V. Volterra had treated the problem of struggle for existence in his book based on the biological interest. Here we will consider some models of struggle for ...
Proc.
854
191.
Japan
Acad.,
47
(1971)
[Vol.
Boltzmann Equation on Some Algebraic Concerning Struggle for Existence By Yoshiaki ITOH of Mathematics, Tokyo Institute
Department (Comm.
by Kinjiro
KuNucu,
47,
Structure
of Technology
M. J. A., May 12, 1971)
V. Volterra had treated the problem of struggle for existence in his book based on the biological interest. Here we will consider some models of struggle for existence. Although the equations are essentially the same to the Volterra's case, the formulations is different from it. The mathematical formulation is done analogously to the case of Kac's caricature of a Maxwellian gas. The Boltzmann equation on some finite algebraic structure will be mentioned. Model I. 1) There are three types of particles, A, B and C, whose numbers are respectively nA, nB and nc. nA+ nB+ nc = n is a constant integer. 2) Each particle is in a chaotic bath of like particles. Uniform distribution of colliding pairs is assumed. 3) Particles vary by the following collision rule : ® ® ® ® 0 0 /\
/\
~~
®
®
o
®
®
o
®
®
®
®
0
0
(1) ® /\
/\ ~~ ® ® ® 0 0 that is to say, A is stronger than B, B is stronger than C and C is stronger than A. Mathematical formulation. On the analogy of Boltzmann's problem, we may have a nA= nA nA+ nBnA+ nA nB- nA t n n n (2)
at °~ n B=nBn +ncn m-" -n n B n B n c B an =ncn+nAn+ncn-flu. at c n c n c n A (2) is rewritten in the following way for sufficiently large n.
c
SuppLi
Boltzmann
Equation
on Some
nA , nB , nC n n n (3) at
Algebraic
= (PA, PBS PC)
= PA2+ PBPA + PAPB - PA A
a P at A = PBZ+ PcPB + PBPc(4)
855
Structure
a P at c = Pct + PP (4) can y
PB
c + PCPA - Pc. from
thederived bemaster more rigorouslequation
the analogy of the way in the book of M. Kac [1]. The collision rule can be represented by the following structure, EA o EA = EA EA o EB = EB o EA = EA (5)
EB o EB=EB Ec o Ec = Ec We can consider a linear sents the state of the system. Define
EB o Ec=Ec OEB=EB Ec o EA = EA o Ec = Ec. space PAEA + PBEB +PcEc,
on
algebraic
which
repre-
(6) (PAEA + PBEB + PcEc) ° (PEA + PBEB + PE) so as to satisfy the distribution law and (5), we find that the product means "quasi-convolution" on a algebraic structure (5). Since the operation is not associative, we are not able to say "convolution". By (5) and (6), (4) takes the form a (P at AEA + PBEC + PcEc) (7)
= (PAEA + PBEB + PE) o (PAEA + PBEB + PcEc) - (PAEA + PBEB + PcEc) . Symbolically
(8) u o u may
(7) is isomorphic to Boltzmann a u =u o u-u. at
equation
be also said "quasi-convolution".
E. Wild's
[3]
solution
Boltzmann's problem can be adapted to (7) as well as (8). Now we consider the trajectory of (4). Since PA + PB + Pc =1 a P at A=PA(PB-Pc) (9)
at a PB = PB(P c - PA) at
c
c
Accordingly (10)
1
a PA + 1
PA at
a PB + 1
PB at
peat
a P=
0.
of
B
Y.
856
ITOH
[Vol. 47,
So °~ log P P P = o. a~ A B C is on the solution of the following PA+PB+PC=1 PA.PB.Pc=kl.
