2014, 11(4): 841-875. doi: 10.3934/mbe.2014.11.841

Effects of nutrient enrichment on coevolution of a stoichiometric producer-grazer system

1. 

School of Mathematics and Statistics, Northeast Normal University, 5268 Renmin Street, Changchun, Jilin, 130024, China, China

Received  September 2013 Revised  December 2013 Published  March 2014

A simple producer-grazer model based on adaptive evolution and ecological stoichiometry is proposed and well explored to examine the patterns and consequences of adaptive changes for the evolutionary trait (i.e., body size), and also to investigate the effect of nutrient enrichment on the coevolutin of the producer and the grazer. The analytical and numerical results indicate that this simple model predicts a wide range of evolutionary dynamics and that the total nutrient concentration in the ecosystem plays a pivotal role in determining the outcome of producer-grazer coevolution. Nutrient enrichment may yield evolutionary branching, trait cycles or sensitive dependence on the initial values, depending on how much nutrient is present in the ecosystem. In the absence of grazing, the lower nutrient density facilitates the continuously stable strategy while the higher nutrient density induces evolutionary branching. When the grazer is present, with the increasing of nutrient level, the evolutionary dynamics is very complicated. The evolutionary dynamics sequentially undergo continuously stable strategy, evolutionary branching, evolutionary cycle, and sensitive dependence on the initial values. Nutrient enrichment asserts not only stabilizing but also destabilizing impact on the evolutionary dynamics. The evolutionary dynamics potentially show the paradox of nutrient enrichment. This study well documents the interplay and co-effect of the ecological and evolutionary processes.
Citation: Lina Hao, Meng Fan, Xin Wang. Effects of nutrient enrichment on coevolution of a stoichiometric producer-grazer system. Mathematical Biosciences & Engineering, 2014, 11 (4) : 841-875. doi: 10.3934/mbe.2014.11.841
References:
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D. M. Anderson, P. M. Glibert and J. M. Burkholder, Harmful algal blooms and eutrophication: Nutrient sources, composition, and consequences,, Estuaties, 25 (2002), 704. doi: 10.1007/BF02804901.

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A. Binzer, C. Guill, U. Brose and B. C. Rall, The dynamics of food chains under climate change and nutrient enrichment,, Phil. Trans. R. Soc. B, 367 (2012), 2935. doi: 10.1098/rstb.2012.0230.

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P. Branco, M. Stomp, M. Egas and J. Huisman, Evolution of nutrient uptake reveals a trade-off in the ecological stoichiometry of plant-herbivore interactions,, Am. Nat., 176 (2010), 162. doi: 10.1086/657036.

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S. Chisholm, Phytoplankton size,, In Primary Productivity and Biogeochemical Cycles in the Sea, 43 (1992), 213. doi: 10.1007/978-1-4899-0762-2_12.

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M. Cortez and S. P. Ellner, Understanding rapid evolution in predator-prey interactions using the theory of fast-slow dynamical systems,, Am. Nat., 176 (2010). doi: 10.1086/656485.

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J. M. Davis, A. D. Rosemond, S. L. Eggert, W. F. Cross and J. B. Wallace, Long-term nutrient enrichment decouples predator and prey production,, Proc. Natl. Acad. Sci. USA, 107 (2010), 121. doi: 10.1073/pnas.0908497107.

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U. Dieckmann and R. Law, The dynamical theory of coevolution: A derivation from stochastic ecological processes,, J. Math. Biol., 34 (1996), 579. doi: 10.1007/BF02409751.

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U. Dieckmann, P. Marrow and R. Law, Evolutionary cycling in predator-prey interactions: Population dynamics and the red queen,, J. Theor. Biol., 176 (1995), 91. doi: 10.1006/jtbi.1995.0179.

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S. Diehl, Paradoxes of enrichment: Effects of increased light versus nutrient supply on pelagic producer-grazer system,, Am. Nat., 169 (2007), 173. doi: 10.1086/516655.

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S. Diehl and M. Feißel, Effects of enrichment on three-level food chainswith omnivory,, Am. Nat., 155 (2000), 200. doi: 10.1086/303319.

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M. Doebeli and U. Dieckmann, Evolutionary branching and sympatric speciation caused by different types of ecological interactions,, Am. Nat., 156 (2000). doi: 10.1086/303417.

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M. R. Droop, Vitamin $b_{12}$ and marine ecology. iv. the kinetics of uptake, growth and inhibition in monochrysis lutheri,, J. Mar. Biol. Assoc. UK, 48 (1968), 689.

