Decoding the spatial spread of cyanobacterial blooms in an epilimnion

Abstract

Cyanobacterial blooms (CBs) pose significant global challenges due to their harmful toxins and socio-economic impacts, with nutrient availability playing a key role in their growth, as described by ecological stoichiometry (ES). However, real-world ecosystems exhibit spatial heterogeneity, limiting the applicability of simpler, spatially uniform models. To address this, we develop a spatially explicit partial differential equation model based on ES to study cyanobacteria in the epilimnion of freshwater systems. We establish the well-posedness of the model and perform a stability analysis, showing that it admits two linearly stable steady states, leading to either extinction or saturation. We use the finite elements method to numerically solve our system on a real lake domain derived from Geographic Information System (GIS) data and realistic wind conditions extrapolated from ERA5-Land. Our numerical results highlight the importance of lake shape and size in CB monitoring, while global sensitivity analysis using Sobol Indices identifies light attenuation and intensity as primary drivers of bloom variation, with water movement influencing early bloom stages and nutrient input becoming critical over time. This model supports continuous water-quality monitoring, informing agricultural, recreational, economic, and public health strategies for mitigating CBs.

Figures (12)

• Figure 1: Time series simulations in one dimension of cyanobacteria, cell quota, and external and internal phosphorus over a 1000-day period. The top row represents simulations when $P_h = 0$, and the bottom when $P_h = 2$. We interpolated a function to a real wind vector from Pigeon Lake, Alberta, in 2023. Once cyanobacteria consume all initial phosphorus, the respective equilibria are stable for all time after.
• Figure 2: Real and imaginary part of eigenvalues for different modes for the extinction equilibrium $E_0$, respectively. Here we set $r = 0.7$ and $P_h = 0$.
• Figure 3: Real and imaginary part of eigenvalues for different modes for the extinction equilibrium $E_0$, respectively. $r=0.7$ and $P_h=0.2$.
• Figure 6: Time series simulations in one dimension of cyanobacteria, cell quota, external and internal phosphorus over short (50 days) and long (365 days) time periods. We interpolated a function to a real wind vector from Pigeon Lake, Alberta, in 2023.
• Figure 7: Time series simulations in two dimensions of cyanobacteria, cell quota, external and internal phosphorus over some day time period when $P_h = 2$. We interpolated a function to a real wind vector from Pigeon Lake, Alberta, in 2023. We used the finite elements method and satellite image data to simulate the system on a realistic lake domain.

Theorems & Definitions (13)

• Lemma 3.1
• proof
• Remark 3.1
• Lemma 3.2: Local existence
• proof
• Lemma 3.3
• proof
• Remark 3.2
• Lemma 3.4
• proof