From: Alan McGowen <alanm@igc.org>
Newsgroups: sci.environment
Date: 11 Apr 92 20:06 PDT
Subject: ECO CENTRAL 8
Nf-ID: #N:cdp:1466601322:000:4250
Nf-From: cdp.UUCP!alanm    Apr 11 20:06:00 1992


ECO CENTRAL 8

          The Community Matrix and Some Terminology

The community matrix, presented in ECO CENTRAL 7, contains a 
great deal of biological information. Recall, this is the matrix

1)  aij  =  partial of Fi wrt Nj  evaluated  at  the  equilibrium 
population sizes Ni

where 

2) dNi/dt = Fi(N1,N2,...,Nm) gives the population dynamics.

Buried in the community matrix is the trophic structure -- all 
the food webs -- that thread the community together. 

Much information can be read from the signs of the elements aij 
alone. For any pair of species, the possibilities are:

      aji
aij   ++     +0     +-
      0+     00     0-
      -+     -0     --

where the first sign is the effect of species i on species j 
(aij), and the second that of species j on i (aji).

These interactions give us six fundamentally different kinds of 
ecological relationships between species:

*Competition* (--) occurs when the two species populations affect 
each other adversely. Competitors are in negative feedback in the 
community. 

*Predation* (+- and -+) occurs when one population affects 
another adversely but itself benefits from the interaction.

*Parasitism* is also a +- type of interaction, except that the 
host organism is not killed outright but exploited over a period 
of time.

*Mutualism* (++) occurs when species benfit from each other. It 
sometimes happens that this beneficial relationship is *obligate*
so that either partner perishes without the other. Obligate 
mutualists are also called *symbionts*, especially when they must 
live in close physical proximity. Mutualists are in 
positive feedback in the community.

*Commensalism* (+0 and 0+) occurs when one species benefits from 
the other which is not affected by it. An example is moose and 
beavers: moose benefit from beavers because the latter's dams 
increase the food supply of the moose, who eat water plants. 
Beavers do not benefit from moose. Moose are thus commensals of 
beavers. Cockroaches, house mice and starlings are commensals of 
humans. In the older literature, commensals are also called 
*inquilines*.

*Amensalism* (-0 and 0-) occurs when one species affects another 
adversely which is not affected by it. Wolves are clearly 
amensals of humans, at least of agricultural and urbanized 
humans. In fact, most species are human amensals, which is why 
there is a mass extinction taking place today.

*Neutralism* (00) is the boring case of no interaction.

The community matrix of a population model 1) and 2) describes 
the *codynamics* of a community, whose equilibrium states are 
*codynamical equilibria.* It is also possible to formulate models 
not of population dynamics, but of changes in gene frequencies 
under the action of natural selection (which will be the subject 
of several future ECO CENTRALs): these are models of evolution. 
Such models can be formulated either for a single species, ala 
the logistic equation, or for interacting populations. When the 
latter is done, an analogue of the community matrix emerges as 
central to the dynamics:

3) aij = Ni* partial of Wi wrt Nj evaluated at Ni* for all i

where Ni* is as above the equilibrium population size of species 
i and Wi is the Wright function for *mean fitness*, which also 
defines the adaptive topography. The dynamics which arises from 
this matrix is called *coevolution*. Here, too, there are 
potentially stable equilibria of the W's, and these are called 
*coevolutionary equilibria* or *coadaptation*. The aij's defined 
by 1) are "tuned" by coevolution governed by 3). 

Actually, almost all evolution is coevolution since every species 
evolves as part of a community, and evolution is constrained by 
community interactions -- by the community matrix. Although the 
concept of coevolution was anticipated by Darwin (what concept in 
evolution wasn't?), the first modern paper on the subject was by 
Paul Ehrlich and Peter Raven who described the coevolution of 
butterfles and plants. [Ehrlich, P.R. and P.H.Raven, 1965. 
Butterflies and Plants: A study in co-evolution. Evolution 18: 
586-608.]  I believe that the title of that paper coined the term 
coevolution, which is now no longer written with a hyphen. 

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