×

Density dependent replicator-mutator models in directed evolution. (English) Zbl 1445.35194

The Cauchy problem for the following parabolic equation is studied \[ \begin{cases} u_{t}(t,x)=\sigma^2\,u_{x x}(t,x)+u(t,x)\,\big(x-\overline{u}(t)\big)\,\big(\overline{u}(t)\big)^{-1},\quad t>0,\,\,x\in\mathbb{R},\\ u(0,x)=u_0(x),\quad x\in \mathbb{R}, \end{cases}\tag{1} \] where \(\displaystyle\overline{u}=\int_{\mathbb{R}} xu(t,x)\,dx\). This is the mathematical replicator-mutator model of a bioligical directed evolution process which is density-dependent. Here \(u(t,\cdot)\) means the probability density on \(\mathbb{R}\) and \(\overline{u}\) is its mean.
Denote by \(\mathcal{A}\) the set of non-negative functions \(f\in L^1(\mathbb{R})\) such that \(\displaystyle\int_{\mathbb{R}}f(x)\,dx=1\) and \(\displaystyle\lim\limits_{x\to\pm \infty}\,f(x) e^{\alpha |x|}=0\), for \(\alpha>0\) \((\star)\). The function \(u\) is defined as a solution of the Problem (1) if: (i) \(u\in C^{\infty} \big((0,\infty)\times \mathbb{R}\big)\); (ii) \(u(t,\cdot)\in \mathcal{A}\) for all \(t\ge 0\); (iii) for all \(t>0\), \(u_t(t,\cdot), u_x(t,\cdot)\) and \(u_{x x}(t,\cdot)\) decrease faster than any exponential function in the sense of \((\star)\); (iv) \(u\) solves (1) in the classical sense; (v)\(u(t,\cdot)\to u_0\) in \(L^1(\mathbb{R})\), as \(t\to 0\).
The authors prove the existence of a unique global solution to the problem (1) for any \(u_0\in\mathcal{A}\) and they give an implicit formula to determine the mean \(\overline{u}\). Also, they establish the large time behavior of \(\overline{u}\). Having these results, the authors establish the well-posedness and the long time behavior of the solution to the following Cauchy problem \[ \begin{cases} v_{t}(t,x)=\sigma^2\,\overline{v}(t)\,v_{xx}(t,x)+v(t,x)\,\big(x-\overline{v}(t)\big),\quad t>0,\,\,x\in\mathbb{R},\\ v(0,x)=v_0(x),\quad x\in \mathbb{R}. \end{cases} \]

MSC:

