Bifurcation control of a minimal model of marine plankton interaction with multiple delays

¹ School of Liberal Arts and Sciences, North China Institute of Aerospace Engineering, Langfang 065000, P.R. China.
² Aerospace Science and Technology, North China Institute of Aerospace Engineering, Langfang 065000, P.R. China.

^* Corresponding author: jzhsuper@163.com

Received: 10 October 2020
Accepted: 13 February 2021

Abstract

Plankton blooms and its control is an intriguing problem in ecology. To investigate the oscillatory nature of blooms, a two-dimensional model for plankton species is considered where one species is toxic phytoplankton and other is zooplankton. The delays required for the maturation time of zooplankton, the time for phytoplankton digestion and the time for phytoplankton cells to mature and release toxic substances are incorporated and the delayed model is analyzed for stability and bifurcation phenomena. It proves that periodic plankton blooms can occur when the delay (the sum of the above three delays) changes and crosses some threshold values. The phenomena described by this mechanism can be controlled through the toxin release rates of phytoplankton. Then, a delay feedback controller with the coefficient depending on delay is introduced to system. It concludes that the onset of the bifurcation can be delayed as negative feedback gain (the decay rate) decreases (increases). Some numerical examples are given to verify the effectiveness of the delay feedback control method and the existence of crossing curve. These results show that the oscillatory nature of blooms can be controlled by human behaviors.