2014, 11(6): 1319-1336. doi: 10.3934/mbe.2014.11.1319

Dynamics of two phytoplankton populations under predation

1. 

Department of Mathematical Sciences, Cameron University, 2800 West Gore Boulevard, Lawton, OK 73505, United States

Received  December 2013 Revised  May 2014 Published  September 2014

The aim of this paper is to investigate the manner in which predation and single-nutrient competition affect the dynamics of a non-toxic and a toxic phytoplankton species in a homogeneous environment (such as a chemostat). We allow for the possibility that both species serve as prey for an herbivorous zooplankton species. We assume that the toxic phytoplankton species produces toxins that affect only its own growth (autotoxicity). The autotoxicity assumption is ecologically explained by the fact that the toxin-producing phytoplankton is not mature enough to produce toxins that will affect the growth of its nontoxic competitor. We show that, in the absence of phytotoxic interactions and nutrient recycling, our model exhibits uniform persistence. The removal rates are distinct and we use general response functions. Finally, numerical simulations are carried out to show consistency with theoretical analysis. Our model has similarities with other food-chain models. As such, our results may be relevant to a wider spectrum of population models, not just those focused on plankton. Some open problems are discussed at the end of this paper.
Citation: Jean-Jacques Kengwoung-Keumo. Dynamics of two phytoplankton populations under predation. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1319-1336. doi: 10.3934/mbe.2014.11.1319
References:
[1]

H. M. Anderson, V. Hutson and R. Law, On the conditions for persistence of species in ecological communities,, Amer. Natur., 139 (1992), 663.   Google Scholar

[2]

R. Aris and A. E. Humphrey, Dynamics of a chemostat in which two organisms compete for a common substrate,, Biotechnol. Bioeng., 19 (1977), 1375.  doi: 10.1002/bit.260190910.  Google Scholar

[3]

M. M. Ballyk and G. S. K. Wolkowicz, Exploitative competition in the chemostat for two perfectly substitutable resources,, Math. Biosci., 118 (1993), 127.  doi: 10.1016/0025-5564(93)90050-K.  Google Scholar

[4]

E. Beretta, G. Bischi and F. Solimano, Stability in chemostat equations with delayed nutrient recycling,, J. Math. Biol., 28 (1990), 99.  doi: 10.1007/BF00171521.  Google Scholar

[5]

B. Boon and H. Laudelout, Kinetics of nitrite oxidation by Nitrobacter winogradskyi,, Biochem. J., 85 (1962), 440.   Google Scholar

[6]

G. Butler, H. I. Freedman and P. Waltman, Uniform persistent systems,, Proc. Amer. Math. Soc., 96 (1986), 425.  doi: 10.1090/S0002-9939-1986-0822433-4.  Google Scholar

[7]

G. Butler and P. Waltman, Persistence in dynamical systems,, J. Diff. Equ., 63 (1986), 255.  doi: 10.1016/0022-0396(86)90049-5.  Google Scholar

[8]

G. J. Butler and G. S. K. Wolkowicz, Predator-mediated competition in the chemostat,, J. Math. Biol., 24 (1986), 167.  doi: 10.1007/BF00275997.  Google Scholar

[9]

G. J. Butler and G. S. K. Wolkowicz, Exploitative competition in a chemostat for two complementary, and possibly inhibitory, resources,, Math. Biosci., 83 (1987), 1.  doi: 10.1016/0025-5564(87)90002-2.  Google Scholar

[10]

S. Chakraborty and J. Chattopadhyay, Nutrient-phytoplankton-zooplankton dynamics in the presence of additional food source-A mathematical study,, J. Biol. Syst., 16 (2008), 547.  doi: 10.1142/S0218339008002654.  Google Scholar

[11]

P. Chesson, J. M. Chase, P. A. Abrams, J. P. Grover, S. Diehl, R. D. Holt, S. A. Richards, R. M. Nisbet and T. J. Case, The interaction between predation and competition: A review and synthesis,, Eco. Let., 5 (2002), 302.   Google Scholar

