Biology Reference
In-Depth Information
(c) Do the equations you obtained look familiar? Compare them with
Eq. (1-12) in Chapter 1.
(d) List the parameters of your models. Explain their biological
meaning.
In Exercise 2-14, you should have found that each population grows to
its carrying capacity in a logistic fashion when undisturbed by
competitors. In the following exercise, we ask you to consider several
preliminary questions and examine a model describing a competitive
interaction.
E
XERCISE
2-15
6¼
(a) Let P
0. With r(N,P) denoting the per capita growth rate of N,
how do you expect r(N,P) to change when P increases? Why?
(b) Let N
0. With k(N,P) denoting the per capita growth rate of P,
how do you expect k(N,P) to change when N increases? Why?
6¼
(c) The following model may be used to describe competition between
N and P, where K and M are the carrying capacities for the
populations N and P respectively:
0
1
dN
dt
¼
N
þ
bP
@
A
N
r
ð
N
;
P
Þ
N
¼
a 1
K
0
1
(2-15)
þ
dP
dt
¼
P
gN
M
@
A
P
k
ð
N
;
P
Þ
P
¼
c 1
and k
N
þ
bP
P
þ
gN
M
where r
ð
N
;
P
Þ¼
a 1
ð
N
;
P
Þ¼
c 1
:
Notice
K
N (i.e.,
0, the first equation becomes
dN
N
K
that when P
¼
dt
¼
a 1
a logistic equation for N with carrying capacity K). The parameter
b measures the competitive effect of P on N, and the parameter g
measures the competitive effect of N on P. Describe the meaning of the
terms N
þ
þ
bP in r(N,P) and P
gN in k(N,P).
E
XERCISE
2-16
Show that the equilibrium states for the model defined by Eqs. (2-15) are:
K
bM
M
gK
(0,0), (0,M), (K,0), and
;
:
1
bg
1
bg
