A reaction–diffusion model for phenotypic evolution

Abstract

We present a reaction–diffusion mathematical model for the evolutionary dynamics of phenotypic evolution. A detailed deduction of the equations is presented for the one-dimensional version, from which a more general model is proposed. Particular cases are studied using analytical approximations and numerical simulations. Results indicate that the approach proposed produces results that are coherent with mainstream models in evolutionary dynamics, suggesting that the reaction–diffusion model could be an alternative tool in the analysis of evolutionary dynamics.

Appendices

Appendix 1: An example

In this appendix, we present the analysis of a necessary condition for the establishment of the fittest individuals in the evolutionary dynamics given by Eq. (2.8). The conditions are obtained using approximations and have the objective of shedding light on general dynamics represented by the model.

The fitness function defined for this particular case is given by:

$$\begin{aligned} f(x)=f_0\mathrm{e}^{-(\varepsilon x)^2} \end{aligned}$$

(7.1)

where $\varepsilon =1{/}\delta$. Since we have a formula for f we can obtain an explicit expression for $f_2$: $f_2=f''(0)=-2f_0\varepsilon ^2$. To use the necessary condition expressed in inequality (3.6), we need an approximation of $N^*$ as a function of $\varepsilon$. For the rest of this section, we will work with approximations of both $N^*$ and $u^*(x)$, the stationary values for the total population and the stationary population distribution. Not to saturate the notation, we will denote N as the approximation for $N^*$ and u(x) the approximation for $u^*(x)$.

We begin by assuming that $u^*(x)$ can be approximated by $u(x)=u_0 \mathrm{e}^{-(\theta x)^2}$. To obtain relations between the variables we use the equation for the stationary solution:

$$\begin{aligned} \displaystyle \frac{\mathrm{d}^2\left[ f(x)u(x)\right] }{\mathrm{d}x^2} +u(x)(f(x)-N^*)=0. \end{aligned}$$

(7.2)

Since $N^*=\int _\Omega u(x) \mathrm{d}x=u_0\sqrt{\pi }{/}\theta$, we can write the above equation as:

$$\begin{aligned} \begin{array}{l} \left( u_0f_0{/}\mathrm{e}^{x^2\varepsilon ^2} - \sqrt{\pi }u_0{/}\theta \right) {/}\mathrm{e}^{x^2\theta ^2} \\ \quad +\,f_0u_0\mathrm{e}^{-x^2(\varepsilon ^2+\theta ^2)}\left( -2\varepsilon ^2 - 2\theta ^2+ (-2x\varepsilon ^2 - 2x\theta ^2)^2\right) =0. \end{array} \end{aligned}$$

(7.3)

Developing the Eq. (7.3) in a power series and grouping the terms of first and second order we obtain:

$$\begin{aligned}&\displaystyle f_0u_0 - 2f_0u_0\varepsilon ^2 - \sqrt{\pi }u_0^2{/}\theta - 2f_0u_0\theta ^2 =0,\end{aligned}$$

(7.4)

$$\begin{aligned}&\displaystyle f_0u_0\left( 6\varepsilon ^4-\varepsilon ^2 - \theta ^2 + 12\varepsilon ^2 \theta ^2+ 6\theta ^4\right) +\sqrt{\pi }u_0^2\theta =0. \end{aligned}$$

(7.5)

From Eq. (7.4) we can write $u_0$ as a function of $\varepsilon$ and $\theta$:

$$\begin{aligned} u_0=\frac{f_0\theta (1 - 2\varepsilon ^2 - 2\theta ^2)}{\sqrt{\pi }}. \end{aligned}$$

(7.6)

Using the Eq. (7.5) of the terms of second order we can relate $\theta$ and $\varepsilon$:

$$\begin{aligned} \theta =\frac{\sqrt{-5\varepsilon ^2 + \sqrt{4\varepsilon ^2 + \varepsilon ^4}}}{2}. \end{aligned}$$

(7.7)

The relation given by (7.7) is obtained through the resolution of an algebraic equation of third degree for $\theta ^2$, from which two solutions are adequate for the objectives of attaining an approximation of $u^*(x)$. The choice of the right root was made by comparing the $u(x)=u_0 \mathrm{e}^{-(\theta x)^2}$ with numerical approximations of $u^*(x)$.

Finally, from the relation $N=u_0 \sqrt{\pi }{/}\theta$ we achieve an approximation of $N^*$ as a function of $\varepsilon$:

$$\begin{aligned} N^*\cong \frac{f_0\left( 2 + \varepsilon ^2 - \sqrt{\varepsilon ^2(4 + \varepsilon ^2)}\right) }{2}. \end{aligned}$$

(7.8)

In Fig. 5, we present a comparison of approximated values given by Eq. (7.8) and those obtained through numerical simulations. In Fig. 6a, b, we compare $u(x)=u_0\mathrm{e}^{-(\theta x)^2}$ with numerical approximations of $u^*(x)$.

Fig. 5

Comparison between the approximation $N=f_0\left( 2 + \varepsilon ^2 - \sqrt{\varepsilon ^2(4 + \varepsilon ^2)}\right) {/}2$ and values of $N^*$ obtained by numerical simulation of Eq. (2.6)

Fig. 6

Comparison between numerical approximation of $u^*(x)$ and approximation using the formula $u(x)=u_0\mathrm{e}^{-(\theta x)^2}$. a $\varepsilon =0.1$, b $\varepsilon =0.05$

Appendix 2: Multiple scale fitness functions

In Sect. 3, we have shown that when the scale of variation of the fitness function ($\Delta$) is significantly longer than the scale of phenotypic change in the timescale of reproduction ($\sqrt{V}$), the fittest individuals are the most frequent in the population. In the particular case analysed, we used a function with just one local maximum and only one scale of variation.

In this Appendix, we present results of simulation in which the fitness function was constructed is such way as to have multiple scales of variation. Our results indicate that, in such cases, the population does not adapt to the shorter scales, while adapting to the longer ones. To simulate the interaction of those scales, we chose the function:

$$\begin{aligned} f(x)=(1-\gamma )\mathrm{e}^{-(x{/}\delta _1)^2}+\gamma \left( 1+\cos (x{/}\delta _2)\right) \end{aligned}$$

(8.1)

where $\gamma \in [0,1]$ is a parameter that controls the contribution of each scale in the fitness function, $\delta _1\gg 2$ is the “long” scale and $\delta _2\ll 2$ is the “short” one.

Fig. 7

Graphs of the stationary solution $u^*(x)$ for $f(x)=(1-\gamma )\mathrm{e}^{-(x{/}\delta _1)^2}+\gamma \left( 1+\cos (x{/}\delta _2)\right)$. In the results presented, $\delta _1=8$ is the ‘long” scale of variation and $\delta _2=1{/}2$ is the “short” scale. The non-dimensional parameter $\gamma$ controls the contribution of each scale for the fitness function. In the graphs above, f was rescaled for a better visualization

The graphs in Fig. 7 indicate that the population shows an adaptation to the long scale, failing to adapt to the shorter one. Note that even when parameter $\gamma$ approaches the value of 1 the model presents an evolutionary dynamics able to create adaptation even at small fitness advantages. Results of our simulations indicate that the evolutionary dynamics described by the simple model is capable of detecting variations in the “long” scale even if they contribute comparatively little to the fitness value.