Dynamics Far from Equilibrium

What happens when a population is well below its equilibrium density or carrying capacity, because it is either invading a new environment or has been reduced to a very sparse density by a calamity?

• From the default drive and path, on which your PAS programs reside, enter [PAS] and press [2] to access the SINGLE-SPECIES MENU, then [2] again to run the Modeling & Simulation program P1b. Use the [F1] key for help and [F10] to move forward.
• Press [N] when asked if you want to build a new model or use an existing one, then enter a name for your organism; e.g., [Organism].
• You will now be asked to assign a sampling unit, say a hectare, and a number of units sampled, say 10 (this means that the area sampled is 10 hectares; however, as this data is not used for calculation you can actually enter anything you like). Enter Hectare then 10, or anything else you want.
• P1b now asks for a value for the maximum per-capita rate of change, A; enter 0.6 (notice the values entered in the * Parameters * window ).
• Next enter a value for the interaction coefficient, C = 0.0001.
• Enter a value for the time delay, d = 1.
• Enter a value for the coefficient of curvature, Q = 1.
• A linear R-function should appear on the screen (see figure). Notice the parameter K, the carrying capacity of the environment is calculated as K = A/C = 0.6/0.0001 = 6000.
• Enter the simulation routine with [F8]. Set the run length at 10 and the initial population density at 2. Next press [N] to avoid simulating trends and steps. Default the random variable to s = 0 (note that default means to accept the current setting by pressing [Enter]).
• Notice how the population grows at an increasing rate, or exponentially. Replot the simulation several times with the [F2] key, adjusting the speed with keys [F4] and [F5] until you can clearly observe the motion of the time trace.
• Use key [F3] to replot the trajectory on the logarithmic scale. Notice the linear growth pattern.
• Use [F8] to re-enter the simulation routine. Enter a time horizon of 10, an initial population of 2, [N] for no trend or step, and a standard deviation of 0.2 (see figure). Use the [F3] key to see a log-plot (see figure). Press [F8] to repeat the simulation several times with the same (default) input values, remembering to plot logarithms.
• Press [F8], default the run length and initial density, press [N], then enter a standard deviation of 0.4. Remember to plot logs.
• Press [F8] and repeat the experiment with a standard deviation of 0.6. Plot logarithms.
• Notice that populations growing exponentially far from equilibrium show a linear trend when plotted on the log scale, even in the presence of considerable environmental variability.