Self-organized pattern synchronization modulated by stochasticity in coupled plankton ecosystems

Abstract

Spatial patterning and synchronization are pervasive features of plankton communities, yet the mechanisms that allow such patterns to persist coherently under environmental noise remain unresolved. In vertically structured aquatic ecosystems, plankton populations are often organized into distinct layers, raising the question of how interactions between layers shape both spatial self-organization and robustness. Here, we develop a spatiotemporal ecosystem model of a two-layer plankton community to examine the role of passive diffusive coupling under stochastic environmental fluctuations. We show that interlayer diffusion induces a sharp transition from independent, layer-specific Turing patterns to fully synchronized spatial patterns once the coupling strength exceeds a critical threshold. Importantly, the same coupling mechanism markedly enhances the stability of spatial patterns against environmental noise, extending their persistence far beyond that of non-coupled layers. Moreover, we uncover a trophic hierarchy in noise sensitivity, with zooplankton exhibiting substantially greater vulnerability than phytoplankton. Together, these results identify passive diffusive coupling as a unifying mechanism that simultaneously promotes spatial synchronization and robustness, providing a mechanistic explanation for the persistence of coherent plankton patterns in fluctuating aquatic environments.

Figures (12)

• Figure 1: Theoretical dispersion relation: maximum real part of eigenvalues max(Re($\lambda_k$)) as a function of horizontal wavenumber $k$ for representative vertical exchange rates $h$. Curves correspond to parameter sets in Table \ref{['tab1']} of the Supplementary Materials (SM).
• Figure 2: Self-organized pattern dynamics in the non-coupled aquatic system (without vertical migration, $h = 0$). (a-d) Spatial patterns in the surface (layer 1) and bottom (layer 2) layers exhibit distinct structural forms. Under different parameter sets but identical initial conditions, the system exhibits a variety of coexisting spatial configurations, each with distinct pattern types in the two layers: (a) stripes in layer 1 and spots in layer 2, (b) spots in both layers, (c) mixed patterns in both layers, and (d) stripes in both layers. The dispersion relation corresponding to the case in (a) is shown in Fig. \ref{['sesanquxian1-4']}; panels (b-d) exhibit analogous dispersion behaviour (not shown). The spatial pattern structures of non-toxic phytoplankton ($N_i$) and zooplankton ($Z_i$) closely resemble those of toxic phytoplankton ($T_i$) (see Fig. \ref{['3DpatternNonSychSM']}). The parameter values are summarized in Table \ref{['tab1']} of the SM.
• Figure 3: Self-organized synchronization pattern dynamics in the coupled aquatic system with vertical migration. (a-d) With the introduction of coupling, the initially independent spatial patterns from Fig. \ref{['3DpatternNonSych']} (e.g., spots, stripes, mixtures) evolve into synchronized states across layers. The merging of the two dispersion-relation peaks at $h = 0$ into a single peak at $h = 0.02$ (see Fig. \ref{['sesanquxian1-4']}) corresponds to this spatiotemporal synchronization shown in Fig. \ref{['3DpatternSych']}a, and Fig. \ref{['3DpatternSych']}b-d exhibit analogous synchronous behavior. The spatial pattern structures of non-toxic phytoplankton ($N_i$) and zooplankton ($Z_i$) closely resemble those of toxic phytoplankton ($T_i$) (see Fig. \ref{['3DpatternSychSM']}). The parameter values are summarized in Table \ref{['tab1']} of the SM.
• Figure 4: Synchronization error analysis for the coupled aquatic system with vertical migration corresponding to Figs. \ref{['3DpatternNonSych']}a, \ref{['3DpatternSych']}a. (a) Temporal evolution of the synchronization error for non-coupled ($h = 0$) and coupled ($h = 0.02$) conditions. The error increases and plateaus without coupling, whereas it decreases and plateaus with coupling. The steady-state synchronization error in the non-coupled system is 3 to 5 times larger than that in the coupled system. (b) Steady-state synchronization error as a function of coupling strength $h$. The error decreases monotonically with increasing $h$ and eventually stabilizes. The parameter values are summarized in Table \ref{['tab1']} of the SM.
• Figure 5: Visual comparison of self-organized patterns and their synchronization errors for non-coupled (Fig. \ref{['3DpatternNonSych']}a) and coupled (Fig. \ref{['3DpatternSych']}a) systems. (a-c) For the non-coupled system ($h = 0$): (a-b) self-organized patterns, (c) pattern of the synchronization error, which exhibits a structure distinctly different from (a-b). (d-f) For the coupled system ($h = 0.02$): (d-e) synchronized patterns, (f) pattern of the synchronization error, which maintains structural similarity to (d-e). The spatial pattern structures of non-toxic phytoplankton ($N_i$) and zooplankton ($Z_i$) closely resemble those of toxic phytoplankton ($T_i$) (see Fig. \ref{['sync_error_FigSM']}). The parameter values are summarized in Table \ref{['tab1']} of the SM.