# 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
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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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