Abstract
Recently it has been show that in some ecosystems fast rates of change of environmental drivers may trigger a critical transition, whereas change of the same magnitude but at slower rates would not. So far, few studies describe this phenomenon of rate-induced tipping, while it is important to understand this phenomenon in the light of the ongoing rapid environmental change. Here, we demonstrate rate-induced tipping in a simple model of cyanobacteria with realistic parameter settings. We explain graphically that there is a range of initial conditions at which a gradual increase in environmental conditions can cause a collapse of the population, but only if the change is fast enough. In addition, we show that a pulse in the environmental conditions can cause a temporary collapse, but that is dependent on both the rate and the duration of the pulse. Furthermore, we study whether the autocorrelation of stochastic environmental conditions can influence the probability of inducing rate-tipping. As both the rate of environmental change, and autocorrelation of the environmental variability are increasing in parts of the climate, the probability for rate-induced tipping to occur is likely to increase. Our results imply that, even though the identification of rate sensitive ecosystems in the real world will be challenging, we should incorporate critical rates of change in our ecosystem assessments and management.
Highlights
In the recent years the notion that ecosystems can have tipping points has received considerable attention
In a simple model with one state variable, this instability can be read from the phase plane (Fig 4), A change with infinite speed from an initial condition to a higher value for the incoming light intensity can be visualized as a horizontal line
We have shown with this model with realistic parameter settings [12] that speed of change can be critical for the effect that environmental conditions have on phytoplankton
Summary
In the recent years the notion that ecosystems can have tipping points has received considerable attention. We increased the light intensity from a low value We exposed the model to a brief pulse in the value of the parameter Iin. We exposed the model to a brief pulse in the value of the parameter Iin The effect of such a pulse is different than a gradual increase in light intensity, because the relative rate, and the duration of the pulse determines whether the system tips or not. We varied both the rate and the duration of pulse. The formula for the pulse in the light intensity is: dIin 1⁄4 < rðIin;max À IinÞ if t > tstart and t < tend ð7Þ dt : rðIin;min À IinÞ otherwise
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