Online Resource: Asymmetric response of early warning indicators of phytoplankton transitions
to and from cycles
A contribution to Theoretical Ecology for the Special Issue on Tipping Points
By
Ryan D. Batt, William A. Brock, Stephen R. Carpenter, Jonathan J. Cole, Michael L. Pace, and
David A. Seekell
All authors made equivalent contributions to this work
Addresses:
Ryan D. Batt (corresponding author; RBatt@wisc.edu), Center for Limnology, University of
Wisconsin, Madison, WI, 53706 USA
William A. Brock, Department of Economics, University of Wisconsin, Madison, WI 53706
USA and Department of Economics, University of Missouri, Columbia, MO 65211 USA
Stephen R. Carpenter, Center for Limnology, University of Wisconsin, Madison, WI, 53706
USA
Jonathan J. Cole, Cary Institute of Ecosystem Studies, Millbrook, NY 12545 USA
Michael L. Pace, Department of Environmental Sciences, University of Virginia, Charlottesville,
VA 22904 USA
David A. Seekell, Department of Environmental Sciences, University of Virginia,
Charlottesville, VA 22904 USA
Keywords: Phytoplankton; Threshold; Early Warning; Algae Bloom, Hopf Bifurcation
1
In the phytoplankton model (eqs. [7-12]), the Hopf bifurcation point occurs when
carrying capacity (K) = 8.33, but oscillations were observed when K was less than 8.33 and
when the system had a single stable equilibrium (Fig. 3). This region of K was where the
variance of the state variables increased sharply (Fig. 4), and where early warning signals (EWS)
were most easily detected. This behavior can be explained by the sluggish rate at which the state
variables approach equilibrium when K is close to the Hopf bifurcation point.
We simulated a deterministic version of the phytoplankton model (i.e., σG, σC, and σZ
equal zero) to show the behavior of algal biomass over time for several values of K (Fig. 6). The
time required for the biomass of colonial algae to reach the stable equilibrium increases as K
approaches 8.33. For example, when K equals 7.33 (Fig. 6A), biomass oscillates only briefly
before reaching an equilibrium value. However, when K=8.08 (Fig. 6D), the biomass slowly
approaches equilibrium, but never reaches an equilibrium value in the simulation. When K ≥
8.33 (Fig. 6E-I), the deterministic system exhibits a stable limit cycle.
When K < 8.33 in the stochastic simulation (Fig. 7A-D), the amplitude of the oscillations
is irregular and does not approach zero because the state variables are constantly subjected to
stochastic shocks. In other words, the dampened oscillations seen in the deterministic system are
sustained in the stochastic system by shocks. These oscillations have a period between 20 and 21
days. When K is increasing and reaches ~8.2, the average amplitude of these oscillations
increases drastically (Fig. 3). Therefore, in the same region of K where the deterministic system
sluggishly approached equilibrium (8.2<K<8.33), the average amplitude of the oscillations in the
stochastic system quickly increases. The amplitude of the oscillations continues to increase until
the system undergoes a Hopf bifurcation and exhibits a stable limit cycle that also has a period of
20 to 21 days (Fig. 7E-I). Consequently, when K is slightly below the bifurcation point, the
2
system will appear to oscillate as though it has entered a stable limit cycle even though a single
stable equilibrium exists (Fig. 3). This behavior results in a sharp increase in variance before the
bifurcation point (Fig. 4), which makes it helpful for anticipating the transition.
3
Figure 6. Simulated time series of the biomass (g C m-3) of colonial algae from a deterministic
version of the phytoplankton model. In each panel biomass changes over time while carrying
capacity (K) is held constant. The duration of the simulation was 50,000 time steps (2,000 days).
K increases from left to right, then from top to bottom. K was equal to 7.333, 7.582, 7.834,
8.083, 8.335, 8.584, 8.832, 9.085, and 9.333 in panels A through I, respectively. In panel E, K is
slightly larger than the bifurcation point (KHopf = 8.33), and the colonial biomass is colored gray.
4
Figure 7. Simulated time series of the biomass (g C m-3) of colonial algae from a stochastic
version of the phytoplankton model. In each panel biomass changes over time while carrying
capacity (K) is held constant. The duration of the simulation was 50,000 time steps (2,000 days).
K increases from left to right, then from top to bottom. K was equal to 7.333, 7.582, 7.834,
8.083, 8.335, 8.584, 8.832, 9.085, and 9.333 in panels A through I, respectively. In panel E, K is
slightly larger than the bifurcation point (KHopf = 8.33), and the colonial biomass is colored gray.
5