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T. H. G. Ezard, S. D. Côté and F. Pelletier, Eco-evolutionary dynamics: Disentangling phenotypic, environmental and population fluctuations,, Phil. Trans. R. Soc. B, 364 (2009), 1491. doi: 10.1098/rstb.2009.0006.

[16]

Z. V. Finkel, M. E. Katz, J. D. Wright, O. M. E Schofield and P. G. Falkowski, Climatically driven macroevolutionary patterns in the size of marine diatoms over the cenozoic,, Proc. Natl. Acad. Sci. USA, 102 (2005), 8927. doi: 10.1073/pnas.0409907102.

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[18]

G. F. Fussmann, S. P. Ellner and N. G. Hairston, Evolution as a critical component of plankton dynamics,, Proc. R. Soc. Lond. B, 270 (2003), 1015. doi: 10.1098/rspb.2003.2335.

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[20]

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S. A. H. Geritz, E. Kisdi, G. Meszéna and J. A. J. Metz, Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree,, Evol. Ecol., 12 (1998), 35. doi: 10.1023/A:1006554906681.

[22]

L. Jiang, O. M. E. Schofield and P. G. Falkowski, Adaptive evolution of phytoplankton cell size,, Am. Nat., 166 (2005), 496. doi: 10.1086/444442.

[23]

M. D. John, A. D. Rosemond, S. L. Eggert, W. F. Cross and J. B. Wallace, Nutrient enrichment differentially affects body sizes of primary consumers and predators in a detritus-based stream,, Limnol. Oceanogr., 55 (2010), 2305. doi: 10.4319/lo.2010.55.6.2305.

[24]

L. E. Jones, L. Becks, S. P. Ellner, N. G. Hairston, T. Yoshida and G. F. Fussmann, Rapid contemporary evolution and clonal food web dynamics,, Phil. Trans. R. Soc. B, 364 (2009), 1579. doi: 10.1098/rstb.2009.0004.

[25]

E. Kisdi, Evolutionary branching under asymmetric competition,, J. Theor. Biol., 197 (1999), 149. doi: 10.1006/jtbi.1998.0864.

[26]

C. A. Klausmeier, E. Litchman and S. A. Levin, A model of flexible uptake of two essential resources,, J. Theor. Biol., 246 (2007), 278. doi: 10.1016/j.jtbi.2006.12.032.

[27]

X. Li, H. Wang and Y. Kuang, Global analysis of a stoichiometric producer-grazer model with Holling type functional responses,, J. Math. Biol., 63 (2011), 901. doi: 10.1007/s00285-010-0392-2.

[28]

N. Loeuille and M. Loreau, Nutrient enrichment and food chains: Can evolution buffer top-down control?, Theor. Popul. Biol., 65 (2004), 285. doi: 10.1016/j.tpb.2003.12.004.

[29]

N. Loeuille and M. Loreau, Evolutionary emergence of size-structured food webs,, Proc. Natl. Acad. Sci. USA, 102 (2005), 5761. doi: 10.1073/pnas.0408424102.

[30]

N. Loeuille, M. Loreau and R. Ferrière, Consequences of plant-herbivore coevolution on the dynamics and functioning of ecosystems,, J. Theor. Biol., 217 (2002), 369. doi: 10.1006/jtbi.2002.3032.

[31]

I. Loladze, Y. Kuang and J. J. Elser, Stoichiometry in producer-grazer systems: Linking energy flow and element cycling,, Bull. Math. Biol., 62 (2000), 1137. doi: 10.1006/bulm.2000.0201.

[32]

M. Loreau, Ecosystem development explained by competition within and between material cycles,, Proc. R. Soc. Lond. B, 265 (1998), 33. doi: 10.1098/rspb.1998.0260.

[33]

A. Mougi and Y. Iwasa, Evolution towards oscillation or stability in a predator-prey system,, Proc. R. Soc. B, 277 (2010), 3163. doi: 10.1098/rspb.2010.0691.

[34]

A. Mougi and Y. Iwasa, Unique coevolutionary dynamics in a predator-prey system,, J. Theor. Biol., 277 (2011), 83. doi: 10.1016/j.jtbi.2011.02.015.

[35]

E. B. Muller, R. M. Nisbet, S. A. L. M Kooijman, J. J. Elser and E. McCauley, Stoichiometric food quality and herbivore dynamics,, Ecol. Lett., 4 (2001), 519. doi: 10.1046/j.1461-0248.2001.00240.x.

[36]

D. Pimentel, Animal population regulation by the genetic feed-back mechanism,, Am. Nat., 95 (1961), 65. doi: 10.1086/282160.

