Frequency domain fits between models and observed adult population time series of length 41 (*A*, *C*, and *E*) and length 213 (*B*, *D*, and *F*) from the three experimental replicates with *c*_{pa} = 0.35. The heavy dashed lines are data log spectra, identical in *A*, *C*, and *E* and identical in *B*, *D*, and *F*. Light solid lines give the minimum, the 5th, 25th, 50th, 75th, and 95th percentiles, and the maximum values at each frequency value of 1,000 log spectra of model-generated time series of length 41 (*A*, *C*, and *E*) and length 213 (*B*, *D*, and *F*). Triangles highlight the 5th and 95th percentiles. Models used were the LSD-LPA model with parameters of *Supporting Text 1.1* (*A* and *B*), the (intentionally incorrect) constrained LSD-LPA model with *c*_{el} = 0 and time-domain-optimized parameters of *Supporting Text 2.2* (*C* and *D*), and the (intentionally incorrect) constrained LSD-LPA model with *c*_{ea} = 0 and time-domain-optimized parameters of *Supporting Text 2.2* (*E* and *F*). All parameters were optimized for length-41 data. Contrast the good fit in *B* with the poor fits in *D* and *F*. Fit of the constrained LSD-LPA model with no further constraint is similar to *A* and *B* and is shown in *Supporting Text 2.2*. Fits were similar when frequency-domain optimized parameters were used (*Supporting Text 2.2*). Aliasing of fundamental frequencies of population fluctuation is unlikely to have occurred because the biology of *Tribolium* suggests that little fluctuation occurs for normalized frequency (nf) >1.

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