PAS P2a, Time Series Analysis reads data from files created, analyzed and saved to disk within the Single-Species Program P1a. After reading this data, P2a asks you to define the kinds of interactions between the two species; i.e., it inquires if the organisms are plants, herbivores, or carnivores and if they benefit or harm each other. This allows P2a to recognize the interaction as competitive (both harm each other), cooperative (both benefit each other), or exploitative (one benefits and the other is harmed). In the latter situation, P2a insists that the resource species (the exploited one that is harmed by the interaction) be entered first as Species 1.
With these preliminaries over, P2a plots the data as a time-series (see figure). Function keys can be used to obtain help [F1], replot the graphs on arithmetic [F2] or logarithmic [F3] scales, speed up [F4] or slow down [F5] the graph trace, back up to a previous screen [F9], or proceed to the next screen [F10]. The program can be terminated by simultaneously pressing the [Alt] + [X] keys.
The program then proceeds to plot the data in the phase space of the two species (see figure). It then fits the LV model to the data for species 1 using linear "least-squares" regression (see figure). The REGRESSION TABLE shows the regression of the rate of change of species 1, R, on its own density, N, with R estimated from
and using the regression model
where A is the intercept and C1 the slope of the simple regression (see column 2 of the table).
P2a then calculates the simple regression of R on the density of the other species, which we will identify by Z(t-1)
with A the intercept and C2 the slope of the regression (see column 3 of the table).
Finally, P2a performs a multiple regression analysis
as seen in column 4 of the regression table. Notice that the statistics of the multiple regression LV model are also summarized in the top-right window of the regression table. In all these models A, the intercept of the regression line, represents the maximum per-capita rate of change of the population in the given environment, C1 the strength of the intraspecific effect, and C2 the strength of the interspecific effect.
The goodness of fit of the regression to the data is given by the "coefficient of determination" (r2, see in the first row of the table). This statistic provides an estimate of the per cent variation in the data explained by the fitted model, or the variation in the data due to the action of intra- and inter-specific interactions. Therefore, the remainder of the variation, 1-r2, represents the contribution of density-independent exogenous (or stochastic) inputs. Finally, the "partial coefficients of determination" (pn and pz) estimate the contribution of intra- and inter-specific effects to variation in the per-capita rate of change when the contribution of the other is removed.
P2a then fits the RD model to each species (see figure). The procedure is identical to that above except that
with N1 being the exploited species and N2 the exploiter. (Note that different ratios may be used in RD competition and cooperation models).
After the parameters of the LV and RD models have been estimated, the user is asked to choose one or the other model (see figure). When this has been done for both species, the feedback structure of the chosen model is displayed, and the user decides to retain or delete each interaction based on its contribution to the model (see figure). However, structural alterations to the model are subject to certain rules:
NOTE that P2a models do not contain explicit time-lags. They assume, instead, that the time-lags found in P1a analyses are due to the two-species interaction. P2a models, therefore, contain implicit time-lags in their interactive structure; i.e., it takes time for one species to respond to another and then to feed back to affect the original species.