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March  2015, 20(2): 469-493. doi: 10.3934/dcdsb.2015.20.469

Influence of a spatial structure on the long time behavior of a competitive Lotka-Volterra type system

1. 

CMAP, Ecole Polytechnique, UMR 7641, route de Saclay, 91128 Palaiseau Cedex, France, France

2. 

Institut de Mathématiques de Toulouse; UMR 5219, Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France

Received  January 2014 Revised  October 2014 Published  January 2015

To describe population dynamics, it is crucial to take into account jointly evolution mechanisms and spatial motion. However, the models which include these both aspects, are not still well-understood. Can we extend the existing results on type structured populations, to models of populations structured by type and space, considering diffusion and nonlocal competition between individuals?
    We study a nonlocal competitive Lotka-Volterra type system, describing a spatially structured population which can be either monomorphic or dimorphic. Considering spatial diffusion, intrinsic death and birth rates, together with death rates due to intraspecific and interspecific competition between the individuals, leading to some integral terms, we analyze the long time behavior of the solutions. We first prove existence of steady states and next determine the long time limits, depending on the competition rates and the principal eigenvalues of some operators, corresponding somehow to the strength of traits. Numerical computations illustrate that the introduction of a new mutant population can lead to the long time evolution of the spatial niche.
Citation: Hélène Leman, Sylvie Méléard, Sepideh Mirrahimi. Influence of a spatial structure on the long time behavior of a competitive Lotka-Volterra type system. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 469-493. doi: 10.3934/dcdsb.2015.20.469
References:
[1]

M. Alfaro, J. Coville and G. Raoul, Travelling waves in a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypic trait, Comm. Partial Differential Equations (CPDE), 38 (2013), 2126-2154. doi: 10.1080/03605302.2013.828069.

[2]

A. Arnold, L. Desvillettes and C. Prévost, Existence of nontrivial steady states for populations structured with respect to space and a continuous trait, Commun. Pure Appl. Anal., 11 (2012), 83-96.

[3]

O. Bénichou , V. Calvez, N. Meunier and R. Voituriez, Front acceleration by dynamic selection in fisher population waves, Phys. Rev. E, 86 (2012), 041908.

[4]

H. Berestycki and G. Chapuisat, Traveling fronts guided by the environment for reaction-diffusion equations, Networks and Heterogeneous Media, 8 (2013), 79-114. doi: 10.3934/nhm.2013.8.79.

[5]

H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844. doi: 10.1088/0951-7715/22/12/002.

[6]

E. Bouin and V. Calvez, Travelling waves for the cane toads equation with bounded traits, Nonlinearity, 27 (2014), 2233-2253, arXiv:1309.4755. doi: 10.1088/0951-7715/27/9/2233.

[7]

E. Bouin, V. Calvez, N. Meunier, S. Mirrahimi, B. Perthame, G. Raoul and R. Voituriez, Invasion fronts with variable motility: Phenotype selection, spatial sorting and wave acceleration, C. R. Math. Acad. Sci. Paris, 350 (2012), 761-766. doi: 10.1016/j.crma.2012.09.010.

[8]

E. Bouin and S. Mirrahimi, A Hamilton-Jacobi approach for a model of population structured by space and trait, preprint, arXiv:1307.8332.

[9]

H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Masson, 1983.

[10]

N. Champagnat, Mathematical Study of Stochastic Models of Evolution Belonging to the Ecological Theory of Adaptive Dynamics, Ph.D thesis, University of Nanterre (Paris X), 2004.

[11]

N. Champagnat, A microscopic interpretation for adaptive dynamics trait substitution sequence models, Stochastic Process. Appl., 116 (2006), 1127-1160. doi: 10.1016/j.spa.2006.01.004.

[12]

N. Champagnat and S. Méléard, Invasion and adaptive evolution for individual-based structured populations, J. Math. Biol., 55 (2007), 147-188. doi: 10.1007/s00285-007-0072-z.

[13]

N. Champagnat, R. Ferrière and S. Méléard, From individual stochastic processes to macroscopic models in adaptive evolution, Stoch. Models, 24 (2008), 2-44. doi: 10.1080/15326340802437710.

[14]

J. Coville, Convergence to equilibrium for positive solutions of some mutation-selection model, preprint, arXiv:1308.6471.

[15]

J. Cronin, Fixed Points and Topological Degree in Nonlinear Analysis, American Mathematical Society, Providence, 1964.

[16]

L. Desvillettes, R. Ferrière and C. Prevost, Infinite dimensional reaction-diffusion for population dynamics, Preprint CMLA, ENS Cachan, 2004.

[17]

U. Dieckmann, R. Law and J. A. J. Metz, The Geometry of Ecological Interactions: Symplifying Spatial Complexity, Cambridge Univ. Press, Cambridge, 2005. doi: 10.1017/CBO9780511525537.

[18]

R. Durrett and S. Levin, Stochastic spatial models: A user's guide to ecological applications, Phil. Trans. Roy. Soc. London, 343 (1994), 329-350.

[19]

J. A. Endler, Geographic Variation, Speciation, and Clines, Princeton university Press, 1977.

[20]

L. C. Evans, Partial Differential Equations, Second edition, American Mathematical Society, 2010.

[21]

D. Futuyama and G. Moreno, The evolution of ecological specialization, Ann. Rev. Ecol. Syst., 19 (1988), 207-233. doi: 10.1146/annurev.ecolsys.19.1.207.

[22]

R. Kassen, The experimental evolution of specialists, generalists and the maintenance of diversity, J. Evol. Biol., 15 (2002), 173-190. doi: 10.1046/j.1420-9101.2002.00377.x.

