# American Institute of Mathematical Sciences

## Density dependent replicator-mutator models in directed evolution

 IMAG, Université de Montpellier, CNRS, Montpellier, 34000, France

* Corresponding author: Matthieu Alfaro

Received  January 2019 Revised  May 2019 Published  September 2019

We analyze a replicator-mutator model arising in the context of directed evolution [24], where the selection term is modulated over time by the mean-fitness. We combine a Cumulant Generating Function approach [14] and a spatio-temporal rescaling related to the Avron-Herbst formula [1] to give of a complete picture of the Cauchy problem. Besides its well-posedness, we provide an implicit/explicit expression of the solution, and analyze its large time behaviour. As a by product, we also solve a replicator-mutator model where the mutation coefficient is socially determined, in the sense that it is modulated by the mean-fitness. The latter model reveals concentration or anti diffusion/diffusion phenomena.

Citation: Matthieu Alfaro, Mario Veruete. Density dependent replicator-mutator models in directed evolution. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019224
##### References:
 [1] M. Alfaro and R. Carles, Explicit solutions for replicator-mutator equations: Extinction versus acceleration, SIAM J. Appl. Math., 74 (2014), 1919-1934. doi: 10.1137/140979411. Google Scholar [2] M. Alfaro and R. Carles, Replicator-mutator equations with quadratic fitness, Proc. Amer. Math. Soc., 145 (2017), 5315-5327. doi: 10.1090/proc/13669. Google Scholar [3] M. Alfaro and R. Carles, Superexponential growth or decay in the heat equation with a logarithmic nonlinearity, Dyn. Partial Differ. Equ., 14 (2017), 343-358. doi: 10.4310/DPDE.2017.v14.n4.a2. Google Scholar [4] M. Alfaro and M. Veruete, Evolutionary Branching Via Replicator–Mutator Equations, J. Dynam. Differential Equations, 2018.Google Scholar [5] F. H. Arnold, Design by Directed Evolution, Acc. Chem. Res, 1998.Google Scholar [6] I. Białynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Physics, 100 (1976), 62-93. doi: 10.1016/0003-4916(76)90057-9. Google Scholar [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 (2014), 1225-1248. doi: 10.1007/s00285-013-0669-3. Google Scholar [8] I. Bomze and R. Burger, Stability by mutation in evolutionary games, Games Econom. Behav., 11 (1995), 146-172. doi: 10.1006/game.1995.1047. Google Scholar [9] R. Bürger, On the maintenance of genetic variation: Global analysis of Kimura's continuum-of-alleles model, J. Math. Biol., 24 (1986), 341-351. doi: 10.1007/BF00275642. Google Scholar [10] R. Bürger, Mutation-selection balance and continuum-of-alleles models, Math. Biosci., 91 (1988), 67-83. doi: 10.1016/0025-5564(88)90024-7. Google Scholar [11] R. Bürger, Perturbations of positive semigroups and applications to population genetics, Math. Z., 197 (1988), 259-272. doi: 10.1007/BF01215194. Google Scholar [12] R. Carles and I. Gallagher, Universal dynamics for the defocusing logarithmic schrödinger equation, Duke Math. J., 167 (2018), 1761-1801. doi: 10.1215/00127094-2018-0006. Google Scholar [13] W. H. Fleming, Equilibrium distributions of continuous polygenic traits, SIAM J. Appl. Math., 36 (1979), 148-168. doi: 10.1137/0136014. Google Scholar [14] M.-E. Gil, F. Hamel, G. Martin and L. Roques, Mathematical properties of a class of integro-differential models from population genetics, SIAM J. Appl. Math., 77 (2017), 1536-1561. doi: 10.1137/16M1108224. Google Scholar [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.Google Scholar [16] K. Hadeler, Stable polymorphisms in a selection model with mutation, SIAM J. Appl. Math., 41 (1981), 1-7. doi: 10.1137/0141001. Google Scholar [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. Google Scholar [18] M. Kimura, A stochastic model concerning the maintenance of genetic variability in quantitative characters, Proc. Natl. Acad. Sci., USA, 54 (1965), 731–736.Google Scholar [19] M. Nowak, N. Komarova and P. Niyogi, Evolution of universal grammar, Science, 291 (2001), 114-118. doi: 10.1126/science.291.5501.114. Google Scholar [20] K. Page and M. Nowak, Unifying evolutionary dynamics, J. Theoret. Biol., 219 (2002), 93-98. doi: 10.1016/S0022-5193(02)93112-7. Google Scholar [21] P. Schuster and K. Sigmund, Replicator dynamics, J. Theoret. Biol., 100 (1983), 533-538. doi: 10.1016/0022-5193(83)90445-9. Google Scholar [22] P. Stadler and P. Schuster, Mutation in autocatalytic reaction networks, J. Math. Biol., 30 (1992), 597-631. doi: 10.1007/BF00948894. Google Scholar [23] P. Taylor and L. Jonker, Evolutionary stable strategies and game dynamics, Math. Biosci., 40 (1978), 145-156. doi: 10.1016/0025-5564(78)90077-9. Google Scholar [24] A. Zadorin and Y. Rondelez, Natural selection in compartmentalized environment with reshuffling, arXiv preprint, arXiv: 1707.07461, 2017.Google Scholar

