A study on the dynamics of a nutrient-plankton-fish interaction model with Caputo fractional operator

Abstract The current study aims to fill a knowledge gap in the dynamics and regulation of algal blooms through fish predation, while accounting for the memory effect. A nutrient-phytoplankton-zooplankton-fish model using fractional derivatives in the Caputo sense is developed. The main conclusions are as follows. Reducing the fractional derivative order from 1.0 to 0.91 improves system stability by suppressing chaotic dynamics. Furthermore, the zooplankton death rate $d_{1}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>d</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> and fish population ingestion rate β were determined as bifurcation parameters. The linear feedback controller is effective at suppressing chaos. Simulations were performed using the Newton polynomial interpolation method. We investigated how the fractional derivative order affects system stability using arbitrarily chosen fractional parameter values. It was discovered that stability deteriorates near the co-axial equilibrium for lower zooplankton death rates. A higher maximal ingestion rate of fish destabilizes the system dynamics. Finally, phytoplankton’s higher nutrient consumption rate reduces stability. Furthermore, higher-order fractional derivatives indicate lower stability, implying that systems with strong memory effects or history dependence are unstable.