Environmental Engineering Reference
In-Depth Information
have explicit formulae defining the effects of the arrows for a food web like
Fig. 13.1 . Equation 13.1 is then able to define the long-term growth of the species
in the focal guild in terms of feedback loops from other species. The a ij are given by
formulae derived from this larger set of equations and depend specifically on how
species forage for their resources, how they are preyed upon by their predators, and
how these resources and predators change in density in response to density changes
in the focal guild [ 17 ]. It is important to note, however, that Eq. 13.1 does not apply
to short-term prediction, but to long-term outcomes such as long-term recovery
of a population from low density, or eventual extinction [ 17 ]. Although the
Lotka-Volterra competition equations are used generically in ecology, it should
be appreciated that they are very specific, and can only be used to illustrate general
principles, not to give precise outcomes in any given system. Fortunately, there are
means of matching Lotka-Volterra models with models for particular systems that
validate their use for general principles when treated appropriately [ 4 , 15 , 23 ].
The key issue to be addressed with the Eq. 13.1 is when species coexistence will
occur, and when some species will be excluded from the community. This is easy
and straightforward in the case of a guild of two species, and this case gives some
key general principles [ 17 ]. The fundamental result in the two-species case is that
species j can exclude i from a community if
a ij > a jj :
(13.2)
The reverse inequality means that species i is always able to invade the system
when dominated by species j. The mutual invasibility criterion [ 24 ] then says that
two species, labeled 1 and 2, will coexist stably whenever they are both able to
recover from low density in the presence of the other species. In the two-species
Lotka-Volterra equations, this criterion leads to the condition
a 11 > a 21 and a 22 > a 12 :
(13.3)
Fundamentally, this means that for stable coexistence, each species must depress its
own growth more strongly than it depresses the growth of the other species as it
increases in population density. It is a very simple and general criterion that ensures
stable coexistence [ 15 ]. If one of the inequalities in ( Eq. 13.3 ) is reversed, then one
species can exclude the other, and not vice versa. This means that one species
always drives the other extinct. On the other hand, if both inequalities are reversed,
then each species can exclude the other. This means that neither species can invade
a system consisting of the other species. Whichever species establishes first remains
the sole occupant in the guild in question in that locality.
The Lotka-Volterra competition Eq. 13.1 can be interpreted directly as meaning
direct interference of individuals of all species with individuals of other species,
harming them by reducing foraging time or in some cases by cannibalism or
intraguild predation , which refers to predation by one species in a guild by another
species in that guild [ 25 ]. In this case of direct interference, resource shortages or
predators need not have a role [ 9 - 11 ]. Indeed,
it
is in this form that
the
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