## The Annals of Applied Probability

### A probabilistic approach to Dirac concentration in nonlocal models of adaptation with several resources

#### Abstract

This work is devoted to the study of scaling limits in small mutations and large time of the solutions $u^{\varepsilon}$ of two deterministic models of phenotypic adaptation, where the parameter $\varepsilon>0$ scales the size or frequency of mutations. The second model is the so-called Lotka–Volterra parabolic PDE in $\mathbb{R}^{d}$ with an arbitrary number of resources and the first one is a version of the second model with finite phenotype space. The solutions of such systems typically concentrate as Dirac masses in the limit $\varepsilon\to0$. Our main results are, in both cases, the representation of the limits of $\varepsilon\log u^{\varepsilon}$ as solutions of variational problems and regularity results for these limits. The method mainly relies on Feynman–Kac-type representations of $u^{\varepsilon}$ and Varadhan’s lemma. Our probabilistic approach applies to multiresources situations not covered by standard analytical methods and makes the link between variational limit problems and Hamilton–Jacobi equations with irregular Hamiltonians that arise naturally from analytical methods. The finite case presents substantial difficulties since the rate function of the associated large deviation principle (LDP) has noncompact level sets. In that case, we are also able to obtain uniqueness of the solution of the variational problem and of the associated differential problem which can be interpreted as a Hamilton–Jacobi equation in finite state space.

#### Article information

Source
Ann. Appl. Probab., Volume 29, Number 4 (2019), 2175-2216.

Dates
Revised: November 2018
First available in Project Euclid: 23 July 2019

https://projecteuclid.org/euclid.aoap/1563869040

Digital Object Identifier
doi:10.1214/18-AAP1446

Mathematical Reviews number (MathSciNet)
MR3983337

#### Citation

Champagnat, Nicolas; Henry, Benoit. A probabilistic approach to Dirac concentration in nonlocal models of adaptation with several resources. Ann. Appl. Probab. 29 (2019), no. 4, 2175--2216. doi:10.1214/18-AAP1446. https://projecteuclid.org/euclid.aoap/1563869040