[12]

W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations,, Heath, (1965).   Google Scholar

[13]

J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics,, Heidelberg, (1977).   Google Scholar

[14]

A. M. Edwards, Adding detritus to a nutrient-phytoplankton-zooplankton model: A dynamical-systems approach,, J. Plankton Res., 23 (2001), 389.  doi: 10.1093/plankt/23.4.389.  Google Scholar

[15]

A. M. Edwards and J. Brindley, Oscillatory behaviour in a three-component plankton population model,, Dyna. Stabi. Syst., 11 (1996), 347.  doi: 10.1080/02681119608806231.  Google Scholar

[16]

A. M. Edwards and J. Brindley, Zooplankton mortality and the dynamical behaviour of plankton population models,, Bull. Math. Biol., 61 (1999), 303.  doi: 10.1006/bulm.1998.0082.  Google Scholar

[17]

J. P. Grover and R. D. Holt, Disentangling resource and apparent competition: Realistic models for plant-herbivore communities,, J. Theor. Biol., 191 (1998), 353.  doi: 10.1006/jtbi.1997.0562.  Google Scholar

[18]

T. G. Hallam, On persistence of aquatic ecosystems., in Ocean. Sound Scat. Predic. (eds. N. R. Anderson and B. G. Zahurance), (1977), 749.   Google Scholar

[19]

T. G. Hallam, Controlled persistence in rudimentary plankton models,, in Proceedings of the First International Conference on Mathematical Modeling (eds. J. R. Avula), (1977), 2081.   Google Scholar

[20]

T. G. Hallam, Structural Sensitivity of grazing formulation in nutrient controlled plankton models,, J. Math. Biol., 5 (1978), 261.  doi: 10.1007/BF00276122.  Google Scholar

[21]

S. R. Hansen and S. P. Hubbell, Single-nutrient microbial competition: Qualitative agreement between experimental and theoretical forecast outcomes,, Sci., 207 (1980), 1491.  doi: 10.1126/science.6767274.  Google Scholar

[22]

R. D. Holt, J. Grover and D. Tilman, Simple rules for interspecific dominance in systems with exploitative and apparent competition,, Amer. Natur., 144 (1994), 741.  doi: 10.1086/285705.  Google Scholar

[23]

S. B. Hsu, Limiting behavior for competing species,, SIAM J. Appl. Math., 34 (1978), 760.  doi: 10.1137/0134064.  Google Scholar

[24]

J. P. Ivlev, Experimental Ecology of the Feeding of Fishes,, Yale University Press, (1961).   Google Scholar

[25]

S. R. J. Jang and J. Baglama, Nutrient-plankton models with nutrient recycling,, Comput. Math. Appl., 49 (2005), 375.  doi: 10.1016/j.camwa.2004.03.013.  Google Scholar

[26]

J. L. Jost, S. F. Drake, A. G. Fredrickson and M. Tsuchiya, Interaction of tetrahymena pyriformis, escherichia, coli, azotobacter vinelandii and glucose in a minimal medium,, J. Bacteriol., 113 (1976), 834.   Google Scholar

[27]

J.-J. Kengwoung-Keumo, Competition Between Two Phytoplankton Species Under Predation and Allelopathic Effects,, Ph.D. dissertation, (2012).   Google Scholar

[28]

J. A. León and D. B. Tumpson, Competition between two species for two complementary or substitutable resources,, J. Theor. Biol., 50 (1975), 185.   Google Scholar

[29]

B. Li and Y. Kuang, Simple Food Chain in a Chemostat with Distinct Removal Rates,, J. Math. Anal. and Appl., 242 (2000), 75.  doi: 10.1006/jmaa.1999.6655.  Google Scholar

[30]

R. K. Miller, Nonlinear Volterra Equation,, W. A. Benjamin, (1971).   Google Scholar

[31]

