A discrete-time nutrients-phytoplankton-oysters mathematical model of a bay ecosystem*

Abstract

Populations are generally censused daily, weekly, monthly or annually. In this paper, we introduce a discrete-time nutrients-phytoplankton-oysters (NPO) model that describes the interactions of nutrients, phytoplankton and oysters in a bay ecosystem. We compute the threshold parameter ${\mathcal{R}}_N$ for persistence of phytoplankton with or without oysters. When ${\mathcal{R}}_N<1$, then both phytoplankton and oysters populations go extinct. However, when ${\mathcal{R}}_N>1$, we show that the model may exhibit two scenarios: (1) a locally asymptotically stable equilibrium with positive values of nutrients and phytoplankton with oysters missing, and (2) a locally asymptotically stable interior equilibrium with positive values of nutrients, phytoplankton and oysters. We use sensitivity analysis to study the impact of human and environmental factors on the model. We use examples to illustrate that some human activities and environmental factors can force the interior equilibrium to undergo a Neimark–Sacker bifurcation which generates phytoplankton blooms with oscillations in oysters population and nutrients level.

Keywords:

• Discrete-time
• nutrients
• phytoplankton
• oysters

Mathematics Subject Classification:

• 92-10

Acknowledgement

I am very thankful to Dr. Abdul-Aziz Yakubu for helpful discussions during the early stages of the project, particularly in the formulation of the model.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Najat Ziyadi was partially supported by a Simons Foundation Grant.