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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

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 & Continuous Dynamical Systems - B, 2013, 18 (9) : 2267-2282. doi: 10.3934/dcdsb.2013.18.2267
References:
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B. Perthame, "Transport Equations in Biology," Birkhäuser, Basel, 2007.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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.  Google Scholar

[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. Google Scholar

[3]

D. Challet, M. Marsili and Y. C. Zhang, "Minority Games: Interacting Agents in Financial Markets," Oxford Finance, 2005. Google Scholar

[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.  Google Scholar

[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, ().  doi: 10.3934/dcdsb.2013.18.891.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. Maynard Smith, "Evolution and the Theory of Games," Cambridge University Press, 1982. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

B. Perthame, "Transport Equations in Biology," Birkhäuser, Basel, 2007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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