Resonance and frequency-locking phenomena in spatially extended phytoplankton-zooplankton system with additive noise and periodic forces

TL;DR

This study investigates how external periodic forces and noise influence spatial phytoplankton-zooplankton dynamics, revealing resonance, frequency-locking, and noise-enhanced oscillations that could explain oceanic plankton blooms.

Contribution

It introduces a spatial phytoplankton-zooplankton model incorporating noise and periodic forces, demonstrating their constructive effects on system oscillations and pattern formations.

Findings

Noise enhances phytoplankton oscillations and clustering.

External periodic forces induce frequency-locking phenomena.

Resonant patterns emerge with combined noise and forcing.

Abstract

In this paper, we present a spatial version of phytoplankton-zooplankton model that includes some important factors such as external periodic forces, noise, and diffusion processes. The spatially extended phytoplankton-zooplankton system is from the original study by Scheffer [M Scheffer, Fish and nutrients interplay determines algal biomass: a minimal model, Oikos \textbf{62} (1991) 271-282]. Our results show that the spatially extended system exhibit a resonant patterns and frequency-locking phenomena. The system also shows that the noise and the external periodic forces play a constructive role in the Scheffer's model: first, the noise can enhance the oscillation of phytoplankton species' density and format a large clusters in the space when the noise intensity is within certain interval. Second, the external periodic forces can induce 4:1 and 1:1 frequency-locking and spatially homogeneous oscillation phenomena to appear. Finally, the resonant patterns are observed in the system when the spatial noises and external periodic forces are both turned on. Moreover, we found that the 4:1 frequency-locking transform into 1:1 frequency-locking when the noise intensity increased. In addition to elucidating our results outside the domain of Turing instability, we provide further analysis of Turing linear stability with the help of the numerical calculation by using the Maple software. Significantly, oscillations are enhanced in the system when the noise term presents. These results indicate that the oceanic plankton bloom may partly due to interplay between the stochastic factors and external forces instead of deterministic factors. These results also may help us to understand the effects arising from undeniable subject to random fluctuations in oceanic plankton bloom.