Structured Model Conserving Biomass for the Size-spectrum Evolution in Aquatic Ecosystems

TL;DR

This paper develops a mathematical model for size-spectrum dynamics in aquatic ecosystems, focusing on predation, biomass conservation, and stability analysis, supported by analytical and numerical results.

Contribution

It introduces a novel non-local quadratic model incorporating predator-prey size ratios and analyzes solution existence, stability, and steady states in aquatic ecosystems.

Findings

Existence of solutions depends on feeding preference and search parameters.

Ecosystems can reach trivial or non-trivial steady states, showing cascade effects.

Numerical simulations confirm analytical stability and convergence results.

Abstract

Mathematical modelling of the evolution of the size-spectrum dynamics in aquatic ecosystems was discovered to be a powerful tool to have a deeper insight into impacts of human- and environmental driven changes on the marine ecosystem. In this article we propose to investigate such dynamics by formulating and investigating a suitable model. The underlying process for these dynamics is given by predation events, causing both growth and death of individuals, while keeping the total biomass within the ecosystem constant. The main governing equation investigated is deterministic and non-local of quadratic type, coming from binary interactions. Predation is assumed to strongly depend on the ratio between a predator and its prey, which is distributed around a preferred feeding preference value. Existence of solutions is shown in dependence of the choice of the feeding preference function as well as the choice of the search exponent, a constant influencing the average volume in water an individual has to search until it finds prey. The equation admits a trivial steady state representing a died out ecosystem, as well as - depending on the parameterregime - steady states with gaps in the size spectrum, giving evidence to the well known cascade effect. The question of stability of these equilibria is considered, showing convergence to the trivial steady state in a certain range of parameters. These analytical observations are underlined by numerical simulations, with additionally exhibiting convergence to the non-trivial equilibrium for specific ranges of parameters.