18
RICHARD S. MILLER
As long as the phenomenon of competition is described only by its
outcome and in the abstract language of the Lotka-Volterra equations,
we are liable to overlook the important fact that any interaction between two individuals or populations is an event which has dimensions
of both space and time. Neither the broad, functional classification nor
the total ecological characterization into mutually exclusive niches is
capable, without modification, of expressing these properties of competition. A partial solution was offered by Elton and Miller (1954) when
they proposed the term “arena” to describe a locus “within which some
temporary relation between the species of animals is arrived at, by a
combination of mutual opposition and various degrees of symbiosis”.
This was an attempt to define species interactions in such a way that
the total ecological characterizations of species express potentials which
may overlap in time and space, leading under appropriate conditions
to interspecies competition.
A more satisfactory, formal description of the relationships between
the ecological niches of potential competitors has been developed by
Hutchinson (1957) using conventional notations of set theory. This
approach allows us to define a niche in terms of all variables relative
to the species, regardless of their exact nature or quality, and we may
at the same time follow the usual set-theoretic practice of representing
an infinite number of variables in a restricted number of dimensions.
Hutchinson’s formalization also includes both the fundamental niche
that might be expressed in the absence of competitive interactions and
the realized niche that is occupied when competition restricts the
expression of the total species potential.
If we consider the total array of variables which limit the survival
of a species S,, we may define an n-dimensional hypervolume N, which
is the fundamental niche of that species. The fundamental niche, as so
defined, may be regarded as a set of points in an abstract N space which
will completely define the ecological properties of S,. Secondly, if B is
a limited volume of physical space comprising the biotope of a collection
of species S,, S,, . . . S,, the biotope is complete relative to S, if all the
points in N, are represented in B. If N, and N, are two fundamental
niches they may have no points in common and are therefore separate,
or they may have points in common and are said to intersect. N,.N, is
the subset of points common to N, and N, and is their intersection
subset.
The following restrictions apply to this mode of expression:
(1) It is assumed that all points in each funda.menta1 niche imply
equal probability of survival, and all points outside the niche imply
zero survival of the relevant species.
(2) It is also assumed that all environmental variables can be linearly
RICHARD S. MILLER
As long as the phenomenon of competition is described only by its
outcome and in the abstract language of the Lotka-Volterra equations,
we are liable to overlook the important fact that any interaction between two individuals or populations is an event which has dimensions
of both space and time. Neither the broad, functional classification nor
the total ecological characterization into mutually exclusive niches is
capable, without modification, of expressing these properties of competition. A partial solution was offered by Elton and Miller (1954) when
they proposed the term “arena” to describe a locus “within which some
temporary relation between the species of animals is arrived at, by a
combination of mutual opposition and various degrees of symbiosis”.
This was an attempt to define species interactions in such a way that
the total ecological characterizations of species express potentials which
may overlap in time and space, leading under appropriate conditions
to interspecies competition.
A more satisfactory, formal description of the relationships between
the ecological niches of potential competitors has been developed by
Hutchinson (1957) using conventional notations of set theory. This
approach allows us to define a niche in terms of all variables relative
to the species, regardless of their exact nature or quality, and we may
at the same time follow the usual set-theoretic practice of representing
an infinite number of variables in a restricted number of dimensions.
Hutchinson’s formalization also includes both the fundamental niche
that might be expressed in the absence of competitive interactions and
the realized niche that is occupied when competition restricts the
expression of the total species potential.
If we consider the total array of variables which limit the survival
of a species S,, we may define an n-dimensional hypervolume N, which
is the fundamental niche of that species. The fundamental niche, as so
defined, may be regarded as a set of points in an abstract N space which
will completely define the ecological properties of S,. Secondly, if B is
a limited volume of physical space comprising the biotope of a collection
of species S,, S,, . . . S,, the biotope is complete relative to S, if all the
points in N, are represented in B. If N, and N, are two fundamental
niches they may have no points in common and are therefore separate,
or they may have points in common and are said to intersect. N,.N, is
the subset of points common to N, and N, and is their intersection
subset.
The following restrictions apply to this mode of expression:
(1) It is assumed that all points in each funda.menta1 niche imply
equal probability of survival, and all points outside the niche imply
zero survival of the relevant species.
(2) It is also assumed that all environmental variables can be linearly