Information Geometry of Evolutionary Dynamics: Fisher–Shahshahani Geometry, Evolutionary Currents, and Response

We develop a mathematical framework for evolutionary dynamics in which population states are represented as probability distributions and evolutionary trajectories are analyzed using the Fisher information metric. The formulation separates the statistical representation of a population from the biological dynamical law governing its evolution, allowing selection, mutation, and other evolutionary processes to be treated within a common geometric setting. For replicator dynamics, we show that the Fisher-geometric speed of evolutionary change is equal to the variance of fitness, \(v_F^2=\operatorname{Var}_p(f), \) providing a direct relation between the rate of change of a population distribution and fitness heterogeneity. We then extend the analysis to mutation--selection dynamics, where the Fisher speed contains distinct contributions from selection, mutation, and their geometric cross term. For evolutionary systems that admit a potential formulation, we express the dynamics as a Fisher-natural-gradient flow and obtain the conditional relation \( -\frac{d\Phi_E}{dt}=v_F^2. \) In Evolutionary Dynamics, systems frequently exhibit non-gradient behavior driven by frequency-dependent selection, mutation, or shifting environments, where the population trajectory cannot be represented as the steepest descent of a scalar fitness landscape. This phenomenon is conceptually analogous to non-gradient currents in physiological systems—such as the active transport of ions across a cell membrane against a passive electrochemical gradient. Just as those biological currents require metabolic energy to oppose natural gradients, non-gradient evolutionary mechanisms introduce persistent biological currents, denoted \(J^{A}\), that drive the population away from passive equilibrium configurations. To mathematically evaluate these complex regimes, we develop a comprehensive local stability and response analysis. For purely gradient systems, local relaxation modes are characterized via the Fisher-relative Hessian operator; however, for general non-gradient dynamics, stability must be inferred from the full Jacobian matrix of the complete evolutionary vector field. The resulting generalized eigenvalues characterize distinct local evolutionary modes and their respective relaxation scales. Concurrently, regimes exhibiting small generalized curvatures identify zones of Fisher-geometric softening, which are uniquely associated with slow relaxation times and enhanced susceptibility to external perturbations.