Differential Equations: The First Steps - Elements of Scientific Computing - page 39

1

r R

<0;

r 0 .0/ D ar 0

and consequently the number of rabbits will decrease. In fact, by (2.19), it will
continue to decrease until we reach r D R:
To summarize, we have observed that if the initial number of rabbits is smaller
than the carrying capacity, then the number of rabbits will be monotonically increas-
ing but will never become larger than the carrying capacity. Similarly, if r 0

>R;

then r.t/ will be monotonically decreasing but will always satisfy r.t/ R: The

carrying capacity is referred to as an equilibrium point because if r.t / D R; then
r 0 .t / D 0, and the solution will remain equal to R for all t

>

t . >

Analytical Solution

To gain further insight about the rabbit population as predicted by (2.19), we can
solve the initial value problem
r 0 .t / D ar .t / 1

;

r.t/
R

(2.20)

r.0/ D r 0;

analytically. Since

D ar 1

;

dr
dt

r
R

we have

dr
r 1

D a dt ;

r
R

andbyintegrationweget

ln r
R r

D at C c;

where c is a constant that we have to determine using the initial condition. At t D 0;
we have

ln r 0
R r 0

D c

and thus

ln Rr
r 0
Rr 0

! D at ;

or

r
R r

r 0
R r 0

e at :

D

By solving this equation with respect to r; we get

r 0
r 0 C e at .R r 0 / R:

r.t/ D

(2.21)