untitled

many times suggests roughly an 8–10% risk of extinction within the next century,

or one extinction event every 10,000–20,000 years (Fig. 17.4).

There is a more mathematically elegant way to estimate extinction risk in popula-

tions with exponential population growth, using what is known as a “diffusion model”

(Lande 1993). We follow the discussion of diffusion models in Morris and Doak (2002).

A diffusion model can be visualized as if a vast number of beads are released at a

single point near the bottom of an infinitely deep river. Over time, the cloud of beads

would spread, due to individual beads bouncing up or down due to turbulence as

they flow downstream, just as the population trajectories for the geometric growth

model tend to spread over time (Fig. 17.4). A probability density formula known

as the inverse Gaussian distribution is used to predict the analogous distribution of

population densities over time, with residual variability in the exponential growth

rate of the population generating the turbulence. This equation generates a bell-shaped

distribution of population densities at each point in time, with the degree of spread

of the bell-shaped curve growing wider with every year. The initial position at which

the beads are released corresponds to the starting population density Nc. This is

important, because more beads will strike the bottom when they are released low

in the water column, just as the probability of extinction is highest for populations

starting at initially low numbers. The quasi-extinction threshold is denoted Nx, μis

the average exponential rate of increase (which equals the arithmetic mean of logeλ),

and σ^2 is the variance of logeλ. The probability of extinction in any given year tand

the cumulative probability of extinction are calculated according to the equations shown

in Box 17.1.

We will illustrate the application of the diffusion equation using population

censuses of females during 1959– 82 from the Yellowstone grizzly bear population

to estimate mean rand the standard deviation in r. As before, we will assume a lower

critical threshold Nc=10 animals and use the 1982 census total as the initial den-

sity Nx=41.

The predicted probability of extinction increases over time, because it takes several

poor years in sequence for a population of 41 grizzlies to crash to below 10 indi-

viduals (Fig. 17.5). The probability declines at very long times, simply because the

losers have already disappeared whereas any winners have likely grown well out of

302 Chapter 17

100

10

0 10 20 30 40 50 60 70 80 90 100

Time

Number of female grizzlies

Fig. 17.4Results of
100 replicate Monte
Carlo simulations of
the Yellowstone grizzly
population, based
on an exponential
growth model with
μ=−0.00086 and
σ=0.08. The lower
critical density is
arbitrarily set at 10
individuals. Between
5% and 10% of the
replicates tend to reach
the critical threshold
within a century.

17.7.2PVA based on
the diffusion model