Information Theory in a Darwinian Evolution Population Dynamics Model

TL;DR

The paper develops a Fisher information–based framework to estimate evolution-trait parameters in a Darwinian population dynamics model, linking the G-function formalism to estimation uncertainty via $I(\Theta)$ and the Cramér–Rao bound. It formulates discrete dynamics $x_{t+1}=x_tG(x_t,\theta_t,u_t)$ and $\theta_{t+1}=\theta_t+\sigma^2 g(x_t,\theta_t,u_t)$, provides closed-form Fisher information expressions for single and multi-trait cases, and analyzes fixed-point structure and ESS convergence. Key results show that for one trait $I(\theta)=\frac{1}{\omega^2}+\kappa^2\frac{b(\theta)}{c_u(\theta)}$ with a maximum at $\theta^*=w^2\kappa$, and that nontrivial equilibria exist according to a set of $\xi$-conditions, often organizing trait values along ellipses in trait space. The framework enables quantification of estimator variance via the Fisher information, guides efficient estimation of trait parameters, and suggests extensions to stochastic dynamics, Bayesian approaches, and multi-species scenarios.

Abstract

Using information theory, we propose an estimation method for traits parameters in a Darwinian evolution model for species with on trait or multiple traits. We use the Fisher's information to obtain the errors on the estimation for one species with one or multiple traits. We perform simulations to illustrate the method.

Figures (8)

• Figure 1: The blue curve represents the Fisher's Information in equation \ref{['eqn:FishInfo']} with its maximum value represented by the red dashed line, for $w=7; \kappa_1=0.5; u=0.02; \frac{b_0}{c_0}=10^{-4}$.
• Figure 2: In (a), represented is the time series of of $x_t$. It shows a convergence to $x^*\approx 20.789\times 10^5$ (blue dashed line). In (b), represented is the time series of $\theta_t$, showing a convergence to $\theta_{*+}\approx 6.113$ (blue dashed line). Figure (c) represents the time series of the Fisher's information $I(\theta_t)$, showing a convergence to $I(\theta_{*+})\approx 345.43\times 10^5$ (red dashed line). Figure (d) is the plot of $I(\theta_t)$ versus $\theta_t$, showing that once the fixed point $\theta_{*+}$ is reached, the Fisher's information is maximized. This is illustrated by the intersection between the blue and red dashed lines.
• Figure 3: This figure shows $F(x,\theta):=xG(x,\theta)$ in blue, $G(x,\theta)$ in black and $\lambda(x,\theta)$ in red for $\theta=1.5$. The green dots represent the intersection between the vertical $x=\frac{1}{\theta}=\frac{2}{3}$ and these curves. We observe that $G(x,\theta)$ is maximized at the same point $x$ where $\lambda(x,\theta)$ is minimized (green dots) and vice versa (red dots).
• Figure 4: In (a), represented is the time series of of $x_t$ in the special case above. It shows a convergence to $x^*=e^{-1}$ (blue dashed line). In (b), represented is the time series of $\theta_t$, showing a convergence to $\theta_{*+}=e$ (blue dashed line). Figure (c) represents the time series of the Fisher's information $I_n(\theta_t)$, showing a convergence to $I(\theta_{*})\approx 0.125$ (red dashed line) as $t\to \infty$. Figure (d) is the plot of $I(\theta_t)$ versus $\theta_t$, showing that once the fixed point $\theta_{*+}=e$ is reached, the Fisher's information ix maximized. This is illustrated by the intersection between the blue and red dashed lines.
• Figure 5: Values of $T_n(\theta_t)$ (dashed line) for two different starting values of $\theta$, each with their 95% confidence bands (colored shaded areas). In each cases, $T_n(\theta_t)$ converges to $e$ (light dashed line) as $t\to \infty$. When $n$ is large as in (c) and (d), $I_n(\theta)^{-1}$ becomes smaller and so is the width of the confidence interval.

Theorems & Definitions (13)

• Definition 2.1
• Definition 2.2
• Theorem 3.1
• Corollary 3.2
• Proposition 3.3
• Remark 3.4
• Remark 3.5
• Theorem 3.6
• Proposition 3.7
• Corollary 3.8