variance evolution analysis

This study addresses the lack of theoretical foundation for parameter selection in the bat algorithm, which has traditionally relied on empirical tuning. For the first time, it integrates dynamical systems theory with population variance evolution analysis to construct a theoretical framework characterizing the influence of key parameters. Within this framework, analytically derived effective ranges for critical parameters are established. Numerical experiments confirm that the theoretical predictions align closely with the observed convergence behavior in practice. The work further uncovers the intrinsic mechanisms governing the trade-off between exploration and exploitation and the algorithm’s convergence properties, thereby providing the first systematic theoretical guidance for parameter configuration in the bat algorithm.

This study investigates the stationary distribution and stability of stochastic differential equation systems with multimodal uncertain parameters exhibiting superposition effects, using the nonlinear Rosenzweig–MacArthur predator–prey model as a case study. For the first time, multimodal mixture-distributed parameters are incorporated into the stationary analysis of stochastic dynamical systems. System stability is quantified through the eigenvalue distribution of the Jacobian matrix, and posterior stationary density estimates are obtained via the Monte Carlo method proposed by Hoegele (2026). The results reveal that under multimodal parameter uncertainty, the system exhibits a multimodal stationary distribution, accurately delineating regions of stability. This demonstrates the effectiveness and novelty of the proposed framework for uncertainty quantification in complex ecological dynamics.

Classical iterative integer-mapping models are valid only under the infinite-population assumption and fail to capture critical individual-level stochasticity—such as demographic noise and extinction risk—in finite populations. Existing noise-perturbation approaches model only environmental stochasticity, neglecting intrinsic noise arising from discrete individual dynamics. Method: We propose a binomial-mapping-based stochastic evolutionary framework that explicitly links deterministic maps—including Logistic and Ricker maps—to individual-based stochastic processes. Contribution/Results: This framework rigorously establishes their non-equivalence and identifies sufficient conditions for equivalence. By embedding discrete-individual dynamics within deterministic map structures, it enables exact modeling of demographic noise, quasi-stationary distributions, and extinction pathways. It provides a unified, analytically tractable paradigm for studying noise-induced phase transitions and extinction risk in population dynamics.

This work addresses the lack of a unified computational framework for analyzing non-ergodicity, modeling heavy-tailed dynamics, and studying decision-making under uncertainty in stochastic processes. To this end, we introduce an open-source Python library that, for the first time, integrates non-ergodicity diagnostics, simulation of heavy-tailed processes—such as multiplicative Lévy growth and memory-dependent mean-reverting dynamics—and agent-based experimentation within a single platform. Built upon the scientific Python ecosystem (NumPy/SciPy), the library supports end-to-end workflows including stochastic process definition, simulation, parameter inference, and partial solution of stochastic differential equations. Through several reproducible examples—ranging from heavy-tailed ensemble diffusion to pre-asymptotic fluctuation analysis—it substantially reduces boilerplate code and enhances both reproducibility and development efficiency in the study of time-averaged behaviors of complex stochastic systems.

Existing mathematical modeling lacks a rigorous, unambiguous ontological foundation, hindering a unified characterization of the mapping between models and real-world phenomena. This paper introduces, for the first time, an axiomatic definition of mathematical models grounded in Hilbert-space operator theory: a model is formalized as a computable operator acting on random variables, systematically unifying theoretical derivation, experimental implementation, and statistical identification. We further establish a geometric correspondence between the model manifold and the prediction surface, exposing intrinsic structural properties and the fundamental nature of model computability. This framework fills a critical gap in the formal ontology of modeling, providing a unified mathematical foundation for interdisciplinary model construction. It significantly enhances the logical rigor of theoretical inference and the reliability of empirical validation.

This work proposes an information-geometric approach based on kernel density estimation (KDE) to approximate output distributions from simulation data for stochastic or agent-based models lacking analytical distributional forms. By computing the Hessian of the symmetric Kullback–Leibler divergence—equivalent to the Fisher information matrix (FIM)—the method identifies sloppy and stiff directions in parameter space. It is the first to reliably recover the characteristic eigenstructure of the FIM in stochastic models without closed-form solutions. Applied to the Kirman ant recruitment model, the KDE-derived FIM eigenvalues and eigenvectors converge to their analytical counterparts as sample size increases, and the identified stiff directions effectively guide efficient parameter exploration across transitions between unimodal and bimodal phases.

This work addresses the challenge faced by beginning graduate students who lack prior exposure to stochastic differential equations and diffusion models by proposing a hierarchical pedagogical framework that systematically constructs the mathematical foundations of diffusion models. Starting from a sampling perspective, it integrates core definitions, key estimates under simplified assumptions, and proof strategies for cutting-edge theorems, thereby bridging classical sampling dynamics with modern diffusion samplers. The material synthesizes probability theory, stochastic differential equations, stochastic numerical methods, and diffusion process theory into a self-contained, proof-oriented curriculum. This approach maintains mathematical rigor while significantly enhancing accessibility, enabling students without prerequisite knowledge to grasp the sampling mechanisms, error analysis, and inference control principles underlying diffusion models.

This work addresses the challenge of reproducing classical Lotka-Volterra oscillations in large-scale agent-based models, which often fail due to sensitivity to local rules and parameters, leading to population collapse or unnatural saturation. By carefully optimizing environmental and population-level parameters, the authors successfully elicit sustained, bounded, and phase-lagged Lotka-Volterra–like oscillations in a predator–prey system composed of agents endowed with local perception, internal energy dynamics, and recurrent neural network (RNN) controllers. A novel feature-based loss function is introduced to guide agents toward theoretically grounded ecological dynamics, and robustness is verified through both stochastic and evolutionary strategies. Leveraging the JAX-based ABMax framework for efficient batched simulation, this study achieves the first stable reproduction of such ecological dynamics in a complex multi-agent system.

This work addresses the lack of a unified convergence analysis framework for population-based optimization algorithms, which hinders systematic comparison and generalization. The authors propose an operator calculus framework that models diverse algorithms as compositions of three fundamental operators—mutation, selection, and recombination—acting on probability measures. By leveraging mean-field limits, they derive a continuous-time transport-reaction-jump partial differential equation governing the algorithmic dynamics. Building upon operator semigroup theory and functional analysis on spaces of probability measures, they develop a modular Lyapunov method that enables dissipativity verification operator by operator. Under explicit stability and regularity conditions, they establish exponential decay of both a state-space Lyapunov functional and the search error, thereby providing a unified guarantee of exponential convergence for a broad class of distributed optimization algorithms.

This study addresses the challenge of parameter identification in Lotka–Volterra predator–prey models under sparse and noisy observations, where conventional optimization methods often fail to converge due to irregular likelihood landscapes and numerical instability or stiffness arising from ordinary differential equation (ODE) solvers across the parameter space. To overcome these issues, the authors propose a computational framework integrating natural gradient ascent. They first employ nondimensionalization to eliminate redundant scale parameters, thereby reducing the dimensionality of the optimization problem. Additionally, they devise a component-separated adaptive ODE integration strategy that dynamically switches between two second-order equations to mitigate numerical instabilities. Experimental results demonstrate that the proposed method achieves stable convergence in fewer iterations compared to standard gradient ascent and BFGS, significantly enhancing both the reliability and efficiency of parameter estimation from sparse data.