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##### References:
 [1] M. Alfaro and R. Carles, Explicit solutions for replicator-mutator equations: Extinction versus acceleration, SIAM J. Appl. Math., 74 (2014), 1919-1934. doi: 10.1137/140979411. Google Scholar [2] M. Alfaro and R. Carles, Replicator-mutator equations with quadratic fitness, Proc. Amer. Math. Soc., 145 (2017), 5315-5327. doi: 10.1090/proc/13669. Google Scholar [3] M. Alfaro and R. Carles, Superexponential growth or decay in the heat equation with a logarithmic nonlinearity, Dyn. Partial Differ. Equ., 14 (2017), 343-358. doi: 10.4310/DPDE.2017.v14.n4.a2. Google Scholar [4] M. Alfaro and M. Veruete, Evolutionary Branching Via Replicator–Mutator Equations, J. Dynam. Differential Equations, 2018.Google Scholar [5] F. H. Arnold, Design by Directed Evolution, Acc. Chem. Res, 1998.Google Scholar [6] I. Białynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Physics, 100 (1976), 62-93. doi: 10.1016/0003-4916(76)90057-9. Google Scholar [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 (2014), 1225-1248. doi: 10.1007/s00285-013-0669-3. Google Scholar [8] I. Bomze and R. Burger, Stability by mutation in evolutionary games, Games Econom. Behav., 11 (1995), 146-172. doi: 10.1006/game.1995.1047. Google Scholar [9] R. Bürger, On the maintenance of genetic variation: Global analysis of Kimura's continuum-of-alleles model, J. Math. Biol., 24 (1986), 341-351. doi: 10.1007/BF00275642. Google Scholar [10] R. Bürger, Mutation-selection balance and continuum-of-alleles models, Math. Biosci., 91 (1988), 67-83. doi: 10.1016/0025-5564(88)90024-7. Google Scholar [11] R. Bürger, Perturbations of positive semigroups and applications to population genetics, Math. Z., 197 (1988), 259-272. doi: 10.1007/BF01215194. Google Scholar [12] R. Carles and I. Gallagher, Universal dynamics for the defocusing logarithmic schrödinger equation, Duke Math. J., 167 (2018), 1761-1801. doi: 10.1215/00127094-2018-0006. Google Scholar [13] W. H. Fleming, Equilibrium distributions of continuous polygenic traits, SIAM J. Appl. Math., 36 (1979), 148-168. doi: 10.1137/0136014. Google Scholar [14] M.-E. Gil, F. Hamel, G. Martin and L. Roques, Mathematical properties of a class of integro-differential models from population genetics, SIAM J. Appl. Math., 77 (2017), 1536-1561. doi: 10.1137/16M1108224. Google Scholar [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.Google Scholar [16] K. Hadeler, Stable polymorphisms in a selection model with mutation, SIAM J. Appl. Math., 41 (1981), 1-7. doi: 10.1137/0141001. Google Scholar [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. Google Scholar [18] M. Kimura, A stochastic model concerning the maintenance of genetic variability in quantitative characters, Proc. Natl. Acad. Sci., USA, 54 (1965), 731–736.Google Scholar [19] M. Nowak, N. Komarova