Sensitivity Analysis

The non-linear, time-delayed, logistic R-function has four parameters, the intercept A, which is the maximum per-capita rate of change (maximum possible value of R), the coefficient of interaction, C, the time delay or lag, d, and the coefficient of curvature, Q. In previous experiments, with the parameters set at A = 0.6, C = 0.0001, d = 1, Q = 1, and with no random variability (s = 0), we saw that trajectories converged smoothly to equilibrium. This is called asymptotic approach to equilibrium or, because such equilibria are obviously stable, just asymptotic stability.

We now pose the question: What happens to the stability of the equilibrium when the parameters are altered? We can answer this question by performing what is called a sensitivity analysis. That is, we alter each parameter by small increments, displace the population from its equilibrium, and observe the subsequent dynamics in a constant environment. This allows us to observe how the system reacts to incremental variations of its parameters; i.e., how sensitive it is to variations of its parameters.

Sensitivity to Changes in C

The parameter C defines the intensity of competitive interactions between members of the population. Thus, C is larger when resources are in short supply because interactions will be stronger. Changes in the availability of food, nesting sites, hiding places, and so on will cause C to change. We now explore what happens when the supply of resources is altered, either naturally or through human actions.

• Starting with the previous model (A = 0.6, C = 0.0001, d = 1, Q = 1), enter the R-function routine by pressing [F7]. Press [Y] to alter the R-function. Default to the current value of A. Change C to 0.001. Default the other parameters. Note the change in K.
• Enter the simulation routine with [F8] and set the run length at 20, the initial density at 500, and the standard deviation at 0.
• Repeat the experiment with C = 0.01, N(0) = 50.
• Repeat the experiment with C = 0.1, N(0) = 5.
• Notice how the system remains asymptotically stable but that the equilibrium point, or carrying capacity K, varies in inverse relationship to C.

Sensitivity to Changes in A

• Press [F7] to alter the R-function and change A to 1.2, C to 0.001, and default all the other parameters. Note the value of K.
• Enter the simulation routine with [F8] and set the run length at 40, the starting density at 1000, and the standard deviation at 0.
• Repeat the experiment with the following values of A: 1.6, 2.0, 2.6, 3.0, 3.8.
• Notice how A affects the stability of the equilibrium, with asymptotic stability when A 1, damped stability when 1 < A < 2, stable limit cycles when 2 A < 2.7, and unpredictable chaotic oscillations when A > 2.7.
• Note that A also affects the carrying capacity.

Sensitivity to Changes in Q

• Enter the R-function routine and set the maximum rate of increase at 2, the competition coefficient at 0.2, the time delay at 1 and the curvature coefficient at 0.3. Default other parameters. Notice the new shape of the R-function.
• Enter the simulation routine and set the run length at 40, the initial density at 1500, and the standard deviation at 0. Compare the trajectory to that in the previous experiment when A = 2. Remember to plot logs.
• Repeat the exercise with the following values for Q: 0.6, 1.3, 1.6 (see dynamics here)

Sensitivity to Changes in d

• Access the R-function routine with [F7] and set the maximum rate of increase at A = 0.6, C = 0.001, d = 2, Q = 1. Notice the new value of d on the R-function abscissa (x-axis).
• Enter the simulation routine and set the run length at 40, the initial densities N(0) = 10 and N(1) = 10 (you have to enter two initial densities when d = 2), and the standard deviation at 0 (see R-function and dynamics).
• Repeat the experiment with d = 3 and compare the results (see R-function and dynamics).
• Notice how larger time lags create cycles of longer period and larger amplitude.
• Enter the R-function routine with [F7] and set A = 0.6, C = 0.001, d = 2, Q = 1. Simulate with run = 50, N(0) = N(1) = 600, and s = 0.2 (see figure). Run several simulations with the same values. Repeat the experiment with s = 0.4 and 0.6.
• Notice how environmental variability causes cycles to persist even though they would damp out in a stable environment