Evolutionary branching patterns in predator-prey structured populations

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November 2013, 18(9): 2267-2282. doi: 10.3934/dcdsb.2013.18.2267

Evolutionary branching patterns in predator-prey structured populations

Marcello Delitala ^1, and Tommaso Lorenzi ^1,

1.	Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino

Received November 2012 Revised July 2013 Published September 2013

Abstract
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Predator-prey ecosystems represent, among others, a natural context where evolutionary branching patterns may arise. Moving from this observation, the paper deals with a class of integro-differential equations modeling the dynamics of two populations structured by a continuous phenotypic trait and related by predation. Predators and preys proliferate through asexual reproduction, compete for resources and undergo phenotypic changes. A positive parameter $\varepsilon$ is introduced to model the average size of such changes. The asymptotic behavior of the solution of the mathematical problem linked to the model is studied in the limit $\varepsilon \rightarrow 0$ (i.e., in the limit of small phenotypic changes). Analytical results are illustrated and extended by means of numerical simulations with the aim of showing how the present class of equations can mimic the formation of evolutionary branching patterns. All simulations highlight a chase-escape dynamics, where the preys try to evade predation while predators mimic, with a certain delay, the phenotypic profile of the preys.

Keywords: facultative predation, Structured populations, integro-differential equations., asymptotic analysis, branching patterns.

Mathematics Subject Classification: Primary: 92D25; Secondary: 45K0.

Citation: Marcello Delitala, Tommaso Lorenzi. Evolutionary branching patterns in predator-prey structured populations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2267-2282. doi: 10.3934/dcdsb.2013.18.2267

References:

[1]	V. Andasari, A. Gerisch, G. Lolas, A. P. South and M. A. J. Chaplain, Mathematical modeling of cancer cell invasion of tissue: Biological insight from mathematical analysis and computational simulation, Journal of Mathematical Biology, 63 (2011), 141-171. doi: 10.1007/s00285-010-0369-1.
[2]	F. Cerretti, B. Perthame, C. Schmeiser, M. Tang and N. Vauchelet, Branching instabilities in Hyperbolic Keller-Segel systems, Math. Models Methods Appl. Sci., 21 (2011), 825-842.
[3]	D. Challet, M. Marsili and Y. C. Zhang, "Minority Games: Interacting Agents in Financial Markets," Oxford Finance, 2005.
[4]	M. Delitala and T. Lorenzi, Asymptotic dynamics in continuous structured populations with mutations, competition and mutualism, Journal of Mathematical Analysis and Applications, 389 (2012), 439-451. doi: 10.1016/j.jmaa.2011.11.076.
[5]	M. Delitala and T. Lorenzi, Recognition and learning in a mathematical model for immune response against cancer, Discrete and Continuous Dynamical Systems - Series B, to appear. doi: 10.3934/dcdsb.2013.18.891.
[6]	L. Desvillettes, P. E. Jabin, S. Mischler and G. Raoul, On selection dynamics for continuous structured populations, Communications in Mathematical Sciences, 6 (2008), 729-747. doi: 10.4310/CMS.2008.v6.n3.a10.
[7]	O. Diekmann, P. E. Jabin, S. Mischler and B. Perthame, The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach, Theoretical Population Biology, 67 (2005), 257-271. doi: 10.1016/j.tpb.2004.12.003.
[8]	M. Gauduchon and B. Perthame, Survival thresholds and mortality rates in adaptive dynamics: conciliating deterministic and stochastic simulations, Mathematical Medicine and Biology, 27 (2010), 195-210. doi: 10.1093/imammb/dqp018.
[9]	S. Genieys, V. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources, Mathematical Modelling of Natural Phenomena, 1 (2006), 65-82. doi: 10.1051/mmnp:2006004.
[10]	S. A. H. Geritz, E. Kisdi, G. Meszena, J. A. J. Metz, Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree, Evol. Ecol., 12 (1998) 35-57.
[11]	S. A. H. Geritz, J. A. J.Metz, E. Kisdi and G. Meszena, Dynamics of adaptation and evolutionary branching, Phys. Rev. Lett., 78 (1997), 2024-2027. doi: 10.1103/PhysRevLett.78.2024.
[12]	D. Hanahan and R. A. Weinberg, Hallmarks of cancer: the next generation, Cell, 144 (2011), 646-674. doi: 10.1016/j.cell.2011.02.013.
[13]	A. Lorz, S. Mirrahimi and B. Perthame, Dirac mass dynamics in multidimensional nonlocal parabolic equations, Communications in Partial Differential Equations, 36 (2011), 1071-1098. doi: 10.1080/03605302.2010.538784.
[14]	J. Maynard Smith, "Evolution and the Theory of Games," Cambridge University Press, 1982.
[15]	S. Mirrahimi, B. Perthame and J. Wakano, Evolution of species trait through resource competition, Journal of Mathematical Biology, 64 (2012), 1189-1223. doi: 10.1007/s00285-011-0447-z.
[16]	G. Naldi, L. Pareschi and G. Toscani (Eds.), "Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences," Birkhäuser, Basel, 2010. doi: 10.1007/978-0-8176-4946-3.
[17]	J. C. Nuno, M. Primicerio and M. A. Herrero, A mathematical model of a criminal-prone society, DCDS-S, 4 (2011), 193-207. doi: 10.3934/dcdss.2011.4.193.
[18]	K. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D: Nonlinear Phenomena, 240 (2011), 363-375. doi: 10.1016/j.physd.2010.09.011.
[19]	B. Perthame, "Transport Equations in Biology," Birkhäuser, Basel, 2007.
[20]	V. Quaranta, K. A. Rejniak, P. Gerlee and A. R. Anderson, Invasion emerges from cancer cell adaptation to competitive microenvironments: quantitative predictions from multiscale mathematical models, Seminars in Cancer Biology, 18 (2008), 338-348. doi: 10.1016/j.semcancer.2008.03.018.
[21]	V. Semeshenko, M. B. Gordon and J. P. Nadal, Collective states in social systems with interacting learning agents, Physica A: Statistical Mechanics and its Applications, 387 (2008), 4903-4916. doi: 10.1016/j.physa.2008.04.019.
[22]	Z. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos, 17 (2007), 037108. doi: 10.1063/1.2766864.

