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), 663-668.

[2]

R. Aris and A. E. Humphrey, Dynamics of a chemostat in which two organisms compete for a common substrate, Biotechnol. Bioeng., 19 (1977), 1375-1386. doi: 10.1002/bit.260190910.

[3]

M. M. Ballyk and G. S. K. Wolkowicz, Exploitative competition in the chemostat for two perfectly substitutable resources, Math. Biosci., 118 (1993), 127-180. doi: 10.1016/0025-5564(93)90050-K.

[4]

E. Beretta, G. Bischi and F. Solimano, Stability in chemostat equations with delayed nutrient recycling, J. Math. Biol., 28 (1990), 99-111. doi: 10.1007/BF00171521.

[5]

B. Boon and H. Laudelout, Kinetics of nitrite oxidation by Nitrobacter winogradskyi, Biochem. J., 85 (1962), 440-447.

[6]

G. Butler, H. I. Freedman and P. Waltman, Uniform persistent systems, Proc. Amer. Math. Soc., 96 (1986), 425-430. doi: 10.1090/S0002-9939-1986-0822433-4.

[7]

G. Butler and P. Waltman, Persistence in dynamical systems, J. Diff. Equ., 63 (1986), 255-263. doi: 10.1016/0022-0396(86)90049-5.

[8]

G. J. Butler and G. S. K. Wolkowicz, Predator-mediated competition in the chemostat, J. Math. Biol., 24 (1986), 167-191. doi: 10.1007/BF00275997.

[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), 1-48. doi: 10.1016/0025-5564(87)90002-2.

[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), 547-564. doi: 10.1142/S0218339008002654.

[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), 302-315.

[12]

W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, Heath, Boston, 1965.

[13]

J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Heidelberg, Springr-Verlag, 1977.

[14]

A. M. Edwards, Adding detritus to a nutrient-phytoplankton-zooplankton model: A dynamical-systems approach, J. Plankton Res., 23 (2001), 389-413. doi: 10.1093/plankt/23.4.389.

[15]

A. M. Edwards and J. Brindley, Oscillatory behaviour in a three-component plankton population model, Dyna. Stabi. Syst., 11 (1996), 347-370. doi: 10.1080/02681119608806231.

[16]

A. M. Edwards and J. Brindley, Zooplankton mortality and the dynamical behaviour of plankton population models, Bull. Math. Biol., 61 (1999), 303-339. doi: 10.1006/bulm.1998.0082.

[17]

J. P. Grover and R. D. Holt, Disentangling resource and apparent competition: Realistic models for plant-herbivore communities, J. Theor. Biol., 191 (1998), 353-376. doi: 10.1006/jtbi.1997.0562.

[18]

T. G. Hallam, On persistence of aquatic ecosystems. in Ocean. Sound Scat. Predic. (eds. N. R. Anderson and B. G. Zahurance), Plenum, New York, 1977, 749-765.

[19]

T. G. Hallam, Controlled persistence in rudimentary plankton models, in Proceedings of the First International Conference on Mathematical Modeling (eds. J. R. Avula), Vol. IV, University of Missouri Press, Rolla, 1977, 2081-2088.

[20]

T. G. Hallam, Structural Sensitivity of grazing formulation in nutrient controlled plankton models, J. Math. Biol., 5 (1978), 261-280. doi: 10.1007/BF00276122.

[21]

S. R. Hansen and S. P. Hubbell, Single-nutrient microbial competition: Qualitative agreement between experimental and theoretical forecast outcomes, Sci., 207 (1980), 1491-1493. doi: 10.1126/science.6767274.

[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), 741-771. doi: 10.1086/285705.

[23]

S. B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763. doi: 10.1137/0134064.

[24]

J. P. Ivlev, Experimental Ecology of the Feeding of Fishes, Yale University Press, New Haven, 1961.

[25]

S. R. J. Jang and J. Baglama, Nutrient-plankton models with nutrient recycling, Comput. Math. Appl., 49 (2005), 375-378. doi: 10.1016/j.camwa.2004.03.013.

[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), 834-840.

[27]

J.-J. Kengwoung-Keumo, Competition Between Two Phytoplankton Species Under Predation and Allelopathic Effects, Ph.D. dissertation, New Mexico State University, Las Cruces, New Mexico, U.S.A., 2012.

