Detection of periodic peaks in Karenia brevis concentration consistent with the time-delay logistic equation

The logistic equation models single-species population growth with a sigmoid curve that begins as exponential and ends with an asymptotic approach to a final population determined by natural system carrying capacity. But the population of a natural system often does not stabilize as it approaches carrying capacity. Instead, it exhibits periodic change, sometimes with very large amplitudes. The time-delay modification of the logistic equation accounts for this behavior by connecting the present rate of population growth to conditions at an earlier time. The periodic change in population with time can progress from a monotonic approach to the carrying capacity; to oscillation around the carrying capacity; to limit-cycle periodic change; and, finally, to chaotic change. The presence of multiple species and inadequate sampling frequency and spatial coverage hinder the application of the time-delay logistic equation to real-world populations. Blooms of Karenia brevis along the southwest Florida Gulf Coast, however, provide a unique opportunity in that blooms are nearly monospecific and are sampled frequently over a wide geographic region; they are good candidates for testing the time-delay logistic equation. We show that these blooms exhibit peaks in concentration with periods in the range of 40-100 days, consistent with that predicted by the time-delay logistic equation. Cell concentrations in the valleys between the peaks are at least 2-3 orders of magnitude lower than peak values, offering predictable windows of opportunity for potential mitigation efforts.