J. Monod, Recherche sur la Croissance des Cultures Bacteriennes,, Hermann et Cie, (1942).   Google Scholar

[32]

B. Mukhopadhyay and R. Bhattacharryya, Modelling phytoplankton allelopathy in a nutrient-plankton model with spatial heterogeneity,, Ecol. Model., 198 (2006), 163.  doi: 10.1016/j.ecolmodel.2006.04.005.  Google Scholar

[33]

L. Perko, Differential Equations and Dynamical Systems,, Third edition, (2001).  doi: 10.1007/978-1-4613-0003-8.  Google Scholar

[34]

D. Rapport, An optimization model of food selection,, Amer. Natur., 105 (1971), 575.  doi: 10.1086/282746.  Google Scholar

[35]

S. Roy, The coevolution of two phytoplankton species on a single resource: Allelopathy as a pseudo-mixotrophy,, Theor. Popul. Biol., 75 (2009), 68.  doi: 10.1016/j.tpb.2008.11.003.  Google Scholar

[36]

S. Ruan, Oscillations in plankton models with recycling,, J. Theor. Biol., 208 (2001), 15.   Google Scholar

[37]

S. Ruan, Persistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrient recycling,, J. Math. Biol., 31 (1993), 633.  doi: 10.1007/BF00161202.  Google Scholar

[38]

A. Sinkkonen, Modelling the effect of autotoxicity on density-dependent phytotoxicity,, J. Theor. Biol., 244 (2007), 218.  doi: 10.1016/j.jtbi.2006.08.003.  Google Scholar

[39]

J. H. Steele and E. W. Henderson, The role of predation in plankton models,, J. Plankton Res., 14 (1992), 157.  doi: 10.1093/plankt/14.1.157.  Google Scholar

[40]

M. A. Tabatabai, W. M. Eby, S. Bae and K. P. Singh, A flexible multivariable model for phytoplankton growth,, Math. Biosci. Eng., 10 (2013), 913.  doi: 10.3934/mbe.2013.10.913.  Google Scholar

[41]

H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations,, J. Math. Biol., 30 (1992), 755.  doi: 10.1007/BF00173267.  Google Scholar

[42]

G. S. K. Wolkowicz and Z. Lu, Global dynamics of a mathematical model of competition in the chemostat: General response functions and differential death rates,, J. Appl. Math., 52 (1992), 222.  doi: 10.1137/0152012.  Google Scholar

[43]

R. D. Yang and A. E. Humphrey, Dynamics and steady state studies of phenol biodegradation in pure and mixed cultures,, Biotechnol. Bioeng., 17 (1975), 1211.   Google Scholar

show all references

References:
[1]

H. M. Anderson, V. Hutson and R. Law, On the conditions for persistence of species in ecological communities,, Amer. Natur., 139 (1992), 663.   Google Scholar

[2]

R. Aris and A. E. Humphrey, Dynamics of a chemostat in which two organisms compete for a common substrate,, Biotechnol. Bioeng., 19 (1977), 1375.  doi: 10.1002/bit.260190910.  Google Scholar

[3]

M. M. Ballyk and G. S. K. Wolkowicz, Exploitative competition in the chemostat for two perfectly substitutable resources,, Math. Biosci., 118 (1993), 127.  doi: 10.1016/0025-5564(93)90050-K.  Google Scholar

[4]

E. Beretta, G. Bischi and F. Solimano, Stability in chemostat equations with delayed nutrient recycling,, J. Math. Biol., 28 (1990), 99.  doi: 10.1007/BF00171521.  Google Scholar

[5]

B. Boon and H. Laudelout, Kinetics of nitrite oxidation by Nitrobacter winogradskyi,, Biochem. J., 85 (1962), 440.   Google Scholar

[6]

G. Butler, H. I. Freedman and P. Waltman, Uniform persistent systems,, Proc. Amer. Math. Soc., 96 (1986), 425.  doi: 10.1090/S0002-9939-1986-0822433-4.  Google Scholar

