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

TL;DR

This paper introduces a novel mathematical model using spin processes to describe long-memory decay and growth in benthic algae populations, capturing complex dynamics observed in river ecosystems.

Contribution

The study develops a new spin process-based framework to model long-memory decay and generic growth in benthic algae, addressing unresolved theoretical questions.

Findings

Model reproduces algebraic decay observed in experiments

Simulations show rate-induced tipping phenomena

Framework offers a tractable approach for long-term prediction

Abstract

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.