# Well-posedness and positivity property for a reaction-diffusion model of plankton communities, involving a rational nonlinearity with singularity

## TL;DR

This paper analyzes a reaction-diffusion model of plankton communities with complex nonlinearities, proving global well-posedness, positivity, and exploring long-term behavior of solutions.

## Contribution

It introduces a novel approach to handle singular rational nonlinearities in a reaction-diffusion system modeling plankton interactions.

## Key findings

- Proved global existence and uniqueness of solutions.
- Established positivity of solutions.
- Analyzed asymptotic behavior over time.

## Abstract

In this work, we consider a reaction-diffusion system, modeling the interaction between nutrients, phytoplanktons and zooplanktons. Using a semigroup approach in $L^2$, we prove global existence, uniqueness and positivity of the solutions. The Holling type 2 nonlinearities, i.e of rational type with singularity, are handled by providing estimates in $L^\infty$. The article finally exhibits some time asymptotics properties of the solutions.