and P. Niyogi, Evolution of universal grammar, Science, 291 (2001), 114-118. doi: 10.1126/science.291.5501.114. Google Scholar [20] K. Page and M. Nowak, Unifying evolutionary dynamics, J. Theoret. Biol., 219 (2002), 93-98. doi: 10.1016/S0022-5193(02)93112-7. Google Scholar [21] P. Schuster and K. Sigmund, Replicator dynamics, J. Theoret. Biol., 100 (1983), 533-538. doi: 10.1016/0022-5193(83)90445-9. Google Scholar [22] P. Stadler and P. Schuster, Mutation in autocatalytic reaction networks, J. Math. Biol., 30 (1992), 597-631. doi: 10.1007/BF00948894. Google Scholar [23] P. Taylor and L. Jonker, Evolutionary stable strategies and game dynamics, Math. Biosci., 40 (1978), 145-156. doi: 10.1016/0025-5564(78)90077-9. Google Scholar [24] A. Zadorin and Y. Rondelez, Natural selection in compartmentalized environment with reshuffling, arXiv preprint, arXiv: 1707.07461, 2017.Google Scholar
Evolution of Gaussian solutions for ${\sigma ^2} = 1$, $a_0 = 1$ and (from left to right) $m_0 = -4$, $m_0 = 0$ and $m_0 = 4$
(A): The first four approximations, for $0\leq t\leq 3$, of the nonlocal term $\overline{u}(t)$ computed via the fixed point iteration (28). (B): Numerical solution obtained by the method described in Section 5, starting from $u_0 = \mathbb{1}_{[1/2,3/2]}$, with ${\sigma ^2} = 1$. The red points are the points on the graph $u(t,\cdot)$ with abscissa $x = \overline u(t)$. The green points are the maxima of $u(t,\cdot)$. This reveals the dissymmetry of the solution
Vector field defined by the differential system (31) with ${\sigma ^2} = 1$, describing the dynamics of Gaussian solutions. In yellow, the set of initial conditions for which $a$ blows up in finite time $T^{\star}$ and in red, dark blue and light blue those for which both $a$ and $m$ are globally defined. The red dashed curve is the set of values defined by $m_0 = -1/{(a_0\sqrt{2{\sigma ^2}})}$, for which $a$ tends to infinity and $m$ tends to zero as time goes to infinity. The dark blue region corresponds to the values leading to an anti-diffusion/diffusion behaviour. The light blue region corresponds to the values leading to a pure diffusion behaviour
Case $(i)$ concentration in finite time. The values of the parameters are $a_0 = 5/64$, $m_0 = -585/64$, ${\sigma ^2} = 1$. It follows that $T^\star\approx 1.878$ and $m(T^\star)\approx -1.277<0$
Case $(ii)$ concentration in infinite time: the solution converges to a Dirac mass at zero. The values of the parameters are $a_0 = 3/16$, $m_0 = -8\sqrt{2}/3$, ${\sigma ^2} = 1$
Case $(iii)$ with $m_0<0$, anti-diffusion/diffusion behaviour. The values of the parameters are $a_0 = 2/10$, $m_0 = -34/10$, ${\sigma ^2} = 1$
Case $(iii)$ with $m_0\geq 0$, the solution is flattening and accelerating. The values of the parameters are $a_0 = 3/2$, $m_0 = 7/2$, ${\sigma ^2} = 1$
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