34 DWYER, NIXON, OVIATT, PEREZ, AND SMAYDA 



coherence squared and the Bode amplitude and phase plots 

 [Fig. 7(b)] . Coherence squared was not significant over most of the 

 frequency range. Since cross-spectral frequency response analysis 

 evaluates the linear dependence, measured by coherence squared, of 

 response on the input, there appears to be little linear relation 

 between the two spectra. The Bode plots show chaotic behavior at 

 frequencies higher than 1 cycle/year; this again indicates a poor 

 linear relation. Cross-spectral analyses of the complete solar-radiation 

 and phytoplankton-abundance time series [Fig. 7(c)] and of the 

 microcosm ammonia and chlorophyll time series [Fig. 7(d)] show 

 similar random frequency responses. 



Bode amplitude and phase plots show very definite forms when a 

 linear relation exists [see Child and Shugart, 1972; Fig. 7(a)] . When 

 calculated from time-series data from a quasilinear system, they can 

 provide information on the minimum complexity needed in an 

 ecosystem model, as well as estimates of system stability properties. 

 Our inability to use them here stems from the nonlinearity of the 

 system and from the fact that significant periodicities are represented 

 in only a very narrow frequency range in the data (1 to 4 

 cycles/year). Variance at higher frequencies represents only the 

 random portion of the data, and, by definition, the coherence 

 between two random time series is not significant. Thus the 

 high-frequency portions of the Bode plots represent essentially 

 random variation. 



DISCUSSION 



A major difficulty in applying cross-spectral methods to data 

 from natural ecosystems, as pointed out by the results of this study, 

 is the fact that natural environmental inputs fluctuate only at a few 

 frequencies. Marine ecosystems receive few stationary inputs at 

 frequencies other than once per year, once per day, or once per tidal 

 cycle. [Although wind-driven turbulence inputs can influence 

 phytoplankton at very high frequencies (Piatt and Denman, 1975, 

 and the literature cited therein), these fluctuations cannot be 

 resolved with the 1-week sampling used here.] Coherence between 

 natural-input time series and ecological responses will be significant 

 only at these input frequencies, regardless of what other periodicities 

 are present in response spectra. The high-frequency portions of the 

 spectra in this study (Figs. 5 and 6) show no significant variance 

 other than that associated with sampling errors. 



In practice, applying artificial periodic inputs to natural ecosys- 

 tems presents many logistical difficulties. The use of laboratory 



