Geoscience Reference
In-Depth Information
prey that were driven entirely by the interaction between the two species. The
model is
dV
/
dt
=
bV
-
aVP
(8.6)
dP
/
dt
=
cVP
-
dP
(8.7)
where
b
is the potential growth rate for the prey in the absence of predation,
a
is the attack rate,
c
is the rate of amelioration of predator population decline
afforded by eating prey, and
d
is the per capita death rate for the predator in
the absence of prey. The right-hand portion of the prey equation (equation
8.6) models the rate at which prey are removed from the population by preda-
tion. The product of
a
´
V
is often called the functional response. Note that
in the first portion of the predator equation we see a similar function of
V
´
P
that models how the rate of predator decline is ameliorated by the conversion
of prey into predator population growth. This portion of the model,
c
´
´
V
P
,
is what we usually call the numerical response.
Although Volterra developed his model independently from basic princi-
ples, an American, Alfred J. Lotka (1925), had already suggested the same
mathematical structure for two-species interactions and presented a full math-
ematical treatment. Lotka was quick to advise Volterra of his priority (Kings-
land 1985). Consequently, most ecologists call the two-species system of dif-
ferential equations the Lotka-Volterra models. Nevertheless, Volterra devel-
oped the analysis of predator-prey interactions in more detail, offered more
examples, and published in several languages, doing much to bring attention
to the approach.
Despite the valuable insight that this simple model affords, the Lotka-
Volterra model has been mercilessly attacked for its unrealistic assumptions
and dynamics (Thompson 1937). The dynamics include neutrally stable oscil-
lations with period length,
T
≈
p
√
bd
, for which the amplitude of oscillations
depends on initial conditions (Lotka 1925). Assumptions include a linear
functional response that essentially says that the number of prey killed per
predator will increase with increasing prey abundance without bound. Yet at
some level we must expect that the per capita rate at which prey are killed
would level off because of satiation or time limitations (Holling 1966).
Another assumption is that neither the predator nor prey has density-depen-
dent limitations other than that afforded by the abundance of the other
species. Furthermore, we have a number of assumptions that are symptomatic
2
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