[37]

J. A. Raven, Physiological consequences of extremely small size for autotrophic organisms on the sea,, Can. Bull. Fish. Aquat. Sci., 214 (1986), 1.

[38]

J. A. Raven, Why are there no picoplanktonic $o_2$ evolvers with volumes less than $10^{-19} m^3$?, J. Plankton. Res., 16 (1994), 565. doi: 10.1093/plankt/16.5.565.

[39]

M. L. Rosenzweig, Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time,, Science, 171 (1971), 385. doi: 10.1126/science.171.3969.385.

[40]

R. W. Sterner and J. J. Elser, Ecological Stoichiometry: The Biology of Elements from Molecules to the Biosphere,, NJ: Princeton University Press, (2002).

[41]

D. Stiefs, G. A. K. Van Voorn, B. W. Kooi, U. Feudel and T. Gross, Food quality in producer-grazer models-: A generalized analysis,, Am. Nat., 176 (2010), 367. doi: 10.1086/655429.

[42]

A. Verdy, M. Follows and G. Flierl, Optimal phytoplankton cell size in an allometric model,, Mar. Ecol. Prog. Ser., 379 (2009), 1. doi: 10.3354/meps07909.

[43]

H. Wang, H. L. Smith, Y. Kuang and J. J. Elser, Dynamics of stoichiometric bacteria-algae interaction in epilimnion,, SIAM J. Appl. Math., 68 (2007), 503. doi: 10.1137/060665919.

[44]

D. Waxman and S. Gavrilets, 20 questions on adaptive dynamics,, J. Evol. Biol., 18 (2005), 1139. doi: 10.1111/j.1420-9101.2005.00948.x.

[45]

T. G. Whitham, J. K. Bailey and J. A. Schweitzer et al, A framework for community and ecosystem genetics: From genes to ecosystems,, Nature Reviews Genetics, 7 (2006), 510. doi: 10.1038/nrg1877.

[46]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos,, Springer-Verlag, (1990).

[47]

T. Yoshida, L. E. Jones, S. P. Ellner, G. F. Fussmann and Jr N. G. Hairston, Rapid evolution drives ecological dynamics in a predator-prey system,, Nature, 424 (2003), 303. doi: 10.1038/nature01767.

[48]

J. Zu, M. Mimura and J. Y. Wakano, The evolution of phenotypic traits in a predator-prey system subject to Allee effect,, J. Theor. Biol., 262 (2010), 528. doi: 10.1016/j.jtbi.2009.10.022.

show all references

References:
[1]

P. A. Abrams and J. D. Roth, The effects of enrichment of three-species food chains with nonlinear functional response,, Ecology, 75 (1994), 1118. doi: 10.2307/1939435.

[2]

A. N. Mizuno and M. Kawata, The effects of the evolution of stoichiometry-related traits on population dynamics in plankton communities,, J. Theor. Biol., 259 (2009), 209. doi: 10.1016/j.jtbi.2009.02.025.

[3]

D. M. Anderson, P. M. Glibert and J. M. Burkholder, Harmful algal blooms and eutrophication: Nutrient sources, composition, and consequences,, Estuaties, 25 (2002), 704. doi: 10.1007/BF02804901.

[4]

A. Binzer, C. Guill, U. Brose and B. C. Rall, The dynamics of food chains under climate change and nutrient enrichment,, Phil. Trans. R. Soc. B, 367 (2012), 2935. doi: 10.1098/rstb.2012.0230.

[5]

P. Branco, M. Stomp, M. Egas and J. Huisman, Evolution of nutrient uptake reveals a trade-off in the ecological stoichiometry of plant-herbivore interactions,, Am. Nat., 176 (2010), 162. doi: 10.1086/657036.

[6]

S. Chisholm, Phytoplankton size,, In Primary Productivity and Biogeochemical Cycles in the Sea, 43 (1992), 213. doi: 10.1007/978-1-4899-0762-2_12.

[7]

M. Cortez and S. P. Ellner, Understanding rapid evolution in predator-prey interactions using the theory of fast-slow dynamical systems,, Am. Nat., 176 (2010). doi: 10.1086/656485.

[8]

J. M. Davis, A. D. Rosemond, S. L. Eggert, W. F. Cross and J. B. Wallace, Long-term nutrient enrichment decouples predator and prey production,, Proc. Natl. Acad. Sci. USA, 107 (2010), 121. doi: 10.1073/pnas.0908497107.