[28]

J. A. León and D. B. Tumpson, Competition between two species for two complementary or substitutable resources, J. Theor. Biol., 50 (1975), 185-201.

[29]

B. Li and Y. Kuang, Simple Food Chain in a Chemostat with Distinct Removal Rates, J. Math. Anal. and Appl., 242 (2000), 75-92. doi: 10.1006/jmaa.1999.6655.

[30]

R. K. Miller, Nonlinear Volterra Equation, W. A. Benjamin, N.Y., 1971.

[31]

J. Monod, Recherche sur la Croissance des Cultures Bacteriennes, Hermann et Cie, Paris, 1942.

[32]

B. Mukhopadhyay and R. Bhattacharryya, Modelling phytoplankton allelopathy in a nutrient-plankton model with spatial heterogeneity, Ecol. Model., 198 (2006), 163-173. doi: 10.1016/j.ecolmodel.2006.04.005.

[33]

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

[34]

D. Rapport, An optimization model of food selection, Amer. Natur., 105 (1971), 575-587. doi: 10.1086/282746.

[35]

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

[36]

S. Ruan, Oscillations in plankton models with recycling, J. Theor. Biol., 208 (2001), 15-26.

[37]

S. Ruan, Persistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrient recycling, J. Math. Biol., 31 (1993), 633-654. doi: 10.1007/BF00161202.

[38]

A. Sinkkonen, Modelling the effect of autotoxicity on density-dependent phytotoxicity, J. Theor. Biol., 244 (2007), 218-227. doi: 10.1016/j.jtbi.2006.08.003.

[39]

J. H. Steele and E. W. Henderson, The role of predation in plankton models, J. Plankton Res., 14 (1992), 157-172. doi: 10.1093/plankt/14.1.157.

[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), 913-923. doi: 10.3934/mbe.2013.10.913.

[41]

H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763. doi: 10.1007/BF00173267.

[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), 222-233. doi: 10.1137/0152012.

[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), 1211-1235.

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), 663-668.

[2]

R. Aris and A. E. Humphrey, Dynamics of a chemostat in which two organisms compete for a common substrate, Biotechnol. Bioeng., 19 (1977), 1375-1386. doi: 10.1002/bit.260190910.

[3]

M. M. Ballyk and G. S. K. Wolkowicz, Exploitative competition in the chemostat for two perfectly substitutable resources, Math. Biosci., 118 (1993), 127-180. doi: 10.1016/0025-5564(93)90050-K.

[4]

E. Beretta, G. Bischi and F. Solimano, Stability in chemostat equations with delayed nutrient recycling, J. Math. Biol., 28 (1990), 99-111. doi: 10.1007/BF00171521.

[5]

B. Boon and H. Laudelout, Kinetics of nitrite oxidation by Nitrobacter winogradskyi, Biochem. J., 85 (1962), 440-447.

[6]

G. Butler, H. I. Freedman and P. Waltman, Uniform persistent systems, Proc. Amer. Math. Soc., 96 (1986), 425-430. doi: 10.1090/S0002-9939-1986-0822433-4.

[7]

G. Butler and P. Waltman, Persistence in dynamical systems, J. Diff. Equ., 63 (1986), 255-263. doi: 10.1016/0022-0396(86)90049-5.

[8]

G. J. Butler and G. S. K. Wolkowicz, Predator-mediated competition in the chemostat, J. Math. Biol., 24 (1986), 167-191. doi: 10.1007/BF00275997.

[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), 1-48. doi: 10.1016/0025-5564(87)90002-2.

[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), 547-564. doi: 10.1142/S0218339008002654.

[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), 302-315.

[12]

W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, Heath, Boston, 1965.

[13]

J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Heidelberg, Springr-Verlag, 1977.

[14]

A. M. Edwards, Adding detritus to a nutrient-phytoplankton-zooplankton model: A dynamical-systems approach, J. Plankton Res., 23 (2001), 389-413. doi: 10.1093/plankt/23.4.389.

