2016, 13(4): 787-812. doi: 10.3934/mbe.2016018

Competition between a nonallelopathic phytoplankton and an allelopathic phytoplankton species under predation

1. 

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

Received  May 2015 Revised  December 2015 Published  May 2016

We propose a model of two-species competition in the chemostat for a single growth-limiting, nonreproducing resource that extends that of Roy [38]. The response functions are specified to be Michaelis-Menten, and there is no predation in Roy's work. Our model generalizes Roy's model to general uptake functions. The competition is exploitative so that species compete by decreasing the common pool of resources. The model also allows allelopathic effects of one toxin-producing species, both on itself (autotoxicity) and on its nontoxic competitor (phytotoxicity). We show that a stable coexistence equilibrium exists as long as (a) there are allelopathic effects and (b) the input nutrient concentration is above a critical value. The model is reconsidered under instantaneous nutrient recycling. We further extend this work to include a zooplankton species as a fourth interacting component to study the impact of predation on the ecosystem. The zooplankton species is allowed to feed only on the two phytoplankton species which are its perfectly substitutable resources. Each of the models is analyzed for boundedness, equilibria, stability, and uniform persistence (or permanence). Each model structure fits very well with some harmful algal bloom observations where the phytoplankton assemblage can be envisioned in two compartments, toxin producing and non-toxic. The Prymnesium parvum literature, where the suppressing effects of allelochemicals are quite pronounced, is a classic example. This work advances knowledge in an area of research becoming ever more important, which is understanding the functioning of allelopathy in food webs.
Citation: Jean-Jacques Kengwoung-Keumo. Competition between a nonallelopathic phytoplankton and an allelopathic phytoplankton species under predation. Mathematical Biosciences & Engineering, 2016, 13 (4) : 787-812. doi: 10.3934/mbe.2016018
References:
[1]

M. An, D. Liu, I. Johnson and J. Lovett, Mathematical modelling of allelopathy. II. The dynamics of allelochemicals from living plants in the environment,, Ecol. Model., 161 (2003), 53.  doi: 10.1016/S0304-3800(02)00289-2.  Google Scholar

[2]

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

[3]

R. Aris and A. E. Humphrey, Dynamics of a chemostat in which two organisms compete for a common substrate,, Biotechnol. Bioeng., 19 (1977), 707.  doi: 10.1002/bit.260190910.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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.  doi: 10.1038/nature07248.  Google Scholar

[11]

A. M. Edwards and J. Brindley, Zooplankton mortality and the dynamical behaviour of plankton population models,, Bull. Math. Biol., 6 (1999), 303.  doi: 10.1006/bulm.1998.0082.  Google Scholar

[12]

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

[13]

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

[14]

T. G. Hallam, On persistence of aquatic ecosystems,, In: Anderson, (1977), 749.   Google Scholar

[15]

T. G. Hallam, Controlled persistence in rudimentary plankton models,, In: Avula, 4 (1977), 2081.   Google Scholar

[16]

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

[17]

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

[18]

G. Hardin, The competitive exclusion principle,, Sci., 131 (1960), 1292.  doi: 10.1126/science.131.3409.1292.  Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

J.-J. Kengwoung-Keumo, Dynamics of two phytoplankton populations under predation,, J. Math. Bio. Eng., 11 (2014), 1319.  doi: 10.3934/mbe.2014.11.1319.  Google Scholar

[25]

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

[26]

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

[27]

J. Maynard-Smith, Models in Ecology,, Cambridge University Press, (1974).   Google Scholar

[28]

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

[29]

H. Molisch, Der Einfluss Einer Pflanze Auf Die Andere-Allelopathie,, Fischer, (1937).   Google Scholar

[30]

J. Monod, Recherche Sur La Croissance Des Cultures Bacteriennes,, Hermann et Cie., (1942).   Google Scholar

[31]

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

[32]

M. Nakamaru and Y. Iwasa, Competition by allelopathy proceeds in travelling waves: colicin-immune strain aids colicin-sensitive strain,, Theor. Popul. Biol., 57 (2000), 131.   Google Scholar

[33]

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

[34]

M. J. Piotrowska, U. Foryś and M. Bodnar, A simple model of carcinogenic mutations with time delay and diffusion,, Math. Biosci. Eng., 10 (2013), 861.  doi: 10.3934/mbe.2013.10.861.  Google Scholar

[35]

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

[36]

E. L. Rice, Allelopathy,, Academic Press, (1984).   Google Scholar

[37]

G. A. Riley, A mathematical model of regional variations in plankton,, Limnol. Oceanog., 10 (1965).   Google Scholar

[38]

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

[39]

P. K Roy, A. N. Chatterjee, D. Greenhalgh and D. Khan, Long term dynamics in a mathematical model of HIV-1 infection with delay in different variants of the basic drug therapy model,, Nonlinear Anal-Real., 14 (2013), 1621.  doi: 10.1016/j.nonrwa.2012.10.021.  Google Scholar