[9]

U. Dieckmann and R. Law, The dynamical theory of coevolution: A derivation from stochastic ecological processes,, J. Math. Biol., 34 (1996), 579. doi: 10.1007/BF02409751.

[10]

U. Dieckmann, P. Marrow and R. Law, Evolutionary cycling in predator-prey interactions: Population dynamics and the red queen,, J. Theor. Biol., 176 (1995), 91. doi: 10.1006/jtbi.1995.0179.

[11]

S. Diehl, Paradoxes of enrichment: Effects of increased light versus nutrient supply on pelagic producer-grazer system,, Am. Nat., 169 (2007), 173. doi: 10.1086/516655.

[12]

S. Diehl and M. Feißel, Effects of enrichment on three-level food chainswith omnivory,, Am. Nat., 155 (2000), 200. doi: 10.1086/303319.

[13]

M. Doebeli and U. Dieckmann, Evolutionary branching and sympatric speciation caused by different types of ecological interactions,, Am. Nat., 156 (2000). doi: 10.1086/303417.

[14]

M. R. Droop, Vitamin $b_{12}$ and marine ecology. iv. the kinetics of uptake, growth and inhibition in monochrysis lutheri,, J. Mar. Biol. Assoc. UK, 48 (1968), 689.

[15]

T. H. G. Ezard, S. D. Côté and F. Pelletier, Eco-evolutionary dynamics: Disentangling phenotypic, environmental and population fluctuations,, Phil. Trans. R. Soc. B, 364 (2009), 1491. doi: 10.1098/rstb.2009.0006.

[16]

Z. V. Finkel, M. E. Katz, J. D. Wright, O. M. E Schofield and P. G. Falkowski, Climatically driven macroevolutionary patterns in the size of marine diatoms over the cenozoic,, Proc. Natl. Acad. Sci. USA, 102 (2005), 8927. doi: 10.1073/pnas.0409907102.

[17]

Z. V. Finkel, J. Beardall, K. J. Flynn, A. Quigg, T. A. Rees and J. Raven, Phytoplankton in a changing world: Cell size and elemental stoichiometry,, J. Plankton. Res., 32 (2010), 119. doi: 10.1093/plankt/fbp098.

[18]

G. F. Fussmann, S. P. Ellner and N. G. Hairston, Evolution as a critical component of plankton dynamics,, Proc. R. Soc. Lond. B, 270 (2003), 1015. doi: 10.1098/rspb.2003.2335.

[19]

G. F. Fussmann, M. Loreau and P. A. Abrams, Eco-evolutionary dynamics of communities and ecosystems,, Funct. Ecol., 21 (2007), 465. doi: 10.1111/j.1365-2435.2007.01275.x.

[20]

S. A. H. Geritz and M. Gyllenberg, Seven answers from adaptive dynamics,, J. EVOL. BIOL., 18 (2005), 1174. doi: 10.1111/j.1420-9101.2004.00841.x.

[21]

S. A. H. Geritz, E. Kisdi, G. Meszéna and J. A. J. Metz, Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree,, Evol. Ecol., 12 (1998), 35. doi: 10.1023/A:1006554906681.

[22]

L. Jiang, O. M. E. Schofield and P. G. Falkowski, Adaptive evolution of phytoplankton cell size,, Am. Nat., 166 (2005), 496. doi: 10.1086/444442.

[23]

M. D. John, A. D. Rosemond, S. L. Eggert, W. F. Cross and J. B. Wallace, Nutrient enrichment differentially affects body sizes of primary consumers and predators in a detritus-based stream,, Limnol. Oceanogr., 55 (2010), 2305. doi: 10.4319/lo.2010.55.6.2305.

[24]

L. E. Jones, L. Becks, S. P. Ellner, N. G. Hairston, T. Yoshida and G. F. Fussmann, Rapid contemporary evolution and clonal food web dynamics,, Phil. Trans. R. Soc. B, 364 (2009), 1579. doi: 10.1098/rstb.2009.0004.

[25]

E. Kisdi, Evolutionary branching under asymmetric competition,, J. Theor. Biol., 197 (1999), 149. doi: 10.1006/jtbi.1998.0864.

[26]

C. A. Klausmeier, E. Litchman and S. A. Levin, A model of flexible uptake of two essential resources,, J. Theor. Biol., 246 (2007), 278. doi: 10.1016/j.jtbi.2006.12.032.

[27]

X. Li, H. Wang and Y. Kuang, Global analysis of a stoichiometric producer-grazer model with Holling type functional responses,, J. Math. Biol., 63 (2011), 901. doi: 10.1007/s00285-010-0392-2.

