Dynamics of concentration in a population model structured by age and a phenotypical trait. (English) Zbl 1398.35251
Summary: We study a mathematical model describing the growth process of a population structured by age and a phenotypical trait, subject to aging, competition between individuals and rare mutations. Our goals are to describe the asymptotic behavior of the solution to a renewal type equation, and then to derive properties that illustrate the adaptive dynamics of such a population. We begin with a simplified model by discarding the effect of mutations, which allows us to introduce the main ideas and state the full result. Then we discuss the general model and its limitations.
Our approach uses the eigenelements of a formal limiting operator, that depend on the structuring variables of the model and define an effective fitness. Then we introduce a new method which reduces the convergence proof to entropy estimates rather than estimates on the constrained Hamilton-Jacobi equation. Numerical tests illustrate the theory and show the selection of a fittest trait according to the effective fitness. For the problem with mutations, an unusual Hamiltonian arises with an exponential growth, for which we prove existence of a global viscosity solution, using an uncommon a priori estimate and a new uniqueness result.
Our approach uses the eigenelements of a formal limiting operator, that depend on the structuring variables of the model and define an effective fitness. Then we introduce a new method which reduces the convergence proof to entropy estimates rather than estimates on the constrained Hamilton-Jacobi equation. Numerical tests illustrate the theory and show the selection of a fittest trait according to the effective fitness. For the problem with mutations, an unusual Hamiltonian arises with an exponential growth, for which we prove existence of a global viscosity solution, using an uncommon a priori estimate and a new uniqueness result.
MSC:
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35F21 | Hamilton-Jacobi equations |
| 49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |
| 92D25 | Population dynamics (general) |
| 35D40 | Viscosity solutions to PDEs |
Keywords:
adaptive evolution; asymptotic behavior; Dirac concentrations; Hamilton-Jacobi equations; mathematical biology; renewal equation; viscosity solutionsReferences:
| [1] | Ackleh, A.S.; Ben Fitzpatrick, G.; Thieme, H.R., Rate distributions and survival of the fittest: a formulation on the space of measures, Discrete Contin. Dyn. Syst., Ser. B, 5, 917-928, (2005) · Zbl 1078.92055 · doi:10.3934/dcdsb.2005.5.917 |
| [2] | Adimy, M.; Crauste, F.; Ruan, S., A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia, SIAM J. Appl. Math., 65, 1328-1352, (2005) · Zbl 1090.34058 · doi:10.1137/040604698 |
| [3] | Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (1997) · Zbl 0890.49011 · doi:10.1007/978-0-8176-4755-1 |
| [4] | Barles, G.: Solutions de viscosité des équations de Hamilton-Jacobi. Springer, Berlin (1994) · Zbl 0819.35002 |
| [5] | Barles, G.; Evans, L.C.; Souganidis, P.E., Wavefront propagation for reaction diffusion systems of PDE, Duke Math. J., 61, 835-858, (1990) · Zbl 0749.35015 · doi:10.1215/S0012-7094-90-06132-0 |
| [6] | Barles, G.; Mirrahimi, S.; Perthame, B., Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result, Methods Appl. Anal., 16, 321-340, (2009) · Zbl 1204.35027 |
