From: Alan McGowen <alanm@igc.org>
Newsgroups: sci.environment
Date: 11 Apr 92 00:00 PDT
Subject: ECO CENTRAL 6
Nf-ID: #N:cdp:1466601315:000:5939
Nf-From: cdp.UUCP!alanm    Apr 11 00:00:00 1992


ECO CENTRAL 6

                    Competition

Competition occurs when the fitness of one organism is a 
decreasing function of the fitness of another. Often the fitness 
of another organism is more or less fixed. In that case, it is 
primarily the *abundance* of the other type of organism which 
decreases the given organism's fitness. The other "type" of 
organism may be conspecifics of the organism or members of 
another species. Both intra- and inter- specific competition 
occur. 

We already have seen a simple model of intraspecific competition: 
the logistic equation. Recall, this is

1) dN/dt = r0(1 - N/K)N.

This is a special case of the most general population model for a 
single species:

2) dN/dt = f(N)

if we expand f(N) in MacLaurin series, we get

3) dN/dt = f(0) + N f'(0) + N^2 f"(0)/2 + O(N^3)

when N = 0, this gives dN/dt = f(0). But since a population of 
size 0 cannot increase or decrease, f(0) = 0. Thus every possible 
population model is of the form

4) dN/dt = r(N)N = N(f'(0) + Nf"(0)/2 + O(N^2))

where r(N) = f'(0) + Nf"(0)/2 + ...

when we ignore all terms above f'(0), r(N) = r(0) = r0 and r is 
independent of the density N. That model is dN/dt = r0, which 
gives exponential growth. The simplest density-dependent model 
includes the next term, and its biologically plausible form is 
the logistic equation. Obviously, we could build models with more 
terms, the general case being given by 4).

In the logistic equation 1), if you add one more member of the 
species

r(N+1) = r0(1 - (N+1)/K) = r0(1 -N/K -1/K) = r(N) - r0(1/K)

the growth rate is decreased by r0/K. This is the extent to which 
one member of the species competes with another. But suppose that 
we add, not another member of the *same* species, but of a 
*different* species. r(N) will still be decreased, but by a 
different amount, say  ar0/K. a measures the competition between 
the two species, and is called a *competition coefficient*. 

We can write down a generalization of 1) for two competing 
species, species 1 and 2:

  
5) dN1/dt = r1(1 - N1/K1 - a12 N2/K1)N1
   dN2/dt = r2(1 - N2/K2 - a21 N1/K2)N2

5) is the Lotka-Volterra competition equation.

As with the predator-prey models, we can look for equilibrium 
solutions by setting the derivatives to zero and solving for N1 
and N2. When we do this, the r1 and r2 drop out of consideration 
and we are left with a pair of linear equations:

6)   K1 - N1 - a12N2 = 0
     K2 - N2 - a21N1 = 0  

For various values of the K's, N's and a's, only four solution 
cases emerge:

Case 1:  K1 > K2/a21, K2 < K1/a12 or a21 > K2/K1, a12 < K1/K2
Case 2:  K1 < K2/a21, K2 > K1/a12 or a21 < K2/K1, a12 > K1/K2
Case 3:  K1 < K2/a21, K2 < K1/a12 or a21 < K2/K1, a12 < K1/K2
Case 4:  K1 > K2/a21. K2 > K1/a12 or a21 > K2/K1, a12 > K1/K2

When K1 = K2 the analysis is very simple. In this case, we have

C1 : a21 > 1, a12 < 1
C2 : a21 < 1, a12 > 1
C3 : a21 < 1, a12 < 1
C4 : a21 > 1, a12 > 1

A graphical analysis of 5) and 6) reveals that in C1, species 1 
excludes species 2, and in C2 species 2 excludes species 1. In 
C3 the two species coexist, and in C4 either species can exclude 
the other, depending upon which happens to have the greater 
initial abundance. When K1 != K2, similar results are obtained 
but scaled by the carrying capacitities. For more details, see 
[Roughgarden] or [Emlen].

The net result of all this is that competing species can exclude 
each other under sufficient competition. Many laboratory 
experiments in competitive exclusion have been done, and it is 
easy to arrive at the result that one species vanishes, and 
rather harder to arrange things so that both species persist. 

All of this led a fellow named G.F. Gause to offer up a "law", 
now known as the *competitive exclusion principle*, which says 
that if two species use exactly the same resources, and hence 
have a12 = a21, and both competition coefficients are strong 
(i.e. > 1), we are in case 4) above, when only one species 
persists -- hence, *species that use the same resources cannot 
coexist*. This led to the pre-Hutchinson definition of the niche: 
one species at a locale = one niche, since competitive exclusion 
would guarantee that if two species exploited the same niche, 
only one would be present. This "law" became something of a dogma 
for quite a while. 

Environmental stochasticity has an interesting effect on the 
stability of competitive systems: species that cannot coexist in 
a constant environment can often coexist in fluctuating 
environments. Species that ought to exclude each other according 
to the above analysis -- and *do* exclude each other in the lab 
-- can coexist in fluctuating reality. Environmental fluctuation 
has a benign side, it is not solely a grim business of increasing 
the chances of extinction: it can permit competitors to coexist. 
Intuitively, as an environment fluctuates, it passes through the 
niche center or optimal requirements of each species (we assume 
that there is still *something* about the environment that can
select for one species in preference to the other -- Gause 
wasn't all wet!) At each "pass through" the favored species gets 
a boost. Under a regime of sufficient "boosts" for each, they can 
coexist, case 4) notwithstanding. 

In a future ECO CENTRAL I hope to discuss the modern ideas of 
"limiting separation" and "niche breadth" established by 
competition in a fluctuating environment. These are tremendously 
interesting concepts, because they determine the maximum number 
of niches which can be crammed into a community, and thus 
potentially place an upper limit to the evolution of diversity. 

-------
Alan McGowen
          
R
Refs

Emlen, John Merit _Population Biology: the Coevolution of Population
Dynamics and Behavior_, NacMillan, New York, 1984

Roughgarden, Johnathan _Theory of Population Genetics and Evolutionary
Ecology: An Introduction_, MacMillan, New York, 1979


Newsgroups: sci.environment