untitled

the danger zone. The peak and spread of this curve depends, obviously, on the mean

rate of growth, as well as the variance.

The diffusion model predicts that the cumulative risk of extinction for Yellow-

stone grizzlies should tend to increase over time, initially at an accelerating rate, then

later at a diminishing rate (Fig. 17.6). For a population whose mean exponential growth

rate μ<0, as this one is, eventual extinction is certain, provided one waits long enough.

The question is: how long might this process take? By year 100, the diffusion model

estimates a 9% probability of extinction (Fig. 17.6), just as we found from our ear-

lier simulations.

In reality the Yellowstone grizzly population managed to recover to much higher

numbers in the 1990s (Fig. 17.7). Hence the risk of imminent extinction in the early

CONSERVATION IN THEORY 303

This diffusion model predicts the probability of extinction in any given year, P(t), as well as the cumu-

lative probability for all years before and including t, G(t). Ncis initial density and Nxis the quasi-

extinction threshold.

where

d=loge(Nc) −loge(Nx)

By integrating the time-specific equation from minus infinity to t, we get the following cumulative

function:

where

Φ( ) zy exp ( )dy

z

=

⎛

⎝⎜

⎞

⎠⎟

−

−∞

1

2

2

π

Gt dt

t

ddt

t

() = −−^ exp^

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

+ −

⎛

⎝⎜

⎞

⎠⎟

⎛−+

⎝

⎜

⎜

⎞

⎠

⎟

⎟

ΦΦμ

σ

μ

σ

μ

(^2) σ
(^22)
2
Pt d
t
dt
t
( ) = exp −+( )
⎡
⎣
⎢
⎢
⎤
⎦
⎥
2 23 2 ⎥
2
2
πσ
μ
σ
Box 17.1Equations for
calculating the
probability of extinction
in any given year tand
the cumulative
probability of
extinction.
0.0020
0.0015
0.0010
0.0005
0
0 20 40 60 80 100
Time
Probability density
Fig. 17.5The predicted
probability of extinction
in any future year,
based on the diffusion
equation approximation
for an exponentially
growing grizzly bear
population with
demographic parameters
identical to the
Yellowstone population
studied during
1959–82.