| [7] | Barles, G.; Perthame, B., Exit time problems in optimal control and vanishing viscosity method, SIAM J. Control Optim., 26, 1133-1148, (1988) · Zbl 0674.49027 · doi:10.1137/0326063 |
| [8] | Barles, G.; Perthame, B., Concentrations and constrained Hamilton-Jacobi equations arising in adpative dynamics, Contemp. Math., 439, 57, (2007) · Zbl 1137.49027 · doi:10.1090/conm/439/08463 |
| [9] | Barles, G.; Perthame, B., Dirac concentrations in Lotka-Volterra parabolic pdes, Indiana Univ. Math. J., 57, 3275-3301, (2008) · Zbl 1172.35005 · doi:10.1512/iumj.2008.57.3398 |
| [10] | Barles, G.; Souganidis, P.E., Front propagation for reaction-diffusion equations arising in combustion theory, Asymptot. Anal., 14, 277-292, (1997) · Zbl 0882.35062 |
| [11] | Busse, J.-E.; Gwiazda, P.; Marciniak-Czochra, A., Mass concentration in a nonlocal model of clonal selection, J. Math. Biol., 73, 1001-1033, (2016) · Zbl 1360.92035 · doi:10.1007/s00285-016-0979-3 |
| [12] | Cai, W.; Jabin, P.-E.; Liu, H., Time-asymptotic convergence rates towards the discrete evolutionary stable distribution, Math. Models Methods Appl. Sci., 25, 1589-1616, (2015) · Zbl 1328.65268 · doi:10.1142/S0218202515500426 |
| [13] | Calsina, A.; Cuadrado, S.; Desvillettes, L.; Raoul, G., Asymptotics of steady states of a selection-mutation equation for small mutation rate, Proc. R. Soc. Edinb., 143, 1123-1146, (2013) · Zbl 1293.35341 · doi:10.1017/S0308210510001629 |
| [14] | Calsina, A.; Palmada, J.M., Steady states of a selection-mutation model for an age structured population, J. Math. Anal. Appl., 400, 386-395, (2013) · Zbl 1262.35200 · doi:10.1016/j.jmaa.2012.11.042 |
| [15] | Calvez, V., Gabriel, P., Mateos González, Á.: Limiting Hamilton-Jacobi equation for the large scale asymptotics of a subdiffusion jump-renewal equation (2016). arXiv:1609.06933 · Zbl 1229.35113 |
| [16] | Champagnat, N.; Ferrière, R.; Ben Arous, G., The canonical equation of adaptive dynamics: a mathematical view, Selection, 2, 73-83, (2002) · doi:10.1556/Select.2.2001.1-2.6 |
| [17] | Champagnat, N.; Jabin, P.-E., The evolutionary limit for models of populations interacting competitively via several resources, J. Differ. Equ., 251, 176-195, (2011) · Zbl 1227.35040 · doi:10.1016/j.jde.2011.03.007 |
| [18] | Crandall, M.G.; Evans, L.C.; Lions, P.-L., Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Am. Math. Soc., 282, 487-502, (1984) · Zbl 0543.35011 · doi:10.1090/S0002-9947-1984-0732102-X |
| [19] | Crandall, M.G.; Ishii, H.; Lions, P.-L., User’s guide to viscosity solutions of second order partial differential equations, Bull. Am. Math. Soc., 27, 1-67, (1992) · Zbl 0755.35015 · doi:10.1090/S0273-0979-1992-00266-5 |
| [20] | Desvillettes, L.; Jabin, P.-E.; Mischler, S.; Raoul, G., On mutation selection dynamics, Commun. Math. Sci., 6, 729-747, (2008) · Zbl 1176.45009 · doi:10.4310/CMS.2008.v6.n3.a10 |
| [21] | Diekmann, O., A beginner’s guide to adaptive dynamics, Banach Cent. Publ., 63, 47-86, (2004) · Zbl 1051.92032 |
| [22] | Diekmann, O.; Jabin, P.-E.; Mischler, S.; Perthame, B., The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach, Theor. Popul. Biol., 67, 257-271, (2005) · Zbl 1072.92035 · doi:10.1016/j.tpb.2004.12.003 |