35K15 Initial value problems for second-order parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B09 Positive solutions to PDEs
92B05 General biology and biomathematics
35R09 Integro-partial differential equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] M. Alfaro; R. Carles, Explicit solutions for replicator-mutator equations: Extinction versus acceleration, SIAM J. Appl. Math., 74, 1919-1934 (2014) · Zbl 1334.92286 · doi:10.1137/140979411
[2] M. Alfaro; R. Carles, Replicator-mutator equations with quadratic fitness, Proc. Amer. Math. Soc., 145, 5315-5327 (2017) · Zbl 1376.92037 · doi:10.1090/proc/13669
[3] M. Alfaro; R. Carles, Superexponential growth or decay in the heat equation with a logarithmic nonlinearity, Dyn. Partial Differ. Equ., 14, 343-358 (2017) · Zbl 1386.35129 · doi:10.4310/DPDE.2017.v14.n4.a2
[4] M. Alfaro and M. Veruete, Evolutionary Branching Via Replicator-Mutator Equations, J. Dynam. Differential Equations, 2018. · Zbl 1426.92047
[5] F. H. Arnold, Design by Directed Evolution, Acc. Chem. Res, 1998.
[6] I. Białynicki-Birula; J. Mycielski, Nonlinear wave mechanics, Ann. Physics, 100, 62-93 (1976) · doi:10.1016/0003-4916(76)90057-9
[7] V. N. Biktashev, A simple mathematical model of gradual darwinian evolution: emergence of a gaussian trait distribution in adaptation along a fitness gradient, J. Math. Biol., 68, 1225-1248 (2014) · Zbl 1288.35273 · doi:10.1007/s00285-013-0669-3
[8] I. Bomze; R. Burger, Stability by mutation in evolutionary games, Games Econom. Behav., 11, 146-172 (1995) · Zbl 0840.90141 · doi:10.1006/game.1995.1047
[9] R. Bürger, On the maintenance of genetic variation: Global analysis of Kimura’s continuum-of-alleles model, J. Math. Biol., 24, 341-351 (1986) · Zbl 0631.92009 · doi:10.1007/BF00275642
[10] R. Bürger, Mutation-selection balance and continuum-of-alleles models, Math. Biosci., 91, 67-83 (1988) · Zbl 0686.92011 · doi:10.1016/0025-5564(88)90024-7
[11] R. Bürger, Perturbations of positive semigroups and applications to population genetics, Math. Z., 197, 259-272 (1988) · Zbl 0618.47036 · doi:10.1007/BF01215194
[12] R. Carles; I. Gallagher, Universal dynamics for the defocusing logarithmic schrödinger equation, Duke Math. J., 167, 1761-1801 (2018) · Zbl 1394.35467 · doi:10.1215/00127094-2018-0006
[13] W. H. Fleming, Equilibrium distributions of continuous polygenic traits, SIAM J. Appl. Math., 36, 148-168 (1979) · Zbl 0436.92012 · doi:10.1137/0136014
[14] M.-E. Gil; F. Hamel; G. Martin; L. Roques, Mathematical properties of a class of integro-differential models from population genetics, SIAM J. Appl. Math., 77, 1536-1561 (2017) · Zbl 1378.35007 · doi:10.1137/16M1108224
[15] M.-E. Gil, F. Hamel, G. Martin and L. Roques, Dynamics of Fitness Distributions in the Presence of a Phenotypic Optimum: An Integro-differential Approach, HAL preprint, 2018. · Zbl 1419.35190
[16] K. Hadeler, Stable polymorphisms in a selection model with mutation, SIAM J. Appl. Math., 41, 1-7 (1981) · Zbl 0443.34028 · doi:10.1137/0141001
[17] J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems: Mathematical Aspects of Selection, London Mathematical Society Student Texts, Cambridge University Press, 1988. · Zbl 0678.92010
[18] M. Kimura, A stochastic model concerning the maintenance of genetic variability in quantitative characters, Proc. Natl. Acad. Sci., USA, 54 (1965), 731-736. · Zbl 0137.14404
[19] M. Nowak; N. Komarova; P. Niyogi, Evolution of universal grammar, Science, 291, 114-118 (2001) · Zbl 1226.91060 · doi:10.1126/science.291.5501.114
[20] K. Page; M. Nowak, Unifying evolutionary dynamics, J. Theoret. Biol., 219, 93-98 (2002) · doi:10.1016/S0022-5193(02)93112-7
[21] P. Schuster; K. Sigmund, Replicator dynamics, J. Theoret. Biol., 100, 533-538 (1983) · doi:10.1016/0022-5193(83)90445-9
[22] P. Stadler; P. Schuster, Mutation in autocatalytic reaction networks, J. Math. Biol., 30, 597-631 (1992) · Zbl 0776.92007 · doi:10.1007/BF00948894
[23] P. Taylor; L. Jonker, Evolutionary stable strategies and game dynamics, Math. Biosci., 40, 145-156 (1978) · Zbl 0395.90118 · doi:10.1016/0025-5564(78)90077-9
[24] A. Zadorin and Y. Rondelez, Natural selection in compartmentalized environment with reshuffling, arXiv preprint, arXiv: 1707.07461, 2017. · Zbl 1425.92146
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.