Simulating Trends and Steps

Up to this point the exercises have assumed that equilibrium points are stationary, or do not change with time. This assumption means that external forces are either constant (s = 0) or act as random variables (s > 0). However, it is possible for environmental factors to cause parameters to change in a more consistent manner. Parameters of the R-function can be affected by exogenous forcing in two basic ways:

1. By gradual and continuous forcing, such as would occur through global warming. This would cause the affected parameter to change smoothly with time, resulting in what is called a trend. We simulate trends by making the affected parameter a continuous function of time.
2. By sudden or discontinuous environmental changes, as would occur from a catastrophe like a hurricane. This causes the affected parameter to change suddenly in what is called a step. We simulate steps by making the parameter a discontinuous function of time.

Simulating Trends

• Access the Modeling & Simulation program P1b and set A = 1, C = 0.001, d = 1, Q = 1.
• Use [F8] to simulate a run of 40, initial density 500, and then press [T] to simulate a trend. Press [K] to simulate a trend in the carrying capacity. Set the forcing parameter to 50 and the random variable at 0.2 (see figure). Repeat the same simulation several times.
• Use [F8] to simulate a run of 40, initial density 500, and press [T] to simulate a trend. Press [A] to simulate a trend in the maximum per-capita rate of change. Set the forcing parameter at 0.1 and random variability at 0. Make sure you replot on the logarithmic scale. Repeat the simulation with s = 0.2.
• Use [F7] to alter the R-function and set A = 1, C = 0.001, d = 2, Q = 1.
• Press [F8] to simulate a run of 50, and initial densities of N(0) = N(1) = 500. Press [T] and then [K] to simulate a trend in the carrying capacity. Set the forcing parameter at 50 and the random variable at 0.2 (see figure). Repeat the same simulation several times.
• Use [F8] to simulate a run of 50, and initial densities of N(0) = N(1) = 500. Press [T] and then [A] to simulate a trend in the maximum per-capita rate of change. Set the forcing parameter at 0.02 and the noise at 0.2 (see figure). Remember to look at log plots.
• Notice how trends in parameters can cause the mean of the trajectory to move (trend in K) and the stability of the equilibrium to change (trend in A).

Simulating Steps

• Use [F7] to alter the R-function and set A = 1, C = 0.001, d = 2, Q = 1.
• Use [F8] to simulate a run of 50, initial densities of N(0) = N(1) = 500, and then press [S] to simulate a step. Press [K] to simulate a trend in the carrying capacity, then set the new K to 500, the time step to 20, and the random variable to 0. Repeat the simulation with s = 0.2 (see figure).
• Use [F8] to simulate a run of 50, initial densities of N(0) = N(1) = 500, and press [S] for step. Press [A] to simulate a step in the maximum per-capita rate of change. Set the new A = 0.5, TSTEP = 20, and the random variable at 0. Repeat with s = 0.2 (see figure).
• Use [F8] to simulate a run of 50, initial densities of N(0) = N(1) = 500. Press [S] and then [d] to simulate a step in the time lag. Set the new d to 1, TSTEP to 20 and random variability at 0. Repeat with s = 0.2 (see figure).
• Notice how steps in the parameters affect the mean and stability of population trajectories.