| [23] | Djidjou-Demasse, R.; Ducrot, A.; Fabre, F., Steady state concentration for a phenotypic structured problem modeling the evolutionary epidemiology of spore producing pathogens, Math. Models Methods Appl. Sci., 27, 385-426, (2017) · Zbl 1360.92077 · doi:10.1142/S0218202517500051 |
| [24] | Doumic, M.; Gabriel, P., Eigenelements of a general aggregation-fragmentation model, Math. Models Methods Appl. Sci., 20, 757-783, (2010) · Zbl 1201.35086 · doi:10.1142/S021820251000443X |
| [25] | Evans, L.C.; Souganidis, P.E., A PDE approach to geometric optics for certain semilinear parabolic equations, Indiana Univ. Math. J., 38, 141-172, (1989) · Zbl 0692.35014 · doi:10.1512/iumj.1989.38.38007 |
| [26] | Fleming, W.H.; Soner, H.M., Controlled Markov processes and viscosity solutions, No. 25, (1993), Berlin · Zbl 0773.60070 |
| [27] | Gyllenberg, M.; Webb, G.F., A nonlinear structured population model of tumor growth with quiescence, J. Math. Biol., 28, 671-694, (1990) · Zbl 0744.92026 · doi:10.1007/BF00160231 |
| [28] | Ishii, H., Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets, Bull. Fac. Sci. Eng. Chuo Univ., 28, 33-77, (1985) · Zbl 0937.35505 |
| [29] | Jabin, P.-E.; Gaël, R., On selection dynamics for competitive interactions, J. Math. Biol., 63, 493-517, (2011) · Zbl 1230.92038 · doi:10.1007/s00285-010-0370-8 |
| [30] | Jabin, P.-E., Schram, R.S.: Selection-mutation dynamics with spatial dependence (2016). arXiv:1601.04553 · Zbl 1328.65268 |
| [31] | Lorenzi, T.; Lorz, A.; Restori, G., Asymptotic dynamics in populations structured by sensitivity to global warming and habitat shrinking, Acta Appl. Math., 131, 44-67, (2014) · Zbl 1305.35149 · doi:10.1007/s10440-013-9849-9 |
| [32] | Lorz, A.; Mirrahimi, S.; Perthame, B., Dirac mass dynamics in multidimensional nonlocal parabolic equations, Commun. Partial Differ. Equ., 36, 1071-1098, (2011) · Zbl 1229.35113 · doi:10.1080/03605302.2010.538784 |
| [33] | Metz, J.A.J., Diekmann, O.: The Dynamics of Physiologically Structured Populations. Lecture Notes in Biomathematics, vol. 68. Springer, Berlin (1986) · Zbl 0614.92014 |
| [34] | Philippe, M., Existence of a solution to the cell division eigenproblem, Math. Models Methods Appl. Sci., 16, 1125-1153, (2006) · Zbl 1094.92023 |
| [35] | Michel, P.; Mischler, S.; Perthame, B., General relative entropy inequality: an illustration on growth models, J. Math. Pures Appl. (9), 84, 1235-1260, (2005) · Zbl 1085.35042 · doi:10.1016/j.matpur.2005.04.001 |
| [36] | Mirrahimi, S., Adaptation and migration of a population between patches, Discrete Contin. Dyn. Syst., Ser. B, 18.3, 753-768, (2013) · Zbl 1263.35021 |
| [37] | Mirrahimi, S.; Perthame, B., Asymptotic analysis of a selection model with space, J. Math. Pures Appl., 104, 1108-1118, (2015) · Zbl 1327.35011 · doi:10.1016/j.matpur.2015.07.006 |
| [38] | Mischler, S.; Perthame, B.; Ryzhik, L., Stability in a nonlinear population maturation model, Math. Models Methods Appl. Sci., 12, 1751-1772, (2002) · Zbl 1020.92025 · doi:10.1142/S021820250200232X |
| [39] | Perthame, B.: Transport Equations in Biology. Frontiers in Mathematics. Birkhäuser, Basel (2007) · Zbl 1185.92006 |
| [40] | Perthame, B.; Souganidis, P.E., Rare mutations limit of a steady state dispersal evolution model, Math. Model. Nat. Phenom., 11, 154-166, (2016) · Zbl 1387.35027 · doi:10.1051/mmnp/201611411 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
