Phase Portrait

The phase portrait is a plot of the per-capita rate of increase R on population density, N(t-d) or ln N(t-d), at lag d (see figure). You can use the PRCF to select the appropriate lag, or the plots themselves can be used to identify the dominant lag (Berryman 1991, 1992, 1993, 1994). Here, the value of R is plotted on N(t-d) or ln N(t-d), changing d until the best correlation is obtained (note that the coefficient of determination r² appears in the *Statistics* window). One can also use the behavior of the trajectory in the phase portrait to help identify the correct lag:

1. If the phase plot has a wide orbit that cycles clockwise, then d is probably too small.
2. If the orbit is wide and cycles in an anti-clockwise direction, then d is probably too large.
3. The best estimate of d is that which creates the narrowest orbit with no obvious direction of motion.

Note that estimates of the dominant time delay in the density dependent negative feedback acting on a population obtained by regression in the phase portrait may be different from those obtained from the PRCF. This is because the phase portrait regressions do not give the partial correlation coefficients. Under these conditions, biological reasoning may be required in choosing the correct time delay.

References:

• Berryman, A. A. 1991. Population theory: an essential ingredient in pest prediction, management and policy-making. American Entomologist 37: 138-142.
• Berryman, A. A. 1992. On choosing models for describing and analyzing ecological time series. Ecology 73: 694-698.
• Berryman, A. A. 1993. Food web connectance and feedback dominance, or does everything really depend on everything else? Oikos 68: 183-185.
• Berryman, A. A. 1994. Population dynamics: forecasting and diagnosis from time series. In S. R. Leather, A. D. Watt, N. J. Mills and K. F. A. Walters (Eds.) Individuals, Populations and Patterns in Ecology. Intercept, Andover, Hampshire.