[28]

N. Loeuille and M. Loreau, Nutrient enrichment and food chains: Can evolution buffer top-down control?, Theor. Popul. Biol., 65 (2004), 285. doi: 10.1016/j.tpb.2003.12.004.

[29]

N. Loeuille and M. Loreau, Evolutionary emergence of size-structured food webs,, Proc. Natl. Acad. Sci. USA, 102 (2005), 5761. doi: 10.1073/pnas.0408424102.

[30]

N. Loeuille, M. Loreau and R. Ferrière, Consequences of plant-herbivore coevolution on the dynamics and functioning of ecosystems,, J. Theor. Biol., 217 (2002), 369. doi: 10.1006/jtbi.2002.3032.

[31]

I. Loladze, Y. Kuang and J. J. Elser, Stoichiometry in producer-grazer systems: Linking energy flow and element cycling,, Bull. Math. Biol., 62 (2000), 1137. doi: 10.1006/bulm.2000.0201.

[32]

M. Loreau, Ecosystem development explained by competition within and between material cycles,, Proc. R. Soc. Lond. B, 265 (1998), 33. doi: 10.1098/rspb.1998.0260.

[33]

A. Mougi and Y. Iwasa, Evolution towards oscillation or stability in a predator-prey system,, Proc. R. Soc. B, 277 (2010), 3163. doi: 10.1098/rspb.2010.0691.

[34]

A. Mougi and Y. Iwasa, Unique coevolutionary dynamics in a predator-prey system,, J. Theor. Biol., 277 (2011), 83. doi: 10.1016/j.jtbi.2011.02.015.

[35]

E. B. Muller, R. M. Nisbet, S. A. L. M Kooijman, J. J. Elser and E. McCauley, Stoichiometric food quality and herbivore dynamics,, Ecol. Lett., 4 (2001), 519. doi: 10.1046/j.1461-0248.2001.00240.x.

[36]

D. Pimentel, Animal population regulation by the genetic feed-back mechanism,, Am. Nat., 95 (1961), 65. doi: 10.1086/282160.

[37]

J. A. Raven, Physiological consequences of extremely small size for autotrophic organisms on the sea,, Can. Bull. Fish. Aquat. Sci., 214 (1986), 1.

[38]

J. A. Raven, Why are there no picoplanktonic $o_2$ evolvers with volumes less than $10^{-19} m^3$?, J. Plankton. Res., 16 (1994), 565. doi: 10.1093/plankt/16.5.565.

[39]

M. L. Rosenzweig, Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time,, Science, 171 (1971), 385. doi: 10.1126/science.171.3969.385.

[40]

R. W. Sterner and J. J. Elser, Ecological Stoichiometry: The Biology of Elements from Molecules to the Biosphere,, NJ: Princeton University Press, (2002).

[41]

D. Stiefs, G. A. K. Van Voorn, B. W. Kooi, U. Feudel and T. Gross, Food quality in producer-grazer models-: A generalized analysis,, Am. Nat., 176 (2010), 367. doi: 10.1086/655429.

[42]

A. Verdy, M. Follows and G. Flierl, Optimal phytoplankton cell size in an allometric model,, Mar. Ecol. Prog. Ser., 379 (2009), 1. doi: 10.3354/meps07909.

[43]

H. Wang, H. L. Smith, Y. Kuang and J. J. Elser, Dynamics of stoichiometric bacteria-algae interaction in epilimnion,, SIAM J. Appl. Math., 68 (2007), 503. doi: 10.1137/060665919.

[44]

D. Waxman and S. Gavrilets, 20 questions on adaptive dynamics,, J. Evol. Biol., 18 (2005), 1139. doi: 10.1111/j.1420-9101.2005.00948.x.

[45]

T. G. Whitham, J. K. Bailey and J. A. Schweitzer et al, A framework for community and ecosystem genetics: From genes to ecosystems,, Nature Reviews Genetics, 7 (2006), 510. doi: 10.1038/nrg1877.

[46]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos,, Springer-Verlag, (1990).

[47]

T. Yoshida, L. E. Jones, S. P. Ellner, G. F. Fussmann and Jr N. G. Hairston, Rapid evolution drives ecological dynamics in a predator-prey system,, Nature, 424 (2003), 303. doi: 10.1038/nature01767.

[48]

J. Zu, M. Mimura and J. Y. Wakano, The evolution of phenotypic traits in a predator-prey system subject to Allee effect,, J. Theor. Biol., 262 (2010), 528. doi: 10.1016/j.jtbi.2009.10.022.

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