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:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[37]

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

[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.  Google Scholar

[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.  Google Scholar

[40]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[46]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[37]

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

[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.  Google Scholar

[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.  Google Scholar

[40]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[46]

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

[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.  Google Scholar

[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.  Google Scholar

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