An NPZ Model with State-Dependent Delay due to Size-Structure in Juvenile Zooplankton

TL;DR

This paper develops a novel NPZ model incorporating size-dependent delays in juvenile zooplankton, transforming complex PDE-ODE systems into delay differential equations for analysis of stability, extinction, and biomass dynamics.

Contribution

It introduces a new size-structured juvenile zooplankton component into NPZ models and transforms the system into delay differential equations for detailed analysis.

Findings

Existence and uniqueness of solutions established.

Conditions for stability and extinction identified.

Numerical simulations confirm analytical results.

Abstract

The study of planktonic ecosystems is important as they make up the bottom trophic levels of aquatic food webs. We study a closed Nutrient-Phytoplankton-Zooplankton (NPZ) model that includes size structure in the juvenile zooplankton. The closed nature of the system allows the formulation of a conservation law of biomass that governs the system. The model consists of a system of nonlinear ordinary differential equation coupled to a partial differential equation. We are able to transform this system into a system of delay differential equations where the delay is of threshold type and is state-dependent. The system of delay differential equations can be further transformed into one with fixed delay. Using the different forms of the model we perform a qualitative analysis of the solutions, which includes studying existence and uniqueness, positivity and boundedness, local and global stability, and conditions for extinction. Key parameters that are explored are the total biomass in the system and the maturity level at which the juvenile zooplankton reach maturity. Numerical simulations are also performed to verify our analytical results.