# American Institute of Mathematical Sciences

December  2019, 24(12): 6693-6724. 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  December 2019 Early access  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 and Continuous Dynamical Systems - B, 2019, 24 (12) : 6693-6724. 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. [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. [3] H. Berestycki, F. 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. [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. [5] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons Ltd., Chichester, UK, 2003. doi: 10.1002/0470871296. [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. [7] R. S. Cantrell, C. 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. [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. [9] R. S. Cantrell, C. Cosner and V. Hutson, Ecological models, permanence and spatial heterogeneity, Rocky Mountain J. Math., 26 (1996), 1-35.  doi: 10.1216/rmjm/1181072101. [10] J. Dockery, V. Hutson, K. 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. [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. [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. [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. [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. [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. [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. [17] B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Springer Berlin Heidelberg, 1985. doi: 10.1007/BFb0075060. [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. [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. [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. [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. [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. [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. [24] B. Perthame and P. E. Souganidis, Rare mutations limit of a steady state dispersion trait model. [25] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, 1997. [26] J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.  doi: 10.1093/biomet/38.1-2.196. [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.

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. [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. [3] H. Berestycki, F. 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. [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. [5] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons Ltd., Chichester, UK, 2003. doi: 10.1002/0470871296. [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. [7] R. S. Cantrell, C. 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. [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. [9] R. S. Cantrell, C. Cosner and V. Hutson, Ecological models, permanence and spatial heterogeneity, Rocky Mountain J. Math., 26 (1996), 1-35.  doi: 10.1216/rmjm/1181072101. [10] J. Dockery, V. Hutson, K. 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. [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. [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. [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. [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. [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. [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. [17] B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Springer Berlin Heidelberg, 1985. doi: 10.1007/BFb0075060. [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. [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. [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. [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. [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. [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. [24] B. Perthame and P. E. Souganidis, Rare mutations limit of a steady state dispersion trait model. [25] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, 1997. [26] J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.  doi: 10.1093/biomet/38.1-2.196. [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.
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