REALISTIC MODELS IN POPULATION ECOLOGY
229
are the density functions for the n species. The growth, death and birth
submodels for each species are then modified to account for the presence
of the other species. For small values of n, this is a reasonable procedure,
but for larger n, especially with many complicated interactions, the
formulation process rapidly becomes impossibly complex. Were it
possible to complete the formulation, the large number of equations
would be intractable even with large-scale digital computers. One may
well ask: What are the possibilities of modeling large, complex, ecological systems with at least some degree of realism?
For very large numbers of species, even the somewhat simplified
population models introduced in the next aection will not be suitable.
One could employ the usual partitioning of an ecosystem into groups of
species. These would be producers, herbivores, carnivores and decomposers (see Hairston et al., 1960; and also Murdoch, 1966). The total
population of each group comprises a set of variables which are insufficient to characterize the system; however, the group populations,
together with the biomass of each group, and some other measures of‘the
group’s ability to reproduce and to utilize the available energy in the
system, may be more nearly adequate. If more detail is desired or
required, the groups could be subdivided into subgroups of species with
somewhat similar ecological characteristics. My thoughts on this subject
are obviously not well formed, and the above remarks are meant only to
stimulate discussion.
For smaller numbers of species the density function models may still
be mathematically intractable. To overcome the mathematical difficulties Streifer and Istock (1973) developed a “critical variable formulation”, which is based on the age-size density function, but contains less
detailed information concerning the population composition.
VI. CRITICAL VARIABLE FORMULATION
A. INTRODUCTION
The critical variables describe the attributes which distinguish individuals in a population. I n most of our discussion these have been age
and mass, but other variables have been considered including, for
example, time from conception for pregnant females in bisexual populations. I n this chapter I deal exclusively with age and mass as critical
variables; the extension to other variables is straightforward.
The population density function, 7, contains a great deal of information about a population; indeed, the knowledge of 7 at all times is all
that is required t o determine all aspects of the population. It is likely
229
are the density functions for the n species. The growth, death and birth
submodels for each species are then modified to account for the presence
of the other species. For small values of n, this is a reasonable procedure,
but for larger n, especially with many complicated interactions, the
formulation process rapidly becomes impossibly complex. Were it
possible to complete the formulation, the large number of equations
would be intractable even with large-scale digital computers. One may
well ask: What are the possibilities of modeling large, complex, ecological systems with at least some degree of realism?
For very large numbers of species, even the somewhat simplified
population models introduced in the next aection will not be suitable.
One could employ the usual partitioning of an ecosystem into groups of
species. These would be producers, herbivores, carnivores and decomposers (see Hairston et al., 1960; and also Murdoch, 1966). The total
population of each group comprises a set of variables which are insufficient to characterize the system; however, the group populations,
together with the biomass of each group, and some other measures of‘the
group’s ability to reproduce and to utilize the available energy in the
system, may be more nearly adequate. If more detail is desired or
required, the groups could be subdivided into subgroups of species with
somewhat similar ecological characteristics. My thoughts on this subject
are obviously not well formed, and the above remarks are meant only to
stimulate discussion.
For smaller numbers of species the density function models may still
be mathematically intractable. To overcome the mathematical difficulties Streifer and Istock (1973) developed a “critical variable formulation”, which is based on the age-size density function, but contains less
detailed information concerning the population composition.
VI. CRITICAL VARIABLE FORMULATION
A. INTRODUCTION
The critical variables describe the attributes which distinguish individuals in a population. I n most of our discussion these have been age
and mass, but other variables have been considered including, for
example, time from conception for pregnant females in bisexual populations. I n this chapter I deal exclusively with age and mass as critical
variables; the extension to other variables is straightforward.
The population density function, 7, contains a great deal of information about a population; indeed, the knowledge of 7 at all times is all
that is required t o determine all aspects of the population. It is likely