Spatiotemporal mapping of Karenia brevis blooms

Measurement of cell concentrations

Measurements of Karenia brevis cell abundance have been compiled by the Fish and Wildlife Research Institute, Florida Fish and Wildlife Conservation Commission, for more than seventy years. Blooms are most frequent along the southwest Florida coast, between Anna Maria Island (N lat 27.54) and Marco Island (N lat 25.80), where newly 120,000 samples have been collected. The sampling rate has varied greatly over this period:

The 1954-57 period of intense sampling was funded by the federal government after severe K. brevis blooms along the Southwest Florida coast in the late 1940s and early 1950s. Sampling rates greatly decreased in following years due to lack of funds, except for the uptick in the late-1970s due to studies proving that fish kills accompanying blooms in 1976-78 were caused by brevetoxins produced by K. brevis (then known as Gymnodinium breve), not oxygen deprivation due decaying organic matter. Very severe blooms (red tides) in 1994-96 and again in 2005-2006 resulted in the Florida Legislature establishing a routine, proactive sampling protocol that continues today.

Data processing: separating signal from noise

Blooms of K. brevis are inherently heterogeneous; even in the midst of a severe bloom, samples taken from the same general area can have cell concentrations ranging from zero to millions of cells per liter. This is caused by current and wind forcing, vertical migration, and localized nutrient sources. As a result, time series of sample cell concentrations form a cloud of points, not a line. The first step to understanding the temporal structure of a bloom is the removal of the noise hiding that structure. This was done with a three-stage cascaded filter:

1. A simple average of the daily measurements to remove sub-daily spatial patchiness;
2. A seven-day, centered moving average acting as a notch filter to remove the weekly sampling protocol artifact; and
3. A 21-day, centered moving average filter to remove transient, short-term events while preserving the lower-frequency oscillations that represent the true temporal structure of the bloom.

Here is the result of this process:

The two peaks are separated by about 100 days. This period is visually evident across all the blooms in the FWC-FWRI dataset. Wavelet analysis shows periods clustering around the 100-day reference line with evidence of period doubling for some blooms. This is consistent with the time-delay logistic equation.

The time-delay logistic equation

$$\frac{dN(t)}{dt} = r N(t) \left( 1 – \frac{N(t-\tau)}{K} \right)$$

The number of cells at the present time is N(t), while N(t – τ) is the number of cell at an earlier time (τ = delay time). The intrinsic rate constant for the increase in the number of cells is r and the system carrying capacity is K.

This equation was developed by Hutchinson in 1948 as a modification to the traditional logistic equation that generates a sigmoid curve of population vs. time, initially exponential (no resource limitation), than gradually slowing (as resources are consumed) to asymptotically approach an upper bound representing system carrying capacity. The time-delay logistic equation recognizes that populations usually don’t approach system carrying capacity asymptomatically. Instead, population increases too rapidly and overshoots carrying capacity, then decreases too rapidly and drops below carrying capacity. The resulting oscillations may be damped so as to settle at the system carrying capacity or may be sustained through many cycles. Sustained oscillations often take the form of limit cycles, where populations rise to levels far in excess of carrying capacity and then crash by orders of magnitude. There is strong evidence that K. brevis blooms behave this way (Kurtz, et al, 2024).

The nature of time-delay logistic equation oscillation depends on the value of the product of r and τ. For rτ ≤ 1/e, no oscillation; 1/e < rt < π/2, damped oscillation; π/2 ≤ rt ≤ 1.9, single period limit-cycle oscillations. For rt > 1.9 period-doubling begins and for rt > 2.06 the system becomes chaotic. For blooms of K. brevis values of rt will be in the range of

Bloom animations

The videos immediately below show animations for blooms starting in September 1954 and July 1957 based on the relatively large number of samples obtained during that period. There was no comparable period in the following years until 1995, when increasingly higher rates of sampling became routine, resulting in the series of eleven bloom animations in the second group below.

The animations are best viewed by immediately expanding them to full screen

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