Biology Reference
In-Depth Information
A. Predator-Prey Models
Suppose we have two species, one of which is a prey for the other. In this
model, we call the predator owls, O, and the prey voles, V. The prey
feeds on an environmental resource, and, if there were no predators, we
assume their numbers would grow according to a single population
model (see Chapter 1). We also assume exclusive predation, meaning the
prey provides the exclusive food resource for the predator. If there were
no prey, the predators would die at a constant per capita rate, and their
population would collapse into extinction.
E
XERCISE
2-10
Under the conditions described above, what are some ways predator
and prey populations could evolve?
1. The Lotka-Volterra Model
If predator and prey are both present, contact between the groups will
provide a food source for the predator, allowing it to extend its survival
and possibly increase, but will cause the population of the prey to
decrease. The first scientists to describe the mathematics of this model
were Alfred James Lotka and Vito Volterra, who studied the problem
independently in the 1920s. The Lotka-Volterra model describes the
dynamics of the interactions between owls and voles as follows:
dV
dt
¼ a
V
g
VO
¼ða g
O
Þ
V
(2-11)
dO
dt
¼d
þ e
¼ðd þ e
Þ
;
O
OV
V
O
where
O) has the
meaning of the net per capita growth rate for the vole population and
that (-
a
,
g
,
d
, and
e
are positive constants. Notice that (
a
-
g
V) is the net per capita growth rate for the owl population.
Thus, the parameters
d þ e
a
,
g
,
d
, and
e
have the following biological
meanings:
a ¼
net per capita growth rate for the voles in the absence of owls.
d ¼
per capita death rate of the owls if there is no food (i.e., voles).
g
per capita death rate of the voles because of predation. The
parameter
O
¼
g
represents the effectiveness of the owls as hunters.
e
per capita birth rate for the owls because of the availability of food
resources (the voles). The parameter
V
¼
represents the per capita rate at
which the increase of the vole population contributes to increasing the
owls' per capita birth rate.
e
