Developing a model for a natural noise-induced phase transition in Aphanizomenon flos-aquae

Background/Question/Methods

Much research has recently been devoted to understanding extinctions that are caused when an environmental driver is changed beyond a tipping point. However, it has not been appreciated that in stochastic systems this can occur when increasing variability in the environmental parameter causes the most probable population size to discontinuously fall to nearly zero, called a noise-induced phase transition. We developed a stochastic phytoplankton model that can be used to represent such a regime shift due to photoinhibition. Self-shading in a mixed water column allows a phytoplankton population to reach a higher biomass than it would without mixing, but as irradiance continues to increase, incident light will arrive at a threshold where self-shading is insufficient, and the population collapses. The subsequent lack of self-shading causes hysteresis. This was simulated in R with a model developed for Aphanizomenon flos-aquae. The light incidence parameter was derived from a random normal distribution around a value at which the population reached its maximum biomass. The variance of this distribution was increased between simulations to show that increasing environmental noise without a change in the environmental state variable can cause population collapse.

Results/Conclusions

A deterministic version of the implemented dynamical model has previously been shown to fit the growth of Aphanizomenon flos-aquae, and our stochastic model exhibits noise-induced phase transitions. The zero-net growth isoclines show catastrophic collapse with hysteresis under certain parameterizations. The inclusion of weak immigration prevents the population from collapsing over the long time scales of the simulation. Fluctuations of light are centered at 600 (mmol photons m^-2 s^-1) and vary randomly around that mean at intervals that are less than the time it takes for the population to adjust. At low variance in light incidence, the population is able to grow and remain stable. The lowest standard deviation at which the model exhibits stochastic switching is 282. The range of light variability which permits bistability is only within 2 mmol photons m^-2 s^-1. Higher light variability causes rapid extinction. The unstable equilibrium shown by the distribution of population sizes for the entire simulation indicates the presence of hysteresis. Because this results from environmental variability, this model suggests the possibility of a natural noise-induced phase transition.