untitled

where μ=0 and σ=0.19. We then combine the random normal deviate at any

point in time (εt) with the rate of increase predicted by the Ricker logistic equation,

rmax(1 −Nt/K), to predict changes in abundance. We use a different symbol (nt) for

the simulated density:

We plot the simulated elk data (nt) in Fig. 8.16. We see that the trends in the

simulated population are completely different from those of the real population

(Fig. 8.14), but the overall magnitude of variability is similar. This similarity occurs

because we have included both the stochastic (environmental and demographic)

processes that tend to perturb the population away from its carrying capacity and

the natural regulatory processes that tend to restore the population, once perturbed.

Both processes are common in the natural world, and therefore we need to accom-

modate them in our management planning.

Such stochastic simulations, sometimes termed Monte Carlo models, offer useful

insights into the degree of variation that wildlife biologists and managers might expect

to see over a long time. Monte Carlo simulation is central to the procedure known

as population viability analysis, which we describe in Chapter 17.

Paradoxically, the same density-dependent processes that are responsible for natural

regulation can also induce population fluctuations, at least under special circumstances.

One way that this can happen is when the maximum rate of growth is particularly

high. For example, consider the dynamics of a hypothetical population whose

maximum rate of increase rmax=3.3 and carrying capacity K=100 (Fig. 8.17). In

this case the population does not increase smoothly over time and level off at the

carrying capacity, but rather the population fluctuates erratically over time, with no

apparent repeated pattern. Such a pattern of population change is known as deter-

ministic chaos(May 1976). It arises because the population grows so fast that it tends

to overshoot the carrying capacity, a process known as overcompensation(May

and Oster 1976). Once above the carrying capacity the net recruitment is negative

nnett

r nKt t

+

⎛⎝⎜− ⎞⎠⎟+

1 =

1

max ε

POPULATION REGULATION, FLUCTUATION, AND COMPETITION WITHIN SPECIES 125

20

15

10

5

0

0 1020304050

t

Simulated elk abundance (thousands)

Fig. 8.16Simulated
dynamics of elk, based
on the Yellowstone
National Park
population. (Data from
Coughenour and Singer
1996.)

8.7 Stability, cycles, and chaos