Massive seasonal fluctuations in the abundance of phytoplankton populations are a common feature of many temperate aquatic ecosystems. While many factors are known to affect algal growth, a well-established body of theory suggests that algal growth is strongly shaped by dynamic interactions with their limiting nutrients (O’Brien 1974; Grover 1989, 1991; DeAngelis 1992; Huisman and Weissing 1995; Huppert et al. 2002, 2005). Here we experimentally test the hypothesis that seasonal algal blooms represent a transient ‘reactive’ response to perturbation of nutrient and algal abundance away from equilibria at the beginning of the growing season (Neubert and Caswell 1997; Neubert et al. 2004; Caswell and Neubert 2005). The defining characteristic of reactive models is that perturbation from stable equilibria can cause short-term amplification of the disturbance, resulting in massive fluctuation in algal abundance before the system settles back into an equilibrium (Neubert and Caswell 1997).
Seasonal perturbation should be routinely expected in temperate freshwater ecosystems due to algal die-off during winter accompanied by seasonal accumulation of nutrients due to fertilizer run-off during precipitation events. The logical consequence of such perturbations is that algal populations at the beginning of the growing season would be substantially depressed at the same time as nutrient abundance is substantially augmented. This combination of perturbations could be particularly potent in generating reactive algal response as algal growth begins anew at the beginning of the next growing season. There could be important practical benefits in better understanding environmental drivers of such reactive nutrient-driven algal blooms, since the frequency and magnitude of harmful algal blooms is thought to have increased in recent years (Michalak et al. 2013, Paerl and Otten 2013). This increasing trend is pronounced in global datasets for large lakes (Ho et al. 2019), although data for smaller lakes in North America show little trend over a 10-year period spanning the beginning of the 21st century (Wilkinson et al. 2021).
Here we use time series trials of 190 days length in 140 L mesocosms to test for reactive transient dynamics using a nutrient-driven model of population dynamics of Chlorella vulgaris, a globally common species of green algae often used for biofuel projects. Detailed experimental data were already available on the effect of nutrient concentration on algal growth rates, which we linked with density-dependent feedbacks parameterized for our experimental system. We used this model to estimate the impact of nutrient input on reactivity, resilience, and stability of the nutrient-Chlorella system. We tested the ability of a low-dimension nutrient-driven algal model to predict the time dynamics of Chlorella populations in response to augmented fertilizer input in replicated mesocosm trials. It is likely that more precise model predictions could be obtained by building a more complex model controlling for the impact of nutrient uptake in relation to cell size and ecosystem concentration, light incidence, water turbidity, nutrient ratio, and spatial distribution of both light and plankton at depth (Grover 1991a; Huisman 1999; Elser et al. 2000). For analytical tractability in this instance, however, we opted for a more stream-lined formulation to simplify the analysis of the impact of initial nutrient and plankton abundance and nutrient influx rates on system reactivity.
Model
Our nutrient-driven model was based on a substantial body of previous theoretical and empirical work on nutrient-phytoplankton dynamics (e.g. O’Brien 1974, Grover 1989; DeAngelis 1992, Huppert et al. 2002, 2005). Our model links a closed population of Chlorella vulgaris (P) whose growth rate is shaped by both density-dependent processes, estimated in our experiments, and potentially limiting levels of nutrients (N), estimated in earlier benchtop experiments (Pannikov and Pirt 1978):
where rmax = maximum per capita phytoplankton growth rate (cells per day), a = daily rate of nutrient influx (μg/ml), bmax = maximum rate of nutrient absorption per phytoplankton cell (μg/cell-day), c = nutrient concentration (μg/ml) at which nutrient absorption = bmax/2, d = nutrient concentration (μg/ml) at which phytoplankton growth rate = rmax/2, f is a scaling constant linking maximum rates of nutrient absorption and phytoplankton growth to local environmental conditions, and g = the log-linear (Gompertz logistic) effect of algal density on the exponential rate of algal population growth. Parameters bmax, c, rmax, and d were obtained from Panikov and Pirt’s (1978) benchtop experiments spanning a wide range of concentrations of nitrogen or phosphorus in a flow cytometer with high flushing rates to maintain extremely low population density of Chlorella vulgaris cells.
Although 6 parameters are required by our model of nutrient-phytoplankton-zooplankton dynamics, one parameter was experimentally controlled (a) and three other parameters have been estimated directly from previously published experimental trials for Chlorella vulgaris (b, c, and d). That left just two parameters to be estimated from our phytoplankton time series experiments: f (specifying the maximum rate of phytoplankton growth and nutrient absorption under our experimental conditions) and g (specifying the density-dependent effect of phytoplankton on per capita mortality rates). For simplicity, we assumed that phytoplankton growth and absorption are scaled in the same way by local environmental conditions, so we applied the same coefficient. We systematically sampled across a wide range of parameter values and used least-squares minimization to identify the most parsimonious set of parameters to explain algal population density over time, when grown in isolation. The likelihood surfaces for these 2 parameters were smooth and unimodal, so the fitting procedure was quite straightforward. Algal density and nutrient levels were initialized in the model with the same starting values as we used in the mesocosm experiments (P0 = 50,000 cells/ml and N0 = 2 μg/ml). R Code for parameter estimation and model simulation is available on Figshare.
A useful first step towards understanding consumer-resource models such as that represented by eq. 1-2 is by evaluating a linearized version of the model in the vicinity of the system equilibrium, whose rates of change are specified by the Jacobian matrix A, whose elements are calculated in the following manner:
Stability and reactivity of the system are then evaluated by applying the Jacobian matrix to the linearized system around the equilibrium:
where n = N - Neq and p = P - Peq represent deviations of nutrient concentration (N) and phytoplankton density (P) from system equilibria (Neq and Peq). If the dominant eigenvalue (λ1) of A < 0, then the system is locally stable and in the long-term the system will recover equilibrium levels of nutrients and algae following perturbation. The speed of long-term recovery in such a stable system is scaled with the magnitude of λ1(A), often referred to as system resilience (Pimm and Lawton 1977; Neubert and Caswell 1997; Ives et al. 2003).
The magnitude of λ1(A) tells us little, however, about the immediate response to perturbation, and this is relevant to seasonal algal blooms because many dynamic systems can experience further deviation in the short-term before they eventually converge on the stable equilibrium over the longer term (Neubert and Caswell 1997). To evaluate that possibility, we calculated the dominant eigenvalue λ1(H) of the Hermitian part of the Jacobian matrix: H = (A + AT)/2, where AT refers to the transpose of matrix A, often referred to as the ‘reactivity’ of the system (Neubert and Caswell 1997). A system will be reactive provided that λ1(H) > 0, implying that at least some perturbations will result in amplified deviation from the stable equilibrium in the short term, rather than reduction. The degree of amplification scales with the magnitude of positive values of λ1(H). We used experimental estimates of the matrices A and H to evaluate the long-term stability and short-term reactivity of nutrient-limited Chlorella populations following perturbation from equilibrium.