2
RICHARD 9. MILLER
species of the same genus usually have, though by no means invariably,
much similarity in habits and constitution, and always in stmcture,
the struggle will generally be more severe between them, if they come
into competition with each other, than between the species of distinct
genera.” He cited examples of the increase of the missel-thrush at the
expense of the song-thrush in Scotland, invasions in which one species
of rat has displaced another, the spread of the Asiatic cockroach a t the
expense of a cogener, and the extermination of the native stingless bee
of Australia by the imported hive bee. Darwin also stated one of the
central problems of competition theory, which exists today as it did
then, when he said “We can dimly see why the competition should be
most severe between allied forms, which fill nearly the same place in
the economy of nature; but probably in no one case could we precisely
say why one species has been victorious over another in the great battle
of life. ’ ’
The well-known mathematical models that were subsequently developed by Volterra, Lotka and Gause from extensions of the logistic
theory, to describe population growth in single- and two-species systems,
were attempts to state in mathematical terms the events that naturalists such as Darwin had recorded from field observation. In logistic
theory, each individual that is added to a population N reduces the
growth capacity of the population by a constant increment. Population
growth is thus described by a sigmoid curve in which N approaches an
upper asymptote K , which represents the carrying capacity of the
environment. If two species with carrying capacities K , and K , comPete for the same resource in the same environment, and if an individual
of either species reduces the growth capacities of both populations, the
growth of the two species can be represented by the following equation
system :
where N , and N , are the numbers of each species, r1 and rz are their
respective rates of increase, and K , and K , are carrying capacities or
saturation values determined by growing each species alone in this
environment. The term B/K2 is the inhibitory effect of N , on the growth
of N , and a/K, is the reciprocal effect of N , on N,. The outcome of
competition will then depend on the inequalities tc > K,/K, and
B > K,/Kl.
RICHARD 9. MILLER
species of the same genus usually have, though by no means invariably,
much similarity in habits and constitution, and always in stmcture,
the struggle will generally be more severe between them, if they come
into competition with each other, than between the species of distinct
genera.” He cited examples of the increase of the missel-thrush at the
expense of the song-thrush in Scotland, invasions in which one species
of rat has displaced another, the spread of the Asiatic cockroach a t the
expense of a cogener, and the extermination of the native stingless bee
of Australia by the imported hive bee. Darwin also stated one of the
central problems of competition theory, which exists today as it did
then, when he said “We can dimly see why the competition should be
most severe between allied forms, which fill nearly the same place in
the economy of nature; but probably in no one case could we precisely
say why one species has been victorious over another in the great battle
of life. ’ ’
The well-known mathematical models that were subsequently developed by Volterra, Lotka and Gause from extensions of the logistic
theory, to describe population growth in single- and two-species systems,
were attempts to state in mathematical terms the events that naturalists such as Darwin had recorded from field observation. In logistic
theory, each individual that is added to a population N reduces the
growth capacity of the population by a constant increment. Population
growth is thus described by a sigmoid curve in which N approaches an
upper asymptote K , which represents the carrying capacity of the
environment. If two species with carrying capacities K , and K , comPete for the same resource in the same environment, and if an individual
of either species reduces the growth capacities of both populations, the
growth of the two species can be represented by the following equation
system :
where N , and N , are the numbers of each species, r1 and rz are their
respective rates of increase, and K , and K , are carrying capacities or
saturation values determined by growing each species alone in this
environment. The term B/K2 is the inhibitory effect of N , on the growth
of N , and a/K, is the reciprocal effect of N , on N,. The outcome of
competition will then depend on the inequalities tc > K,/K, and
B > K,/Kl.