Simulating Two Basins of Attraction

The models considered so far only have one convergent equilibrium, although this equilibrium point can change with time (trends or steps). However, some natural systems appear to have 2, and perhaps more, basins of attraction. For example, many insect herbivores are kept at low densities by negative feedback with their natural enemies. However, if these enemies become satiated by high prey densities, the pest population can escape and rise to the limits set by its resources. In other words, this system has 2 basins of attraction, one to a low-density equilibrium determined by negative feedback with natural enemies, and another to a high-density equilibrium determined by negative feedback with food supplies or other essential resources.

To simulate systems with two basins of attraction we create a model with two R-functions, one governing the high-density dynamics and the other the low-density dynamics. We must also specify an escape threshold, a particular density where the population escapes from (or returns to) low-density regulation, which in turn defines the unstable separatrix between the two basins of attraction.

Two Stable Equilibria

• Access the PAS routine P1b and set the parameters of the R-function at A = 1, C = 0.0001, d = 1, Q = 1 (these will become the parameters of the high-density R-function). Press [F10] to proceed, then [Y] to build a lower R-function. Set the parameters Cl = 0.001, dl = 1, Ql = 1 and the escape threshold U = 3000. Notice that the new double R-function is separated by a dotted curve representing the set of N, R values that will give rise to a population of density of U. The curve is called a separatrix because it separates the basins of attraction of the two attractors (see figure).
• Enter the simulation routine with [F8] and set run at 20, initial density at 2999 (just below the escape threshold), and the random variable at 0.
• Press [F8] to repeat with the initial density at 3001 (just above the escape threshold). Default the other condition.
• Notice the dynamics on each side of the escape threshold.
• Use [F8] to repeat the simulation with a run length of 100 and random variability of 0.6. Run 10 or more 100 year simulations and observer the dynamics (see figure). Remember to examine the log plots.

Time Lags at the Upper Equilibrium

• Press [F9] twice to return to the high-density R-function. Press [F7] to enter the R-function routine. Change the time lag to 2 and default the other parameters. Press [F10] to proceed to the low-density R-function and set Cl = 0.001, dl = 1, Ql = 1, U = 2000.
• Press [F8] to simulate runs of 100, initial densities of N(0) = N(1) = 1000, and random variation of 0.4. Run the same simulation many times.
• Note the complex, unpredictable dynamics exhibited by systems with two basins of attraction in noisy environments.