[7]

G. Butler and P. Waltman, Persistence in dynamical systems,, J. Diff. Equ., 63 (1986), 255.  doi: 10.1016/0022-0396(86)90049-5.  Google Scholar

[8]

G. J. Butler and G. S. K. Wolkowicz, Predator-mediated competition in the chemostat,, J. Math. Biol., 24 (1986), 167.  doi: 10.1007/BF00275997.  Google Scholar

[9]

G. J. Butler and G. S. K. Wolkowicz, Exploitative competition in a chemostat for two complementary, and possibly inhibitory, resources,, Math. Biosci., 83 (1987), 1.  doi: 10.1016/0025-5564(87)90002-2.  Google Scholar

[10]

S. Chakraborty and J. Chattopadhyay, Nutrient-phytoplankton-zooplankton dynamics in the presence of additional food source-A mathematical study,, J. Biol. Syst., 16 (2008), 547.  doi: 10.1142/S0218339008002654.  Google Scholar

[11]

P. Chesson, J. M. Chase, P. A. Abrams, J. P. Grover, S. Diehl, R. D. Holt, S. A. Richards, R. M. Nisbet and T. J. Case, The interaction between predation and competition: A review and synthesis,, Eco. Let., 5 (2002), 302.   Google Scholar

[12]

W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations,, Heath, (1965).   Google Scholar

[13]

J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics,, Heidelberg, (1977).   Google Scholar

[14]

A. M. Edwards, Adding detritus to a nutrient-phytoplankton-zooplankton model: A dynamical-systems approach,, J. Plankton Res., 23 (2001), 389.  doi: 10.1093/plankt/23.4.389.  Google Scholar

[15]

A. M. Edwards and J. Brindley, Oscillatory behaviour in a three-component plankton population model,, Dyna. Stabi. Syst., 11 (1996), 347.  doi: 10.1080/02681119608806231.  Google Scholar

[16]

A. M. Edwards and J. Brindley, Zooplankton mortality and the dynamical behaviour of plankton population models,, Bull. Math. Biol., 61 (1999), 303.  doi: 10.1006/bulm.1998.0082.  Google Scholar

[17]

J. P. Grover and R. D. Holt, Disentangling resource and apparent competition: Realistic models for plant-herbivore communities,, J. Theor. Biol., 191 (1998), 353.  doi: 10.1006/jtbi.1997.0562.  Google Scholar

[18]

T. G. Hallam, On persistence of aquatic ecosystems., in Ocean. Sound Scat. Predic. (eds. N. R. Anderson and B. G. Zahurance), (1977), 749.   Google Scholar

[19]

T. G. Hallam, Controlled persistence in rudimentary plankton models,, in Proceedings of the First International Conference on Mathematical Modeling (eds. J. R. Avula), (1977), 2081.   Google Scholar

[20]

T. G. Hallam, Structural Sensitivity of grazing formulation in nutrient controlled plankton models,, J. Math. Biol., 5 (1978), 261.  doi: 10.1007/BF00276122.  Google Scholar

[21]

S. R. Hansen and S. P. Hubbell, Single-nutrient microbial competition: Qualitative agreement between experimental and theoretical forecast outcomes,, Sci., 207 (1980), 1491.  doi: 10.1126/science.6767274.  Google Scholar

[22]

R. D. Holt, J. Grover and D. Tilman, Simple rules for interspecific dominance in systems with exploitative and apparent competition,, Amer. Natur., 144 (1994), 741.  doi: 10.1086/285705.  Google Scholar

[23]

S. B. Hsu, Limiting behavior for competing species,, SIAM J. Appl. Math., 34 (1978), 760.  doi: 10.1137/0134064.  Google Scholar

[24]

J. P. Ivlev, Experimental Ecology of the Feeding of Fishes,, Yale University Press, (1961).   Google Scholar

[25]

