Biology Reference
In-Depth Information
determined in Exercise (2-16)
K
bM
M
gK
The equilibrium state
;
1
bg
1
bg
will be of interest only when both of its coordinates are non-negative.
There are various cases that need to be examined, and the computations
become technical and somewhat tedious. Complete classification of the
equilibrium states as stable or unstable will also require the examination
of all such cases. It can be shown that (0,0) is always unstable (see
Exercise 2-17) and that the other three equilibrium states could be stable
or unstable depending on the values of the parameters. Figure 2-24
gives examples that demonstrate convergence to the
2000
1800
1600
1400
1200
K
bM
M
gK
1000
states
;
;
(0,M), and (K,0), respectively. Convergence to
1
bg
1
bg
800
demonstrates coexistence of the
600
K
bM
M
gK
400
;
the state
200
1
bg
1
bg
0
A
0
100
200
300
400
500
600
populations, while convergence to either of the other states corresponds
to one of the populations dying out. Notice that the initial conditions, the
carrying capacities, and the values of the parameters a
Time
1200
1000
¼
0.04 are the same for all fours panels in the figure. The difference in the
long-term behavior of the models is due to the difference in the
competition parameters b and g.
¼
0.03 and c
800
600
400
200
0
0
100
200
300
400
500
600
B
E
XERCISE
2-17
Time
1200
Show that the equilibrium state (0,0) is always unstable, regardless of the
values of the parameters.
1000
800
600
400
200
E
XERCISE
2-18
0
0
100
200
300
400
500
600
C
Time
In the model defined by Eq. (2-15), for K
0.51,
prove that the equilibrium state (K, 0) is asymptotically stable, regardless
of the values of a, b, and c.
¼
2000, M
¼
1000, and g
¼
2500
2000
1500
1000
Before leaving this topic, we note there are many other models
describing competition among species. Though the model we considered
is quite simple, one could contemplate further refinements. The
methods we used to develop and analyze this model were familiar and
could also be used to develop models of symbiotic interactions
between species.
500
0
0
D
100
200
300
400
500
600
Time
FIGURE 2-24.
Numerical solutions for N (black line) and P (gray
line) of the competition model defined by
Eq. (2-15) with initial conditions N(0) ¼ 500 and
P(0) ¼ 300 and carrying capacities for the two
populations of K¼ 2000 and M ¼ 1000. The
values for a ¼ 0.03 and c¼ 0.04 are the same for
all four panels. Panel A: b ¼ 0.25, g ¼ 0.04; panel
B: b ¼ 1.5; g ¼ 0.04; panel C: b ¼ 2.7; g ¼ 0.04;
panel D: b ¼ 2.7; g ¼ 0.8.
VIII. APPENDIX: VALIDATION OF A
MATHEMATICAL CLAIM
The epidemic models models considered in this chapter were developed
assuming the groups are uniformly mixed and the amount of contact
between two groups is proportional to the product of the number of
