the stable state, this will be a single point at (1000, 10). For, say, a starting N2 value of 11 it will show
a whorl of expanding population oscillations, ending when either N1 or N2 < 0 (Fig. 4.1).Any values satisfying those equalities will be at equilibrium, but we will ignore the
case N1 = N2 = 0. We try arbitrary values that meet the equilibrium conditions by
running VOLTERRA on a computer with Matlab software. The screen interface of
Matlab is convenient. Type the program into the editor window, hit the “run” button
and the results will appear in a new window as a time-series plot. All the constants
and starting values are in the program. To change any of them, just type in different
numbers, push run again and the new results will appear. Thus, you can (and should)
readily experiment. The program in Box 4.1 uses some equilibrium values: N1 =
1000, N2 = 10, D = 0.1, K2 = 0.0001, B = 0.1, K1 = 0.01, ΔT = 1. Looking at the
results every 10 time-steps, the boring output is:
(^) TIMEN 1 N 2
10 100010
20 100010
30 100010
40 100010
: : :
: : :
90 100010
(^) The model is at equilibrium, as predicted, and a single point appears in the plot.
Now, edit the program, changing N2 = 11 and run it without changing any other state
variable or parameter. The time-series of N1 and N2 values will start to oscillate,
getting further and further from equilibrium until the model “crashes”, with one of the
species (and then the dependent other) going to extinction or “exploding”. A graphical
presentation of such models is often made in “phase space” (Fig. 4.1), a plot on axes
scaled by N1 and N2. The time course is traced on this phase plane. By starting trials
on various parts of the plane, the stability properties of the relationship between N 1
and N2 can be investigated. As formulated, the model has only the one equilibrium
point, which is unstable because even a slight displacement from it will result in a
crash.
Fig. 4.1 Phase diagram for a Lotka–Volterra predator–prey model. Initial prey and
predator numbers are 1000 and 11, just off the stable point (1000,10) for the
parameters. The diagram shows an expanding oscillation that is said to “crash” when
N2 becomes less than 1.