[23]

J. McGlad, Advanced Ecological Theory: Principles and Applications, Blackwell Science, Oxford, 1999.

[24]

E. Mayr, Animal Species and Evolution, Harvard University Press, Cambridge, 1963. doi: 10.4159/harvard.9780674865327.

[25]

J. D. Murray, Mathematical Biology, I: An Introduction, Springer, New York, NY, USA, 2002.

[26]

B. L. Phillips, G. P. Brown, J. K. Webb and R. Shine, Invasion and the evolution of speed in toads, Nature, 439 (2006), p803. doi: 10.1038/439803a.

[27]

G. Teschl, Ordinary Differential Equations and Dynamical Systems, American Mathematical Society, 2012.

[28]

D. Tilman and P. Kareiva, Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions, Princeton University Press, Princeton, NJ, 1996.

show all references

References:
[1]

M. Alfaro, J. Coville and G. Raoul, Travelling waves in a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypic trait, Comm. Partial Differential Equations (CPDE), 38 (2013), 2126-2154. doi: 10.1080/03605302.2013.828069.

[2]

A. Arnold, L. Desvillettes and C. Prévost, Existence of nontrivial steady states for populations structured with respect to space and a continuous trait, Commun. Pure Appl. Anal., 11 (2012), 83-96.

[3]

O. Bénichou , V. Calvez, N. Meunier and R. Voituriez, Front acceleration by dynamic selection in fisher population waves, Phys. Rev. E, 86 (2012), 041908.

[4]

H. Berestycki and G. Chapuisat, Traveling fronts guided by the environment for reaction-diffusion equations, Networks and Heterogeneous Media, 8 (2013), 79-114. doi: 10.3934/nhm.2013.8.79.

[5]

H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844. doi: 10.1088/0951-7715/22/12/002.

[6]

E. Bouin and V. Calvez, Travelling waves for the cane toads equation with bounded traits, Nonlinearity, 27 (2014), 2233-2253, arXiv:1309.4755. doi: 10.1088/0951-7715/27/9/2233.

[7]

E. Bouin, V. Calvez, N. Meunier, S. Mirrahimi, B. Perthame, G. Raoul and R. Voituriez, Invasion fronts with variable motility: Phenotype selection, spatial sorting and wave acceleration, C. R. Math. Acad. Sci. Paris, 350 (2012), 761-766. doi: 10.1016/j.crma.2012.09.010.

[8]

E. Bouin and S. Mirrahimi, A Hamilton-Jacobi approach for a model of population structured by space and trait, preprint, arXiv:1307.8332.

[9]

H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Masson, 1983.

[10]

N. Champagnat, Mathematical Study of Stochastic Models of Evolution Belonging to the Ecological Theory of Adaptive Dynamics, Ph.D thesis, University of Nanterre (Paris X), 2004.

[11]

N. Champagnat, A microscopic interpretation for adaptive dynamics trait substitution sequence models, Stochastic Process. Appl., 116 (2006), 1127-1160. doi: 10.1016/j.spa.2006.01.004.

[12]

N. Champagnat and S. Méléard, Invasion and adaptive evolution for individual-based structured populations, J. Math. Biol., 55 (2007), 147-188. doi: 10.1007/s00285-007-0072-z.

[13]

N. Champagnat, R. Ferrière and S. Méléard, From individual stochastic processes to macroscopic models in adaptive evolution, Stoch. Models, 24 (2008), 2-44. doi: 10.1080/15326340802437710.

[14]

J. Coville, Convergence to equilibrium for positive solutions of some mutation-selection model, preprint, arXiv:1308.6471.

[15]

J. Cronin, Fixed Points and Topological Degree in Nonlinear Analysis, American Mathematical Society, Providence, 1964.

[16]

L. Desvillettes, R. Ferrière and C. Prevost, Infinite dimensional reaction-diffusion for population dynamics, Preprint CMLA, ENS Cachan, 2004.

[17]

U. Dieckmann, R. Law and J. A. J. Metz, The Geometry of Ecological Interactions: Symplifying Spatial Complexity, Cambridge Univ. Press, Cambridge, 2005. doi: 10.1017/CBO9780511525537.

[18]

R. Durrett and S. Levin, Stochastic spatial models: A user's guide to ecological applications, Phil. Trans. Roy. Soc. London, 343 (1994), 329-350.

[19]

J. A. Endler, Geographic Variation, Speciation, and Clines, Princeton university Press, 1977.

[20]

L. C. Evans, Partial Differential Equations, Second edition, American Mathematical Society, 2010.

[21]

D. Futuyama and G. Moreno, The evolution of ecological specialization, Ann. Rev. Ecol. Syst., 19 (1988), 207-233. doi: 10.1146/annurev.ecolsys.19.1.207.

[22]

R. Kassen, The experimental evolution of specialists, generalists and the maintenance of diversity, J. Evol. Biol., 15 (2002), 173-190. doi: 10.1046/j.1420-9101.2002.00377.x.

[23]

J. McGlad, Advanced Ecological Theory: Principles and Applications, Blackwell Science, Oxford, 1999.

[24]

E. Mayr, Animal Species and Evolution, Harvard University Press, Cambridge, 1963. doi: 10.4159/harvard.9780674865327.

[25]

J. D. Murray, Mathematical Biology, I: An Introduction, Springer, New York, NY, USA, 2002.

[26]

B. L. Phillips, G. P. Brown, J. K. Webb and R. Shine, Invasion and the evolution of speed in toads, Nature, 439 (2006), p803. doi: 10.1038/439803a.

[27]

G. Teschl, Ordinary Differential Equations and Dynamical Systems, American Mathematical Society, 2012.

[28]

D. Tilman and P. Kareiva, Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions, Princeton University Press, Princeton, NJ, 1996.

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