Stability and Bifurcations in a Nutrient–Phytoplankton–Zooplankton Model with Delayed Nutrient Recycling with Gamma Distribution

Mihaela Sterpu (), Carmen Rocşoreanu, Raluca Efrem and Sue Ann Campbell
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Mihaela Sterpu: Department of Mathematics, University of Craiova, 200585 Craiova, Romania
Carmen Rocşoreanu: Department of Statistics and Economic Informatics, University of Craiova, 200585 Craiova, Romania
Raluca Efrem: Department of Mathematics, University of Craiova, 200585 Craiova, Romania
Sue Ann Campbell: Department of Applied Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada

Mathematics, 2023, vol. 11, issue 13, 1-24

Abstract: Two nutrient–phytoplankton–zooplankton (NZP) models for a closed ecosystem that incorporates a delay in nutrient recycling, obtained using the gamma distribution function with one or two degrees of freedom, are analysed. The models are described by systems of ordinary differential equations of four and five dimensions. The purpose of this study is to investigate how the mean delay of the distribution and the total nutrients affect the stability of the equilibrium solutions. Local stability theory and bifurcation theory are used to determine the long-time dynamics of the models. It is found that both models exhibit comparable qualitative dynamics. There are a maximum of three equilibrium points in each of the two models, and at most one of them is locally asymptotically stable. The change of stability from one equilibrium to another takes place through a transcritical bifurcation. In some hypotheses on the functional response, the nutrient–phytoplankton–zooplankton equilibrium loses stability via a supercritical Hopf bifurcation, causing the apparition of a stable limit cycle. The way in which the results are consistent with prior research and how they extend them is discussed. Finally, various application-related consequences of the results of the theoretical study are deduced.

Keywords: plankton; nutrient recycling; delay; gamma distribution; closed ecosystem; dynamics; bifurcation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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