(11)
The trajectory (12)
[2]
Model II. 1), 2) are the same to Model I. 3) If an A collides with a B, the A varies to a B with probability 1-PAB and remains unchanged with probability PAB, while the B varies to an A with probability PAB and remains unchanged with probability 1- PAB. In other words, EA ° EB = EB ° EA = PABEA + (1- PAB)EB. The collision between the same types makes no change. For another combinations, the system is ruled by similar laws. If n is sufficiently large, it can be formulated as a Boltzmann eqation (7) on a randomized algebraic structure. (13)
braic
EA ° EA=EA
EA ° EB=EB
EB ° EB = EB Ec ° Ec=Ec
EB ° Ec = Ec ° EB = PBCEB + (1- PBC)EC Ec ° EA=EA ° Ec=PCAEc+(1-PCA)EA
° EA=PABEA+
(1-PAB)EB
The difference between Model I and Model II is caused structure. From (7) and (13), we can derive
(14)
1 a P P A=PA + 2PBPAB + 2Pc(1A at
PCA) -1
Pat 1
PAB) -1
a PB = PB + 2PCPBC + 2PA(1-
by the alge-
1 a P P c = Pc + 2PAPCA + 2PB(1- PBC) -1. c at We can reform (14) to 1 a P P A= PB(2PAB-1)+Pc(1-2PCA) A at (15)
P1
ata PB=PA(1-2PAB)
+Pc(2PBC-1)
1 a P P c=PA(2PCA-1)+PB(1-2Bc). c at Consider P+ /31 aP+ 1 aP (16) Pa1 a A A at PB at Pc at For appropriate time independent constants a, 3, r, we can make (16) equal zero. Because (17) has non-trivial solution.
r
n
SuppL]
Boltzmann
Equation
on Some
Algebraic
Structure
857
(1-2PAB)/3+(2PCA-1)r=0 (2PAB-1)a +(1-2PBC)r=o (1-2PCA)a+(2PBC-1)/3 =0.
(17) For
0 1-2P AB 2PCA-1 (18) 2PAB-1 0 1-2PBC 1- 2PCA 2PBC-1 0 is a skew symmetric matrix, the determinant is zero. (The determinant of an odd dimensional skew symmetric matrix is zero .) The trajectory is on the solution of (19)
P
PAPBPc= k2 A+PB+Pc=1.
By (a, /3, r), many types of trajectory can be considered . The trajectory of Model II is on a curve on PA+ PB + PC =1 (PA, 'B P>0). The author made Model II from an actual problem of operations research which was suggested by Prof. K. Kunisawa. This type of phenomena often occurs in economics, for example , the change of the market share. In this case collision means comparison, and uniform distribution of comparing pairs is assumed. We can extend the above discussion to the struggle for existence among n=2k+ 1 species. It will be formulated as a Boltzmann equation (20) on a randomized algebraic structure (21).
(20)
__ a (i=1PIEi= i=1 nPEI)ni=1P2E2- ~ji=1 nPE) at 2 2 Ei .E~=P13Ei+(1-P23)EJ
(21) For appropriate jectory satisfies (22)
time independent
n=2k+1. constants a1, a2, ... , an, the tra-
~Pi=1 2=1 Pr=k3
n=2k+1.
2=1
Acknowledgement. I am much indebted to Prof. K. Kunisawa for his valuable suggestion and advice. Prof. H. Tanaka has kindly taught me his method about Boltzmann-type problem. Dr . Y. Horibe and Mr. T. Fujimagari have encouraged me throughout the preparation of this paper.
858
Y. ITOH
[Vol. 47,
References [1] 12] 13] 14]
Kac, M.: Probability and Related Topics in Physical Sciences. New York (1959). Kimura, M.: Mathematical Theory of Genetics. Iwanami-Koza, GendaiOyo-Sugaku, Tokyo (1957) (in Japanese). Mckean, H. P. Jr.: Speed of approach to equilibrium for Kac's caricature of Maxwellian gas. Arch. Rational Mech. Anal., 21, 343-367 (1966). Volterra, V.: Lecons sur la theorie mathematique de la lutte pour la vie. Cahiers scientifiques. VII. Paris, Gauthier-Villas (1931).
