doi: 10.3934/dcdsb.2019163

Trait selection and rare mutations: The case of large diffusivities

Laboratoire Jacques-Louis Lions, CNRS UMR 7598, Paris Sorbonne Université, 4 place Jussieu, 75005 Paris, France

* Corresponding author: Idriss Mazari

Received  July 2018 Revised  March 2019 Published  July 2019

Fund Project: The author was partially supported by the Project "Analysis and simulation of optimal shapes - application to lifesciences" of the Paris City Hall

We consider a system of $ N $ competing species, each of which can access a different resources distribution and who can disperse at different speeds. We fully characterize the existence and stability of steady-states for large diffusivities. Indeed, we prove that the resources distribution yielding the largest total population size at equilibrium is, broadly speaking, always the winner when species disperse quickly. The criterion also uses the different dispersal rates. The methods used rely on an expansion of the solutions of the Lotka-Volterra sytem for large diffusivities, and is an extension of the "slowest diffuser always wins" principle.

Using this method, we also study the case of an equation modelling a trait structured population, with small mutations. We assume that each trait is characterized by its diffusivity and the resources it can access. We similarly derive a criterion mixing these diffusivities and the total population size functional for the single species model to show that for rare mutations and large diffusivities, the population concentrates in a neighbourhood of a trait maximizing this criterion.

Citation: Idriss Mazari. Trait selection and rare mutations: The case of large diffusivities. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019163
References:
[1]

O. Bénichou, V. Calvez, N. Meunier and R. Voituriez, Front acceleration by dynamic selection in fisher population waves, Phys. Rev. E, 86 (2012), 041908, https://link.aps.org/doi/10.1103/PhysRevE.86.041908.Google Scholar

[2]

H. Berestycki and T. Lachand-Robert, Some properties of monotone rearrangement with applications to elliptic equations in cylinders, Mathematische Nachrichten, 266 (2004), 3-19. doi: 10.1002/mana.200310139. Google Scholar

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H. BerestyckiF. Hamel and L. Roques, Analysis of the periodically fragmented environment model: Ⅰ. species persistence, Journal of Mathematical Biology, 51 (2005), 75-113. doi: 10.1007/s00285-004-0313-3. Google Scholar

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E. Bouin and S. Mirrahimi, A hamilton-jacobi approach for a model of population structured by space and trait, Communications in Mathematical Sciences, 13 (2015), 1431-1452. doi: 10.4310/CMS.2015.v13.n6.a4. Google Scholar

[5]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons Ltd., Chichester, UK, 2003. doi: 10.1002/0470871296. Google Scholar

[6]

R. S. Cantrell and C. Cosner, On the effects of spatial heterogeneity on the persistence of interacting species, Journal of Mathematical Biology, 37 (1998), 103-145. doi: 10.1007/s002850050122. Google Scholar

[7]

R. S. CantrellC. Cosner and V. Hutson, Permanence in ecological systems with spatial heterogeneity, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 123 (1993), 533-559. doi: 10.1017/S0308210500025877. Google Scholar

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R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments. Ⅱ, SIAM J. Math. Anal., 22 (1991), 1043-1064. doi: 10.1137/0522068. Google Scholar

[9]

R. S. CantrellC. Cosner and V. Hutson, Ecological models, permanence and spatial heterogeneity, Rocky Mountain J. Math., 26 (1996), 1-35. doi: 10.1216/rmjm/1181072101. Google Scholar

[10]

J. DockeryV. HutsonK. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction diffusion model, Journal of Mathematical Biology, 37 (1998), 61-83. doi: 10.1007/s002850050120. Google Scholar

[11]

R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x. Google Scholar

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L. Girardin, Competition in periodic media: I-Existence of pulsating fronts, Discrete and Continuous Dynamical Systems Series B, 22 (2017), 1341-1360, https://hal.archives-ouvertes.fr/hal-01328421. doi: 10.3934/dcdsb.2017065. Google Scholar

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X. Q. He and W. -M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: Diffusion and spatial heterogeneity Ⅰ, Communications on Pure and Applied Mathematics, 69 (2016), 981-1014. doi: 10.1002/cpa.21596. Google Scholar

[14]