#### References

• [1] Bardi, M. and Capuzzo-Dolcetta, I. (1997). Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Systems & Control: Foundations & Applications. Birkhäuser, Boston, MA.
• [2] Barles, G., Evans, L. C. and Souganidis, P. E. (1990). Wavefront propagation for reaction-diffusion systems of PDE. Duke Math. J. 61 835–858.
• [3] Barles, G., Mirrahimi, S. and Perthame, B. (2009). Concentration in Lotka–Volterra parabolic or integral equations: A general convergence result. Methods Appl. Anal. 16 321–340.
• [4] Barles, G. and Perthame, B. (2007). Concentrations and constrained Hamilton–Jacobi equations arising in adaptive dynamics. In Recent Developments in Nonlinear Partial Differential Equations. Contemp. Math. 439 57–68. Amer. Math. Soc., Providence, RI.
• [5] Billingsley, P. (1971). Weak Convergence of Measures: Applications in Probability. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics 5. SIAM, Philadelphia, PA.
• [6] Bogachev, V. I. (2007). Measure Theory. Vol. I, II. Springer, Berlin.
• [7] Calvez, V. and Lam, K.-Y. (2018). Uniqueness of the viscosity solution of a constrained Hamilton–Jacobi equation. ArXiv e-prints.
• [8] Catoni, O. and Cerf, R. (1995/97). The exit path of a Markov chain with rare transitions. ESAIM Probab. Stat. 1 95–144.
• [9] Champagnat, N. (2015). Stochastic and deterministic approaches in Biology: Adaptive dynamics, ecological modeling, population genetics and molecular dynamics; well-posedness for ordinary and stochastic differential equations. Habilitation à diriger des recherches, Univ. Lorraine.
• [10] Champagnat, N. and Jabin, P.-E. (2011). The evolutionary limit for models of populations interacting competitively via several resources. J. Differential Equations 251 176–195.
• [11] Champagnat, N., Jabin, P.-E. and Méléard, S. (2014). Adaptation in a stochastic multi-resources chemostat model. J. Math. Pures Appl. (9) 101 755–788.
• [12] Champagnat, N., Jabin, P.-E. and Raoul, G. (2010). Convergence to equilibrium in competitive Lotka–Volterra and chemostat systems. C. R. Math. Acad. Sci. Paris 348 1267–1272.
• [13] Cirillo, E. N. M., Nardi, F. R. and Sohier, J. (2015). Metastability for general dynamics with rare transitions: Escape time and critical configurations. J. Stat. Phys. 161 365–403.
• [14] Clarke, F. (2013). Functional Analysis, Calculus of Variations and Optimal Control. Graduate Texts in Mathematics 264. Springer, London.
• [15] Demazure, M. (2000). Bifurcations and Catastrophes: Geometry of Solutions to Nonlinear Problems. Universitext. Springer, Berlin. Translated from the 1989 French original by David Chillingworth.
• [16] Dembo, A. and Zeitouni, O. (1993). Large Deviations Techniques and Applications. Jones and Bartlett, Boston, MA.
• [17] Desvillettes, L., Jabin, P.-E., Mischler, S. and Raoul, G. (2008). On selection dynamics for continuous structured populations. Commun. Math. Sci. 6 729–747.
• [18] Dieckmann, U. and Doebeli, M. (1999). On the origin of species by sympatric speciation. Nature 400 354–357.
• [19] Diekmann, O., Jabin, P.-E., Mischler, S. and Perthame, B. (2005). The dynamics of adaptation: An illuminating example and a Hamilton–Jacobi approach. Theor. Popul. Biol. 67 257–271.
• [20] Engel, K.-J. and Nagel, R. (2000). One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics 194. Springer, New York.
• [21] Feng, J. and Kurtz, T. G. (2006). Large Deviations for Stochastic Processes. Mathematical Surveys and Monographs 131. Amer. Math. Soc., Providence, RI.
• [22] Fleming, W. H. and Soner, H. M. (1993). Controlled Markov Processes and Viscosity Solutions. Applications of Mathematics (New York) 25. Springer, New York.
• [23] Fleming, W. H. and Souganidis, P. E. (1986). PDE-viscosity solution approach to some problems of large deviations. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 13 171–192.
• [24] Freidlin, M. (1985). Limit theorems for large deviations and reaction-diffusion equations. Ann. Probab. 13 639–675.
• [25] Freidlin, M. I. (1987). Tunneling soliton in the equations of reaction-diffusion type. In Reports from the Moscow Refusnik Seminar. Ann. New York Acad. Sci. 491 149–156. New York Acad. Sci., New York.
• [26] Freidlin, M. I. (1992). Semi-linear PDEs and limit theorems for large deviations. In École D’Été de Probabilités de Saint-Flour XX—1990. Lecture Notes in Math. 1527 1–109. Springer, Berlin.
• [27] Jabin, P.-E. and Raoul, G. (2011). On selection dynamics for competitive interactions. J. Math. Biol. 63 493–517.
• [28] Kraut, A. and Bovier, A. (2018). From adaptive dynamics to adaptive walks. Available at arXiv:1810.13188.
• [29] Lions, P.-L. (1982). Generalized Solutions of Hamilton–Jacobi Equations. Research Notes in Mathematics 69. Pitman, Boston, MA.
• [30] Lions, P.-L. and Perthame, B. (1987). Remarks on Hamilton–Jacobi equations with measurable time-dependent Hamiltonians. Nonlinear Anal. 11 613–621.
• [31] Lorz, A., Mirrahimi, S. and Perthame, B. (2011). Dirac mass dynamics in multidimensional nonlocal parabolic equations. Comm. Partial Differential Equations 36 1071–1098.
• [32] Lunardi, A. (1995). Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications 16. Birkhäuser, Basel.
• [33] Metz, J. A. J., Geritz, S. A. H., Meszéna, G., Jacobs, F. J. A. and van Heerwaarden, J. S. (1996). Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction. In Stochastic and Spatial Structures of Dynamical Systems (Amsterdam, 1995). Konink. Nederl. Akad. Wetensch. Verh. Afd. Natuurk. Eerste Reeks 45 183–231. North-Holland, Amsterdam.
• [34] Mirrahimi, S., Perthame, B. and Wakano, J. Y. (2012). Evolution of species trait through resource competition. J. Math. Biol. 64 1189–1223.
• [35] Mirrahimi, S. and Roquejoffre, J.-M. (2015). Uniqueness in a class of Hamilton–Jacobi equations with constraints. C. R. Math. Acad. Sci. Paris 353 489–494.
• [36] Mirrahimi, S. and Roquejoffre, J.-M. (2016). A class of Hamilton–Jacobi equations with constraint: Uniqueness and constructive approach. J. Differential Equations 260 4717–4738.
• [37] Pazy, A. (1974). Semi-groups of linear operators and applications to partial differential equations. Lecture Note, No. 10, Dept. Mathematics, Univ. Maryland, College Park, MD.
• [38] Perthame, B. (2015). Parabolic Equations in Biology: Growth, Reaction, Movement and Diffusion. Lecture Notes on Mathematical Modelling in the Life Sciences. Springer, Cham.
• [39] Perthame, B. and Barles, G. (2008). Dirac concentrations in Lotka–Volterra parabolic PDEs. Indiana Univ. Math. J. 57 3275–3301.
• [40] Perthame, B. and Génieys, S. (2007). Concentration in the nonlocal Fisher equation: The Hamilton–Jacobi limit. Math. Model. Nat. Phenom. 2 135–151.
• [41] Raoul, G. (2011). Long time evolution of populations under selection and vanishing mutations. Acta Appl. Math. 114 1–14.
• [42] Vinter, R. B. and Wolenski, P. (1990). Hamilton–Jacobi theory for optimal control problems with data measurable in time. SIAM J. Control Optim. 28 1404–1419.