show all references

References:

[1]	V. Andasari, A. Gerisch, G. Lolas, A. P. South and M. A. J. Chaplain, Mathematical modeling of cancer cell invasion of tissue: Biological insight from mathematical analysis and computational simulation, Journal of Mathematical Biology, 63 (2011), 141-171. doi: 10.1007/s00285-010-0369-1.
[2]	F. Cerretti, B. Perthame, C. Schmeiser, M. Tang and N. Vauchelet, Branching instabilities in Hyperbolic Keller-Segel systems, Math. Models Methods Appl. Sci., 21 (2011), 825-842.
[3]	D. Challet, M. Marsili and Y. C. Zhang, "Minority Games: Interacting Agents in Financial Markets," Oxford Finance, 2005.
[4]	M. Delitala and T. Lorenzi, Asymptotic dynamics in continuous structured populations with mutations, competition and mutualism, Journal of Mathematical Analysis and Applications, 389 (2012), 439-451. doi: 10.1016/j.jmaa.2011.11.076.
[5]	M. Delitala and T. Lorenzi, Recognition and learning in a mathematical model for immune response against cancer, Discrete and Continuous Dynamical Systems - Series B, to appear. doi: 10.3934/dcdsb.2013.18.891.
[6]	L. Desvillettes, P. E. Jabin, S. Mischler and G. Raoul, On selection dynamics for continuous structured populations, Communications in Mathematical Sciences, 6 (2008), 729-747. doi: 10.4310/CMS.2008.v6.n3.a10.
[7]	O. Diekmann, P. E. Jabin, S. Mischler and B. Perthame, The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach, Theoretical Population Biology, 67 (2005), 257-271. doi: 10.1016/j.tpb.2004.12.003.
[8]	M. Gauduchon and B. Perthame, Survival thresholds and mortality rates in adaptive dynamics: conciliating deterministic and stochastic simulations, Mathematical Medicine and Biology, 27 (2010), 195-210. doi: 10.1093/imammb/dqp018.
[9]	S. Genieys, V. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources, Mathematical Modelling of Natural Phenomena, 1 (2006), 65-82. doi: 10.1051/mmnp:2006004.
[10]	S. A. H. Geritz, E. Kisdi, G. Meszena, J. A. J. Metz, Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree, Evol. Ecol., 12 (1998) 35-57.
[11]	S. A. H. Geritz, J. A. J.Metz, E. Kisdi and G. Meszena, Dynamics of adaptation and evolutionary branching, Phys. Rev. Lett., 78 (1997), 2024-2027. doi: 10.1103/PhysRevLett.78.2024.
[12]	D. Hanahan and R. A. Weinberg, Hallmarks of cancer: the next generation, Cell, 144 (2011), 646-674. doi: 10.1016/j.cell.2011.02.013.
[13]	A. Lorz, S. Mirrahimi and B. Perthame, Dirac mass dynamics in multidimensional nonlocal parabolic equations, Communications in Partial Differential Equations, 36 (2011), 1071-1098. doi: 10.1080/03605302.2010.538784.
[14]	J. Maynard Smith, "Evolution and the Theory of Games," Cambridge University Press, 1982.
[15]	S. Mirrahimi, B. Perthame and J. Wakano, Evolution of species trait through resource competition, Journal of Mathematical Biology, 64 (2012), 1189-1223. doi: 10.1007/s00285-011-0447-z.
[16]	G. Naldi, L. Pareschi and G. Toscani (Eds.), "Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences," Birkhäuser, Basel, 2010. doi: 10.1007/978-0-8176-4946-3.
[17]	J. C. Nuno, M. Primicerio and M. A. Herrero, A mathematical model of a criminal-prone society, DCDS-S, 4 (2011), 193-207. doi: 10.3934/dcdss.2011.4.193.
[18]	K. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D: Nonlinear Phenomena, 240 (2011), 363-375. doi: 10.1016/j.physd.2010.09.011.
[19]	B. Perthame, "Transport Equations in Biology," Birkhäuser, Basel, 2007.
[20]	V. Quaranta, K. A. Rejniak, P. Gerlee and A. R. Anderson, Invasion emerges from cancer cell adaptation to competitive microenvironments: quantitative predictions from multiscale mathematical models, Seminars in Cancer Biology, 18 (2008), 338-348. doi: 10.1016/j.semcancer.2008.03.018.
[21]	V. Semeshenko, M. B. Gordon and J. P. Nadal, Collective states in social systems with interacting learning agents, Physica A: Statistical Mechanics and its Applications, 387 (2008), 4903-4916. doi: 10.1016/j.physa.2008.04.019.
[22]	Z. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos, 17 (2007), 037108. doi: 10.1063/1.2766864.

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