X. Q. He and W. -M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, Ⅱ, Calculus of Variations and Partial Differential Equations, 55 (2016), Art. 25, 20 pp. doi: 10.1007/s00526-016-0964-0. Google Scholar

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X. Q. He and W. -M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, Ⅲ, Calculus of Variations and Partial Differential Equations, 56 (2017), Art. 132, 26 pp. doi: 10.1007/s00526-017-1234-5. Google Scholar

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V. Hutson, Y. Lou and K. Mischaikow, Spatial heterogeneity of resources versus Lotka-Volterra dynamics, Journal of Differential Equations, 185 (2002), 97-136, http://www.sciencedirect.com/science/article/pii/S0022039601941579. doi: 10.1006/jdeq.2001.4157. Google Scholar

[17]

B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Springer Berlin Heidelberg, 1985. doi: 10.1007/BFb0075060. Google Scholar

[18]

K. -Y. Lam and Y. Lou, An integro-PDE model for evolution of random dispersal, Journal of Functional Analysis, 272 (2017), 1755-1790. doi: 10.1016/j.jfa.2016.11.017. Google Scholar

[19]

J. Lamboley, A. Laurain, G. Nadin and Y. Privat, Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions, Calc. Var. Partial Differential Equations, 55 (2016), Art. 144, 37 pp. doi: 10.1007/s00526-016-1084-6. Google Scholar

[20]

Y. Lou, Tutorials in athematical biosciences Ⅳ: Evolution and ecology, Chapter Some Challenging Mathematical Problems in Evolution of Dispersal and Population Dynamics, 171-205, Springer Berlin Heidelberg, Berlin, Heidelberg, 2008.Google Scholar

[21]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, Journal of Differential Equations, 223 (2006), 400-426. doi: 10.1016/j.jde.2005.05.010. Google Scholar

[22]

I. Mazari, G. Nadin and Y. Privat, Optimal location of resources maximizing the total population size in logistic models, Working paper or preprint, (2017), https://hal.archives-ouvertes.fr/hal-01607046.Google Scholar

[23]

S. W. Pacala and J. Roughgarden, Spatial heterogeneity and interspecific competition, Theoretical Population Biology, 21 (1982), 92-113, http://www.sciencedirect.com/science/article/pii/0040580982900089. doi: 10.1016/0040-5809(82)90008-9. Google Scholar

[24]

B. Perthame and P. E. Souganidis, Rare mutations limit of a steady state dispersion trait model.Google Scholar

[25] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, 1997. Google Scholar
[26]

J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218. doi: 10.1093/biomet/38.1-2.196. Google Scholar

[27]

O. Turanova, On a model of a population with variable motility, Mathematical Models and Methods in Applied Sciences, 25 (2015), 1961-2014, https://www.worldscientific.com/doi/abs/10.1142/S0218202515500505. doi: 10.1142/S0218202515500505. Google Scholar

show all references

References:
[1]

O. Bénichou, V. Calvez, N. Meunier and R. Voituriez, Front acceleration by dynamic selection in fisher population waves, Phys. Rev. E, 86 (2012), 041908, https://link.aps.org/doi/10.1103/PhysRevE.86.041908.Google Scholar

[2]

H. Berestycki and T. Lachand-Robert, Some properties of monotone rearrangement with applications to elliptic equations in cylinders, Mathematische Nachrichten, 266 (2004), 3-19. doi: 10.1002/mana.200310139. Google Scholar

[3]

H. BerestyckiF. Hamel and L. Roques, Analysis of the periodically fragmented environment model: Ⅰ. species persistence, Journal of Mathematical Biology, 51 (2005), 75-113. doi: 10.1007/s00285-004-0313-3. Google Scholar

[4]

E. Bouin and S. Mirrahimi, A hamilton-jacobi approach for a model of population structured by space and trait, Communications in Mathematical Sciences, 13 (2015), 1431-1452. doi: 10.4310/CMS.2015.v13.n6.a4. Google Scholar

[5]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons Ltd., Chichester, UK, 2003. doi: 10.1002/0470871296. Google Scholar

[6]

R. S. Cantrell and C. Cosner, On the effects of spatial heterogeneity on the persistence of interacting species, Journal of Mathematical Biology, 37 (1998), 103-145. doi: 10.1007/s002850050122. Google Scholar

[7]