S. R. J. Jang and J. Baglama, Nutrient-plankton models with nutrient recycling,, Comput. Math. Appl., 49 (2005), 375.  doi: 10.1016/j.camwa.2004.03.013.  Google Scholar

[26]

J. L. Jost, S. F. Drake, A. G. Fredrickson and M. Tsuchiya, Interaction of tetrahymena pyriformis, escherichia, coli, azotobacter vinelandii and glucose in a minimal medium,, J. Bacteriol., 113 (1976), 834.   Google Scholar

[27]

J.-J. Kengwoung-Keumo, Competition Between Two Phytoplankton Species Under Predation and Allelopathic Effects,, Ph.D. dissertation, (2012).   Google Scholar

[28]

J. A. León and D. B. Tumpson, Competition between two species for two complementary or substitutable resources,, J. Theor. Biol., 50 (1975), 185.   Google Scholar

[29]

B. Li and Y. Kuang, Simple Food Chain in a Chemostat with Distinct Removal Rates,, J. Math. Anal. and Appl., 242 (2000), 75.  doi: 10.1006/jmaa.1999.6655.  Google Scholar

[30]

R. K. Miller, Nonlinear Volterra Equation,, W. A. Benjamin, (1971).   Google Scholar

[31]

J. Monod, Recherche sur la Croissance des Cultures Bacteriennes,, Hermann et Cie, (1942).   Google Scholar

[32]

B. Mukhopadhyay and R. Bhattacharryya, Modelling phytoplankton allelopathy in a nutrient-plankton model with spatial heterogeneity,, Ecol. Model., 198 (2006), 163.  doi: 10.1016/j.ecolmodel.2006.04.005.  Google Scholar

[33]

L. Perko, Differential Equations and Dynamical Systems,, Third edition, (2001).  doi: 10.1007/978-1-4613-0003-8.  Google Scholar

[34]

D. Rapport, An optimization model of food selection,, Amer. Natur., 105 (1971), 575.  doi: 10.1086/282746.  Google Scholar

[35]

S. Roy, The coevolution of two phytoplankton species on a single resource: Allelopathy as a pseudo-mixotrophy,, Theor. Popul. Biol., 75 (2009), 68.  doi: 10.1016/j.tpb.2008.11.003.  Google Scholar

[36]

S. Ruan, Oscillations in plankton models with recycling,, J. Theor. Biol., 208 (2001), 15.   Google Scholar

[37]

S. Ruan, Persistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrient recycling,, J. Math. Biol., 31 (1993), 633.  doi: 10.1007/BF00161202.  Google Scholar

[38]

A. Sinkkonen, Modelling the effect of autotoxicity on density-dependent phytotoxicity,, J. Theor. Biol., 244 (2007), 218.  doi: 10.1016/j.jtbi.2006.08.003.  Google Scholar

[39]

J. H. Steele and E. W. Henderson, The role of predation in plankton models,, J. Plankton Res., 14 (1992), 157.  doi: 10.1093/plankt/14.1.157.  Google Scholar

[40]

M. A. Tabatabai, W. M. Eby, S. Bae and K. P. Singh, A flexible multivariable model for phytoplankton growth,, Math. Biosci. Eng., 10 (2013), 913.  doi: 10.3934/mbe.2013.10.913.  Google Scholar

[41]

H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations,, J. Math. Biol., 30 (1992), 755.  doi: 10.1007/BF00173267.  Google Scholar

[42]

G. S. K. Wolkowicz and Z. Lu, Global dynamics of a mathematical model of competition in the chemostat: General response functions and differential death rates,, J. Appl. Math., 52 (1992), 222.  doi: 10.1137/0152012.  Google Scholar

[43]

R. D. Yang and A. E. Humphrey, Dynamics and steady state studies of phenol biodegradation in pure and mixed cultures,, Biotechnol. Bioeng., 17 (1975), 1211.   Google Scholar

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