N O TIZEN
132
NOTIZEN
An analytic solution o f the L o t k a V o l t e r r a
equations
E.
H
o f e l ic h
A
reads:
oo n —1
x(t ) = x ( 0 ) e£'f
(x(O)e*‘0 A( —y(0)e «**)*
and
H
21 b, 132
y ( t ) = y ( 0 ) e ~ s‘(
( - y ( 0 ) e - ‘* 0 AX n,h(t)
(3 b)
where
[1969] ; ein gegan gen am 28. Juni 1968)
K n .h
It is usual to solve nonlinear differential equations
with aid of an analog computer, a procedure yielding
results that prove to be satisfactory for most purposes.
Although, in general, analytic solutions to nonlinear
differential equations are hard to obtain, such solutions
would, nevertheless, have the advantage that they could
be evaluated ivithout a computer.
In connexion with a problem of laser physics we
have solved the following system of two coupled non­
linear differential equations of first order 1
x = —a x + c x y ;
V= - b y - d x y ;
a !> 0, c > 0,
6 > 0, d > 0 ,
(1)
deriving the exact solution as well as a rather simple
approximate solution by means of G r ö b n e r ’s method
of L i e series 2. Since we have nowhere made use of the
assumption that all coefficients are positive, these solu­
tions are valid for arbitrary real coefficients as well.
To the problems, which may thus be solved, belong
also the L o t k a - V o l t e r r a equations 3
x — exx —x y ,
y = - £ 2 V+xy;
> 0,
e2 > o .
(2 )
Although for system (2) the corresponding differen­
tial equation for the trajectories in the phase plane can
be solved in closed form, explicit solutions
and
y( t ) have to the authors’ knowledge nowhere been
given. In view of the importance of the L o t k a - V o l ­
t e r r a equations as one of the basic models in m athe­
matical ecology3 and for the study of oscillating
reactions 4, connected esDecially with recent discoveries
in biochemistry, explicit solutions to (2) should be of
particular interest for biochemists and biologist.
Since the details may be found in our cited paper *,
we will give here only the result. The solution to (2)
1 E. H o f e l i c h - A b a t e and F. H o f e l i c h , Z. Physik. 209, 13
[1968],
2 W. G r ö b n e r , Die L i e - Reihen und ihre Anwendungen, 2 .
Aufl. Berlin: VEB Deutscher Verlag der Wissenschaften,
1967.
Theorie Mathematique de la lutte pour la vie,
Gautiers-Villars, Paris 1931.
3 V . V o lte r r a ,
(3 a)
h K n, h ( t )
oo «—1
1 + 2 2 (x(Q) e£'t) n~ h
n
— h—0
«=1
o f e l ic h
Battelle Institute, Advanced Studies Center,
Geneva, Switzerland
(Z. Naturforschg.
2
n=1h=0
bate
Institut de Physique Theorique, University of Geneva,
Geneva, Switzerland
F.
i+ 2
(■t )
=/
d rh
. . ./
0
d rt
U
‘*12 2 y j?..*
2
s h= /« + l
s* = 3 s ,= 2
exp (e2 T n ) • .. exp ( - Et
. .. exp ( —
Tsi). ■• exp ( —ej r*,). . . exp (e, r x) ,
n
h ^ 1
t
T2
Xn.o (t) = f dr„ . . . J d r t exp (e2 r n) . .. exp (e2 Tj)
0
T s„ ) ..
.
(4 a)
0
(4 b)
and
7sh--s! = (h + l ) n~ Sb( s h - h )
(sh- i - Ä - f 1 ) .. .
2*.-*1- i ( 5 1- l ) ; n > h
(5)
h (t) has the same form af (4), but with the sub­
stitution
£l
^
—
82 9
£2
^
—
^1
•
The solutions (3) are rather cumbersome to deal with
because of the complicated expressions -Kn.h(i) and
K n,h (i) , although these coefficient functions do not
depend on the initial conditions and the integrations
involved can be carried out. In practice, however, it
in —1'
I h
integrals in (4) by an average integral. K n.h and
X n,h are then approximated to:
1
K n,h(t)~ -n j y
/ \ - e ~ s A h ! e ^ t - l \ n —h
— — j
\
e2
)
—hf
eet t —1
5 n.h
(6 a)
h
(6 b)
and
1 —e —
with
h+ 1
2
s„.h= K= 1
n+1
(7)
A. J. L o t k a , Elements of Mathematical Biology, Dover
Publications, New York 1956.
4 I. P r i g o g i n e and R. B a l e s c u , Acad. roy. Belgique, Bull. CI.
Sei. 42. 256 [1956].
J. H i g g i n s , Ind. Eng. Chem. 59, 19 [1967].
B. C h a n c e , K. P y e . and J. H i g g i n s . TEEE Spectrum 4, 79
[1967].
Unauthenticated
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