[40]

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

[41]

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

[42]

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

[43]

H. L. Smith and P. Waltman, The theory of the chemostat: Dynamics of microbial competition,, Vol. 13 of Cambridge Studies in Mathematical Biology, (1995).  doi: 10.1017/CBO9780511530043.  Google Scholar

[44]

J. Solé, E. García-Ladona, P. Ruardij and M. Estrada, Modelling allelopathy among marine algae,, Ecol. Model., 183 (2005), 373.   Google Scholar

[45]

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

[46]

H. V. Thurman, Introductory Oceanography,, 8th edition, (1997).  doi: 10.5962/bhl.title.59942.  Google Scholar

[47]

R. Whittaker and P. Feeny, Allolechemicals: chemical interactions between species,, Sci., 171 (1971), 757.   Google Scholar

[48]

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

[49]

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]

M. An, D. Liu, I. Johnson and J. Lovett, Mathematical modelling of allelopathy. II. The dynamics of allelochemicals from living plants in the environment,, Ecol. Model., 161 (2003), 53.  doi: 10.1016/S0304-3800(02)00289-2.  Google Scholar

[2]

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

[3]

R. Aris and A. E. Humphrey, Dynamics of a chemostat in which two organisms compete for a common substrate,, Biotechnol. Bioeng., 19 (1977), 707.  doi: 10.1002/bit.260190910.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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.  doi: 10.1038/nature07248.  Google Scholar

[11]

A. M. Edwards and J. Brindley, Zooplankton mortality and the dynamical behaviour of plankton population models,, Bull. Math. Biol., 6 (1999), 303.  doi: 10.1006/bulm.1998.0082.  Google Scholar

[12]

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

[13]

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

[14]

T. G. Hallam, On persistence of aquatic ecosystems,, In: Anderson, (1977), 749.   Google Scholar

[15]

T. G. Hallam, Controlled persistence in rudimentary plankton models,, In: Avula, 4 (1977), 2081.   Google Scholar

[16]

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

[17]

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

[18]

G. Hardin, The competitive exclusion principle,, Sci., 131 (1960), 1292.  doi: 10.1126/science.131.3409.1292.  Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

J.-J. Kengwoung-Keumo, Dynamics of two phytoplankton populations under predation,, J. Math. Bio. Eng., 11 (2014), 1319.  doi: 10.3934/mbe.2014.11.1319.  Google Scholar

[25]

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

[26]

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

[27]

J. Maynard-Smith, Models in Ecology,, Cambridge University Press, (1974).   Google Scholar

[28]

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

[29]

H. Molisch, Der Einfluss Einer Pflanze Auf Die Andere-Allelopathie,, Fischer, (1937).   Google Scholar

[30]

J. Monod, Recherche Sur La Croissance Des Cultures Bacteriennes,, Hermann et Cie., (1942).   Google Scholar

[31]

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

[32]

M. Nakamaru and Y. Iwasa, Competition by allelopathy proceeds in travelling waves: colicin-immune strain aids colicin-sensitive strain,, Theor. Popul. Biol., 57 (2000), 131.   Google Scholar

[33]

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

[34]

M. J. Piotrowska, U. Foryś and M. Bodnar, A simple model of carcinogenic mutations with time delay and diffusion,, Math. Biosci. Eng., 10 (2013), 861.  doi: 10.3934/mbe.2013.10.861.  Google Scholar

[35]

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

[36]

E. L. Rice, Allelopathy,, Academic Press, (1984).   Google Scholar

[37]

G. A. Riley, A mathematical model of regional variations in plankton,, Limnol. Oceanog., 10 (1965).   Google Scholar

[38]

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

[39]

P. K Roy, A. N. Chatterjee, D. Greenhalgh and D. Khan, Long term dynamics in a mathematical model of HIV-1 infection with delay in different variants of the basic drug therapy model,, Nonlinear Anal-Real., 14 (2013), 1621.  doi: 10.1016/j.nonrwa.2012.10.021.  Google Scholar

[40]

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

[41]

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

[42]

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

[43]

H. L. Smith and P. Waltman, The theory of the chemostat: Dynamics of microbial competition,, Vol. 13 of Cambridge Studies in Mathematical Biology, (1995).  doi: 10.1017/CBO9780511530043.  Google Scholar

[44]

J. Solé, E. García-Ladona, P. Ruardij and M. Estrada, Modelling allelopathy among marine algae,, Ecol. Model., 183 (2005), 373.   Google Scholar

[45]

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

[46]

H. V. Thurman, Introductory Oceanography,, 8th edition, (1997).  doi: 10.5962/bhl.title.59942.  Google Scholar

[47]

R. Whittaker and P. Feeny, Allolechemicals: chemical interactions between species,, Sci., 171 (1971), 757.   Google Scholar

[48]

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

[49]

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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