Ecosystem models

According to [31] , we can distinguish two main categories of models of Harmful Algal Bloom (HAB), that we will call “minimal dynamic models” [36] and “complex dynamic models”.

“Minimal dynamic models” are “heuristic, examining the likelihood of certain processes generating a HAB” [31] . These models generally take into account between 2 and 5 variables like in the well-known Nutrient-Phytoplankton-Zooplankton (NPZ) models [37] . Several hypotheses on the origin of blooms have been studied with these models. In [38] , the relaxation of the zooplankton grazing pressure compared to the phytoplankton growth rate is identified as a potential key mechanism of bloom formation. Conversely, the control of the initiation and demise of the blooms by nutrients levels (bottom-up control) is numerically investigated in [39] . Most of these models are deterministic differential equations but the stochastic nature of the events and pathways leading to cyanobacterial blooms [40], [41] has more recently advocated for the development of stochastic models [42] . 

“Complex dynamic models” are designed to be used as a virtual reality [36] for the simulation of the whole ecosystem (e.g., CAEDYM, PCLake, DELWAQ). They are therefore generally coupled with a detailed hydrodynamic model and validated on field data. They are used to better understand the cyanobacterial blooms dynamics [43] , to support the lake management strategies by forecasting the blooms occurrence at short term [44] or to anticipate the future behaviour of the ecosystem in response to different local management or global changes scenarios [45] . The main approaches used for lake ecosystem modelling are reviewed in [36], [46], [47] . For short-term (day to week) forecast, genetic models or artificial neural networks [48] can be used. But for the simulation of annual cycles, such kind of models would not be able to infer the shift in the phytoplankton succession whereas deterministic models could [46] . Individual based models have also been considered, but as the number of individuals is limited by the computational cost, it is necessary to develop some derived approaches [36] .

“Complex dynamic models” generally include numerous biogeochemical variables and processes (biological and physical). Consequently, the outcomes of simulated scenarios are hampered with great uncertainty because of the huge set of parameters (sometimes more then 100) to be calibrated. Similarly with weather forecast approach, several “ensemble approaches” have been proposed to reduce this uncertainty. In [49] , [50] an ensemble of model runs obtained with several “equivalent” set of parameters are used to improve the predictive power of the model. In [45] , the same scenario is simulated using an ensemble of models. Another effective way to improve the performances of 3D hydrodynamic-ecological models is to conduct cross-comparisons of similar models on the same study sites. This type of comparative studies has been performed recently for 1D models [51]–[53] . To our knowledge very few attempts have been made for hydrodynamic-ecological models [45] and none for 3D models. To do so, portability of the model software is required. This type of concern is just starting to be addressed by the modeller community (e.g. [54], [55] ). Defining a statistical framework to compare the simulations would therefore be very helpful [56], [57] .

Modelling of specific processes

Despite the complexity of the models, some processes are still poorly represented and the models need therefore to be improved. For example, most of the models are not able to simulate the shifts in the phytoplankton community and the variation of the stoichiometry of living organisms [58] . Recently, it has been shown that the universally used Michaelis-Menten kinetics is not able to fully represent the variable stoichiometry of biomass and of nutrient uptake [59] . Promising size- and trait-based kinetics of nutrient uptake have been proposed but not yet tested against experimental or field data [60] . Some recent process-based approaches, which take into account the major eco-physiological traits of phytoplankton [61], [62] , have also recently been proposed in order to well represent the adaptive shifts of the phytoplankton community. In particular, light utilization efficiency which is a contrasted trait between cyanobacteria and other phytoplankton groups [63] can be used to improve phytoplankton succession modelling.

Regarding the cyanobacteria specific processes, modelling their migration in the water column remains an essential research issue. The key processes driving the sinking or floating of the cells are still poorly simulated [64] . Experimental [65], [66] and numerical studies [67] have also proved that the microbial loop can have a great impact on the cyanobacteria dynamics. Yet, the process of mineralization of the organic matter by bacteria is often represented as a simple chemical reaction with a rate either constant or dependent on the bacteria concentration at best [68], [69] . Physical processes, occurring in lakes at bi- or three-dimensional scales, such as hydrodynamics, sediment resuspension and internal waves, play also a key role (e.g. [70] ). Yet, simulations with 2D or 3D ecological models and their confrontation with consistent field data set are still very rare [71] . Until now, 3D models were principally used to assess the effect of environmental changes on lake hydrodynamics [72] .

Impacts of changes in C/N/P ratio and water temperature on cyanobacteria population dynamics

The increase in the occurrence of cyanobacterial blooms in freshwater ecosystems is due for a large part to an increase in nutrient (P and N) availability, these two elements being limiting for the primary producers of these ecosystems (e.g. [73] ). Rising CO₂ concentrations and water temperature also enhance the growth rate of cyanobacteria and consequently the potential occurrence of blooms in eutrophic ecosystems (e.g. [24], [74] ). Other processes, also influenced by global changes, contribute indirectly to the increase of cyanobacterial blooms. For example, climate warming enhances the release of nutrients from the sediments and the mineralization of the organic matter conducting to an increase of P and N availability [75], [76] .

However, little is known about the consequences of the changes occurring in C/N/P ratio on cyanobacterial blooms while freshwater ecosystems are subject to different trajectories concerning this ratio. Indeed, as mentioned previously, P concentrations are decreasing in Europe but increasing in China and carbonate equilibrium is changing due to the increase in CO₂ concentrations. We therefore expect changes in the ecological stoichiometry of freshwater ecosystems, whose consequences are largely unknown. It has been shown [77] that rising pCO₂ will change elemental stoichiometry of phytoplankton with putative consequences on zooplankton feeding on phytoplankton. In the same way, a recent study conducted on 24 freshwater bacterial strains [78] has revealed that the stoichiometric flexibility of heterotrophic bacteria depends on the species and on their cellular phosphorus quota, suggesting that changes in C/N/P ratio can lead to changes in the composition of the bacterial communities. The same authors [79] also demonstrated that the aquatic bacterial communities display very contrasting values in their P content and stoichiometry. These authors proposed that anthropogenic changes in C/N/P ratio might have consequences on aquatic ecosystem productivity and on the extent and periodicity of nutrient fluctuations.

To the best of our knowledge, no experimental study has been conducted to test the combined impact of changes occurring in P, N and CO₂ concentrations and in water temperature on the mineralization of the organic matter, the nutrient release from the sediment and their consequence on the blooms of cyanobacteria.