[15]

A. M. Edwards and J. Brindley, Oscillatory behaviour in a three-component plankton population model, Dyna. Stabi. Syst., 11 (1996), 347-370. doi: 10.1080/02681119608806231.

[16]

A. M. Edwards and J. Brindley, Zooplankton mortality and the dynamical behaviour of plankton population models, Bull. Math. Biol., 61 (1999), 303-339. doi: 10.1006/bulm.1998.0082.

[17]

J. P. Grover and R. D. Holt, Disentangling resource and apparent competition: Realistic models for plant-herbivore communities, J. Theor. Biol., 191 (1998), 353-376. doi: 10.1006/jtbi.1997.0562.

[18]

T. G. Hallam, On persistence of aquatic ecosystems. in Ocean. Sound Scat. Predic. (eds. N. R. Anderson and B. G. Zahurance), Plenum, New York, 1977, 749-765.

[19]

T. G. Hallam, Controlled persistence in rudimentary plankton models, in Proceedings of the First International Conference on Mathematical Modeling (eds. J. R. Avula), Vol. IV, University of Missouri Press, Rolla, 1977, 2081-2088.

[20]

T. G. Hallam, Structural Sensitivity of grazing formulation in nutrient controlled plankton models, J. Math. Biol., 5 (1978), 261-280. doi: 10.1007/BF00276122.

[21]

S. R. Hansen and S. P. Hubbell, Single-nutrient microbial competition: Qualitative agreement between experimental and theoretical forecast outcomes, Sci., 207 (1980), 1491-1493. doi: 10.1126/science.6767274.

[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), 741-771. doi: 10.1086/285705.

[23]

S. B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763. doi: 10.1137/0134064.

[24]

J. P. Ivlev, Experimental Ecology of the Feeding of Fishes, Yale University Press, New Haven, 1961.

[25]

S. R. J. Jang and J. Baglama, Nutrient-plankton models with nutrient recycling, Comput. Math. Appl., 49 (2005), 375-378. doi: 10.1016/j.camwa.2004.03.013.

[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), 834-840.

[27]

J.-J. Kengwoung-Keumo, Competition Between Two Phytoplankton Species Under Predation and Allelopathic Effects, Ph.D. dissertation, New Mexico State University, Las Cruces, New Mexico, U.S.A., 2012.

[28]

J. A. León and D. B. Tumpson, Competition between two species for two complementary or substitutable resources, J. Theor. Biol., 50 (1975), 185-201.

[29]

B. Li and Y. Kuang, Simple Food Chain in a Chemostat with Distinct Removal Rates, J. Math. Anal. and Appl., 242 (2000), 75-92. doi: 10.1006/jmaa.1999.6655.

[30]

R. K. Miller, Nonlinear Volterra Equation, W. A. Benjamin, N.Y., 1971.

[31]

J. Monod, Recherche sur la Croissance des Cultures Bacteriennes, Hermann et Cie, Paris, 1942.

[32]

B. Mukhopadhyay and R. Bhattacharryya, Modelling phytoplankton allelopathy in a nutrient-plankton model with spatial heterogeneity, Ecol. Model., 198 (2006), 163-173. doi: 10.1016/j.ecolmodel.2006.04.005.

[33]

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

[34]

D. Rapport, An optimization model of food selection, Amer. Natur., 105 (1971), 575-587. doi: 10.1086/282746.

[35]

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

[36]

S. Ruan, Oscillations in plankton models with recycling, J. Theor. Biol., 208 (2001), 15-26.

[37]

S. Ruan, Persistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrient recycling, J. Math. Biol., 31 (1993), 633-654. doi: 10.1007/BF00161202.

[38]

A. Sinkkonen, Modelling the effect of autotoxicity on density-dependent phytotoxicity, J. Theor. Biol., 244 (2007), 218-227. doi: 10.1016/j.jtbi.2006.08.003.

[39]

J. H. Steele and E. W. Henderson, The role of predation in plankton models, J. Plankton Res., 14 (1992), 157-172. doi: 10.1093/plankt/14.1.157.

[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), 913-923. doi: 10.3934/mbe.2013.10.913.

[41]

H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763. doi: 10.1007/BF00173267.

[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), 222-233. doi: 10.1137/0152012.

[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), 1211-1235.

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