the solution $ x_2(t) $. (d) Temporal dynamic of the solution $ x_3(t) $
Figure 7.  Numerical simulation of the the solution for the case that $ D> 0 $ and $ E> 0 $. (a) Portrait phase of $ S(t) $ and $ x_1(t) $. (b) Portrait phase of $ S(t) $ and $ x_2(t) $. (c) Portrait phase of $ S(t) $ and $ x_3(t) $. (d) Portrait phase of $ x_1(t), x_2(t) $ and $ x_3(t) $
Table 1.  Biological meanings of the variables and the parameters
Term Biological meaning
$ S(t) $ Concentration of nutrient at time $ t $
$ x_i(t) $ Concentration of prey, predator and superpredator at time $ t $ ($ i = 1, 2, 3 $)
$ T_0(t) $ Concentration of toxicant in the population at time $ t $
$ T_e(t) $ Concentration of toxicant in the environment at time $ t $
$ S^0 $ Concentration of input nutrient
$ d $ Dilution rate of the system
$ d_i $ Sum of dilution rate of system and
mortality rate of populations ($ i = 1, 2, 3 $)
$ m_i $ Maximum growth rate ($ i = 1, 2, 3 $)
$ n_i $ Half saturation constant ($ i = 1, 2, 3 $)
$ r_i $ Rate of decrease of the growth rate for population $ x_i $ ($ i = 1, 2, 3 $)
$ k $ Environmental toxicant uptake rate per unit mass population
$ g $ Population net ingestion and depuration rates of toxicant
$ v $ Loss rate of toxicants from the environment through volatilization
$ \Lambda $ Impulsive input constant for toxicant at time $ nT, \ n\in Z^+ $
Term Biological meaning
$ S(t) $ Concentration of nutrient at time $ t $
$ x_i(t) $ Concentration of prey, predator and superpredator at time $ t $ ($ i = 1, 2, 3 $)
$ T_0(t) $ Concentration of toxicant in the population at time $ t $
$ T_e(t) $ Concentration of toxicant in the environment at time $ t $
$ S^0 $ Concentration of input nutrient
$ d $ Dilution rate of the system
$ d_i $ Sum of dilution rate of system and
mortality rate of populations ($ i = 1, 2, 3 $)
$ m_i $ Maximum growth rate ($ i = 1, 2, 3 $)
$ n_i $ Half saturation constant ($ i = 1, 2, 3 $)
$ r_i $ Rate of decrease of the growth rate for population $ x_i $ ($ i = 1, 2, 3 $)
$ k $ Environmental toxicant uptake rate per unit mass population
$ g $ Population net ingestion and depuration rates of toxicant
$ v $ Loss rate of toxicants from the environment through volatilization
$ \Lambda $ Impulsive input constant for toxicant at time $ nT, \ n\in Z^+ $
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