From: Alan McGowen <alanm@igc.org>
Newsgroups: sci.environment
Date: 12 Apr 92 00:47 PDT
Subject: ECO CENTRAL 9
Nf-ID: #N:cdp:1466601325:000:3908
Nf-From: cdp.UUCP!alanm    Apr 12 00:47:00 1992


ECO CENTRAL 9

                    Environmental Stochasticity
                    and Minimum Viable Populations

Real environments fluctuate, and this induces fluctuation in the 
parameters of population models such as intrinsic growth rate r 
and carrying capacity K. To incorporate environmental 
fluctuation, we replace the deterministic models of a population 
size as a function of time, N(t), with a probability distribution 
f(n,t), the probability that the population is of size n at time 
t. Just as the population models we have looked at previously had 
equilibrium solutions Ni*, so in stochastic models there are 
equilibrium probability distributions f*(n). The strength of the 
environmental fluctuation is measured by the coefficient of 
variance, CV, of the distribution f*, also called sigma^2. 

In the deterministic case, we saw in ECO CENTRAL 7 that the 
stability of the equilibrium was controlled by the real part of 
the eigenvalues of the community matrix: if the largest of these 
was negative the equilibrium is neighborhood stable:

1) lambda > 0 where -lambda = max[real(Li)],
   the Li given by solving det|A -LI| = 0, A is the community matrix.

The stability of the stochastic models requires that lambda > CV. 
Otherwise, if CV > lambda, the fluctuations bring the model into 
regions of instability, and the population will eventually become 
extinct. If lambda < CV, we say that environmental stochasticity 
is strong, and if lambda > CV, we say that it is weak. The 
requirement

2) lambda > CV

for stability of a population model in the face of environmental 
fluctuation is that we have only weak environmental 
stochasticity. When environmental stochasticity is strong

3) lambda <= CV

the population will eventually become extinct.

Stochastic models are produced from deterministic ones by way of 
the diffusion equation:

4) Df(n,t)/Dt = -D/Dn (M(n)f(n,t)) + (1/2)D^2/Dn^2 (V(n)f(n,t))

[here D is partial differentiation]

where M(n) = <F(n)> is the expected value of F, and
V(n) = <(F(n) - M)^2> is the variance. F is the dynamical
function of the deterministic model dN(t)/dt = F(N(t)).

The equilibrium probability distribution (if it exists) is found 
by setting 4) to zero:

5) 0 = -d/dn(M(n)f*(n)) + (1/2)d^2/dn^2 (V(n)f*(n))

The generalization to m species is straightforward. For the 
deterministic model dNi(t)/dt = Fi(N1,N2,...,Nm), define 
Mi(n1,n2,...nm) = <Fi>, and Vij(n1,n2,...nm) = <(Fi - Mi)(Fj - 
Mj)>, and the diffusion equation becomes

6) Df/Dt = - SUM(i=1,m) D/Dni (Mi f) 
           + (1/2) SUM(i,j=1,m)D^2/DniDnj(Vij f).

Again, the equilibrium distribution, if it exists, is found by 
setting 6) to 0. 

The stability condition is

7) lambda > V(n)

For biologically realistic models, V(n) is often a decreasing 
function of K, or lambda an increasing function of K. If so, there is 
a value of K called MVP such that

8) lambda <= V(n)  when K < MVP
   lambda > V(n)   when K > MVP

i.e. the population will become extinct if it is smaller than MVP 
and will not if it is greater. MVP is the *minimum viable 
population*. Determining an MVP is known as *population 
vulnerability analysis.* The MVP is attained in a habitat with a 
carrying capacity equal to the MVP. Other things being equal 
(i.e. assuming constant habitat quality), carrying capacity is 
proportional to the area of the habitat -- to increase K, 
increase the area. Ensuring that a population does not fall 
beneath MVP means choosing a preserve size large enough that its 
carrying capacity is >= MVP. Such an analysis, for example, has 
been done for the Northern Spotted Owl [in the Jack Ward Thomas 
report]. Of course, *protecting* the required preserve size is a 
political problem. When it cannot be done, eventually human 
pressure on the habitat will cause the K to reduce below MVP, and
extinction is almost inevitable.   

-----------
Alan McGowen
    

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