236
4. Stability Theory
r(1−N /K)/acN
1
−exp(−r(1−H /K))
N
H
*
*
*
*
*
1
1
N
*
2
FIGURE 4.30.
The nontrivial equilibrium points are given by
H
∗
=
λ
ln λ
(λ
− 1) ac
,
P
∗
=
1
a
ln λ.
It can be shown by linearization that (H
∗
, P
∗
) is unstable. Thus this model
is too simple for any practical applications except possibly under contrived
laboratory conditions.
It is reasonable to modify the H(n) equation (4.7.4) to incorporate some
saturation of the prey population, or, in terms of predator encounters, a
prey-limiting model. Hence a more realistic model is given by
H(n + 1) = H(n) exp
r
1
−
H(n)
k
− aP (n)
,
r > 0,
P (n + 1) = cH(n)(1
− exp(−aP (n))).
(4.7.21)
The equilibrium points are solutions of
1 = exp
r
1
−
H
∗
K
− aP
∗
,
P
∗
= cH
∗
(1
− exp(−aP
∗
)).
Hence
P
∗
=
r
a
1
−
H
∗
K
=
r
a
(1
− q), H
∗
=
P
∗
(1
− ¯e
ap
∗
)
.
(4.7.22)
Thus
r(1
−
H
∗
K
)
acH
∗
= 1
− exp
−r
1
−
H
∗
K
.
(4.7.23)
Clearly, H
∗
1
= K, P
∗
1
= 0 is an equilibrium state. The other equilibrium
point may be obtained by plotting the left- and right-hand sides of (4.7.19)
against H
∗
. From Figure 4.30 we see that there is another equilibrium point
with 0 < H
∗
2
< K. Then we may find P
∗
2
from (4.7.18).