R. S. CantrellC. Cosner and V. Hutson, Permanence in ecological systems with spatial heterogeneity, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 123 (1993), 533-559. doi: 10.1017/S0308210500025877. Google Scholar

[8]

R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments. Ⅱ, SIAM J. Math. Anal., 22 (1991), 1043-1064. doi: 10.1137/0522068. Google Scholar

[9]

R. S. CantrellC. Cosner and V. Hutson, Ecological models, permanence and spatial heterogeneity, Rocky Mountain J. Math., 26 (1996), 1-35. doi: 10.1216/rmjm/1181072101. Google Scholar

[10]

J. DockeryV. HutsonK. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction diffusion model, Journal of Mathematical Biology, 37 (1998), 61-83. doi: 10.1007/s002850050120. Google Scholar

[11]

R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x. Google Scholar

[12]

L. Girardin, Competition in periodic media: I-Existence of pulsating fronts, Discrete and Continuous Dynamical Systems Series B, 22 (2017), 1341-1360, https://hal.archives-ouvertes.fr/hal-01328421. doi: 10.3934/dcdsb.2017065. Google Scholar

[13]

X. Q. He and W. -M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: Diffusion and spatial heterogeneity Ⅰ, Communications on Pure and Applied Mathematics, 69 (2016), 981-1014. doi: 10.1002/cpa.21596. Google Scholar

[14]

X. Q. He and W. -M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, Ⅱ, Calculus of Variations and Partial Differential Equations, 55 (2016), Art. 25, 20 pp. doi: 10.1007/s00526-016-0964-0. Google Scholar

[15]

X. Q. He and W. -M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, Ⅲ, Calculus of Variations and Partial Differential Equations, 56 (2017), Art. 132, 26 pp. doi: 10.1007/s00526-017-1234-5. Google Scholar

[16]

V. Hutson, Y. Lou and K. Mischaikow, Spatial heterogeneity of resources versus Lotka-Volterra dynamics, Journal of Differential Equations, 185 (2002), 97-136, http://www.sciencedirect.com/science/article/pii/S0022039601941579. doi: 10.1006/jdeq.2001.4157. Google Scholar

[17]

B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Springer Berlin Heidelberg, 1985. doi: 10.1007/BFb0075060. Google Scholar

[18]

K. -Y. Lam and Y. Lou, An integro-PDE model for evolution of random dispersal, Journal of Functional Analysis, 272 (2017), 1755-1790. doi: 10.1016/j.jfa.2016.11.017. Google Scholar

[19]

J. Lamboley, A. Laurain, G. Nadin and Y. Privat, Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions, Calc. Var. Partial Differential Equations, 55 (2016), Art. 144, 37 pp. doi: 10.1007/s00526-016-1084-6. Google Scholar

[20]

Y. Lou, Tutorials in athematical biosciences Ⅳ: Evolution and ecology, Chapter Some Challenging Mathematical Problems in Evolution of Dispersal and Population Dynamics, 171-205, Springer Berlin Heidelberg, Berlin, Heidelberg, 2008.Google Scholar

[21]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, Journal of Differential Equations, 223 (2006), 400-426. doi: 10.1016/j.jde.2005.05.010. Google Scholar

[22]

I. Mazari, G. Nadin and Y. Privat, Optimal location of resources maximizing the total population size in logistic models, Working paper or preprint, (2017), https://hal.archives-ouvertes.fr/hal-01607046.Google Scholar

[23]

S. W. Pacala and J. Roughgarden, Spatial heterogeneity and interspecific competition, Theoretical Population Biology, 21 (1982), 92-113, http://www.sciencedirect.com/science/article/pii/0040580982900089. doi: 10.1016/0040-5809(82)90008-9. Google Scholar

[24]

B. Perthame and P. E. Souganidis, Rare mutations limit of a steady state dispersion trait model.Google Scholar

[25] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, 1997. Google Scholar
[26]

J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218. doi: 10.1093/biomet/38.1-2.196. Google Scholar

[27]

O. Turanova, On a model of a population with variable motility, Mathematical Models and Methods in Applied Sciences, 25 (2015), 1961-2014, https://www.worldscientific.com/doi/abs/10.1142/S0218202515500505. doi: 10.1142/S0218202515500505. Google Scholar

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