Kunihiko Hamagami | Searcharxiv

arXiv2026

Tractable infinite-dimensional model for long-term environmental impact assessment of long-memory deacay processes

Focusing on the assessment of benthic algae blooms that decay subexponentially, we propose a tractable (solvable in a semi-explicit form) and well-defined (that does not diverge) environmental index for the impact assessment of long-memory processes under model uncertainties. Our target system generates long-memory decay through an infinite superposition of multiscale processes. The sensitivity of the environmental index can be controlled by the degree of model uncertainty in terms of the relative entropy and nonexponential discount; hence, we apply a long-memory discount to evaluate long-memory processes. In our framework, the evaluation of the environmental index is reduced to finding a proper solution to an infinite-dimensional extended Hamilton-Jacobi-Bellman system. We solve this system, verify a reasonable solution, and numerically handle it by using a quantization technique. Finally, we present a demonstrative application of the proposed framework to benthic algae population dynamics in river environments based on a laboratory experiment. This paper offers a tractable framework for the assessment of persistent environmental phenomena.

math.OC↗

arXiv2026

Micro-macro population dynamics models of benthic algae with long-memory decay and generic growth

Benthic algae as a primary producer in riverine ecosystems develop biofilms on the riverbed. Their population dynamics involve growth and decay processes, the former owing to the balance between biological proliferation and mortality, while the latter to mechanical abrasion because of the transport of sediment particles. Contrary to the assumptions of previous studies, the decay has experimentally been found to exhibit long-memory behavior, where the population decreases at an algebraic rate. However, the origin and mathematical theory of this phenomenon remain unresolved. The objective of this study is to introduce a novel mathematical model employing spin processes to describe microscopic biofilm dynamics. A spin process is a continuous-time jump process transitioning between states 0 and 1, and the continuum limit of these processes captures the long-memory decay and generates generic growth. The proposed framework leverages heterogeneous spin rates, achieved by appropriately superposing spin processes with distinct rates, to reproduce the long-memory decay. Computational simulations demonstrate the behavior of the model, particularly emphasizing rate-induced tipping phenomena. This mathematical model provides a computationally tractable interpretation of benthic algae dynamics and their long-term prediction, relevant to river-engineering applications.

math.PR↗