From: Alan McGowen <alanm@igc.org>
Newsgroups: sci.environment
Date: 05 Apr 92 22:25 PDT
Subject: ECO CENTRAL 5
Nf-ID: #N:cdp:1466601311:000:5626
Nf-From: cdp.UUCP!alanm    Apr  5 22:25:00 1992


ECO CENTRAL 5

              Demographic Stochasticity

Populations grow and shrink by individual births and deaths. 
There is a branch of the theory of Markov processes which deals 
with such *birth and death processes*, which are characterized by 
a matrix of transition probabilities Qij (the probability that a 
population of size i becomes one of size j in an interval of time 
dt) such that

Qij  =  b(i) dt    if j = i+1
     =  d(i) dt    if j = i-1
     =  1 - d(i) dt - b(i) dt if j = i
     =  0        otherwise

where b(i) is the birth rate given a population of size i
and d(i) is the death rate given a population of size i. The 
matrix Qij says that the population grows or shrinks by units 
of one or zero in each interval dt.

If the population should ever make a transition to size 0 it will 
remain there forever, since zero individuals cannot reproduce. 
The state 0 is called an *absorbing state* of the Markov process 
for this reason. The absorbing state zero is the state in which 
the population has become extinct.

A simple continous time model of a birth-death process can be 
obtained for a population of size N by letting the probability of 
a birth in the interval dt be Nldt and of a death Nmdt (birth and 
death probabilities are here assumed to be density independent). 
If the population is of size N at t, its size at t+dt is either 
N-1,N, or N+1, and

p(N,t+dt) = p(N-1,t)l(N-1)dt      the population was at N-1
            + p(N,t)[1 -Nldt -Nmdt]   the pop was at N
            + p(N+1,t)m(N+1)dt    the pop was at N+1

thus,

0) dp(N,t)/dt = N(l+m)p(N,t) +l(N-1)p(N-1,t) +m(N+1)p(N+1,t)

The expected size of the population in this model is N0exp(l-m)t,
i.e. we have simple exponential growth as would be expected since 
no density dependence has been included.

The probability that a population of initial size N0 will become 
extinct by time t is given by [Pielou]

1) p0(t) = [(m exp(l-m)t - m)/(l exp(l-m)t - m)]^N0

for m < l, lim t-->inf p0(t)= lim[(m exp(l-m)t)/(lexp(l-m)t)]^N0
by L'Hopital's rule, and this is   

2) (m/l)^N0

if l < m as t-->infinity the exponential terms of 1) vanish and 
p0(t) --> 1. But by 2), even if l > m -- even if the population 
is growing -- there is still a finite chance of extinction due to 
stochasitic effects. The larger the initial population N0, the 
smaller this chance is. Even a growing population, as it 
fluctuates in the birth and death process, has a chance of 
becoming extinct. Such fluctuations are called *demographic 
stochasticity*.

What of a stable population, i.e. one for which m = l? Setting
r = l-m and expanding the exponentials in 1) as series, we get

p0(t) = {[m (rt +O(r^2))]/[l(1 +rt + O(r^2)) -m]}^N0

as r->0 discard the terms of O(r^2) and we get

p0(t) --> [mrt/(l-m +lrt)]^N0 = [mrt/(r + lrt)]^N0 =

[mt/(1 + lt)]^N0 --> [lt/(1 + lt)]^N0 for l = m. As

t-->  infinity,  this  approaches 1.  Thus even for a  population 
which is stable in expected size, its eventual fate is to succumb 
to demographic stochasticity,  in this simple model.  However, it 
is important to ask what the expected time to extinction is.

Extinction by demographic stochasticity can be thought of in 
terms of a run of coin tosses: if the probabilities of birth and 
death are each 1/2, the extinction of a population of initial 
size N0 is like a run of coin tosses in which the number of heads 
exceeds that of tails by N0. The expected length of such a run 
grows very rapidly with N0, in fact, even moderate sized 
populations would take longer to become extinct by demographic 
fluctuation alone than they would be expected to persist on other 
grounds.   

When modeling geometric growth up to carrying capacity K, a 
population with intrinsic rate of increase r will on average last 
the following number of generations (starting from size K) 
[Diamond]:

 |   K    10        20        30        40
--------------------------------------------------------
r| 

0.1       46        197       925       4,897

0.2      109       2,844     103,750    4,386,342

0.3      414      92,184     29,218,650 1.057 E10

0.4     2,312     5,222,731  1.64 E10   5.843 E13


Actually, populations fluctuate more than this because of
environmental fluctuation, this so the above table overestimates 
the times to extinction, because of environmental fluctuations. 
When these are taken into account, the expected lifetime 
increases more slowly with K. [Diamond] However, this table makes 
it clear that extinction by demographic stochasticity is not a 
significant risk for populations of hundreds or thousands. 
However, environmental fluctuations can bring such populations 
down to levels at which extinction by demographic fluctuation is 
possible.

Small populations with r close to 0 are at great risk from 
extinction by demographic stochasticity. For example, 10 ha 
patches of Amazonian forest can support only one or two groups of 
howler monkeys consisting of about 10 individuals in each group. 
These monkeys, which have r (birth rate - death rate) close to 0, 
can last only a few generations in such fragments. [Lovejoy]
    
Refs.

Diamond, J. "Normal" extinctions of isolated populations. In 
_Extinctions_, ed. Matthew H. Nitecki,University of Chicago 
Press, Chicago, 1984.

Lovejoy, T.E. et al. Edge and other effects of isolation on 
Amazon Forest Fragments. in _Conservation Biology: The Science of 
Scarcity and Diversity_, ed Michael E. Soule, Sinauer, 
Sunderland, Massachusetts, 1986.

Pielou, E. C. _An Introduction to Mathematical Ecology_, Wiley-
Interscience, New York, 1969.
----------
Alan McGowen