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.

💡 Research Summary

This paper presents a novel mathematical framework for describing the population dynamics of benthic algae (periphyton) on riverbeds, explicitly incorporating the experimentally observed long‑memory (algebraic) decay caused by sediment abrasion together with generic growth mechanisms. The authors begin by highlighting the ecological importance of benthic algae as primary producers and the inadequacy of traditional models that assume simple exponential decay for abrasion. Recent laboratory measurements show that algae coverage declines as a power law (t^{-\alpha}) (with (\alpha>0)), indicating a memory effect that cannot be captured by ordinary differential equations (ODEs) with constant decay rates.

To address this gap, the authors introduce spin processes as the microscopic building blocks. A spin process is a continuous‑time two‑state jump process that switches from state 1 (algae present) to state 0 (algae absent) with a rate (R>0). The riverbed domain (D) is partitioned into (M) sub‑domains of equal area; each sub‑domain (D_i) carries its own spin rate (R_i). The heterogeneity of flow and shear stress over a non‑flat bed is modeled by assuming that the rates (R_i) are random draws from a probability distribution (F(R)). For a single sub‑domain the expected occupancy decays exponentially, (\mathbb{E}