BULLETIN OF MATHEMATICAL BIOLOGY

VOLV~E 38, 1976

NOTE B O U N D S F O R G E N E R A L I Z E D DREITLEIN-SMOES MODELS

9 PAV-LE. RUBr~ Systems Analysis and Engineering Department, Naval Air Development Center, Warminster, Pennsylvania, U.S.A.

The analysis of a previous paper obtaining bounds on the total population number of species (chemical or biological) described by the recently proposed Dreitlein-Smoes model of oscillatory kinetic systems, including diffusion, is extended to generalized models of the Dreitlein-Smoes type, describing a system of S components with S > 2. The results for such generalized models are analogous to those of the previous case. It is found that the effects of diffusion serve to restrict the region in the concentration space available to limitcycle type oscillations.

1. Introduction. The recently proposed model of Dreitlein and Smoes (Smoes and Dreitlein, 1973; Dreitlein and Smoes, 1974) for oscillatory kinetic systems has been analyzed in a previous paper (Fizell and Rubin, 1976) for bounds on the total population of the interacting species (chemical or biological) where a full account of diffusion of the oscillating species is taken. I t was found t h a t the effect of diffusion is to limit the m a x i m u m possible variations in limit-cycle t y p e behavior. In particular, it was shown t h a t in some cases such limit-cycle behavior is precluded. In this paper, the analysis is e x t e n d e d to certain types of generalized Dreitlein-Smoes models (Dreitlein and Smoes, 1974) where again diffusion will be included. After certain transformations on the original equations, a form is arrived at t h a t allows the immediate application of the preceding (Fizell and Rubin, 1976) results. 739

740

P A U L E. R U B I N .

2. Generalized Models.

We consider the system (Dreitlein and Smoes, 1974)

0Xt -

D V2XI + ~ ~I~jXj + [E - l(X)]Xi

(1)

in a closed, bounded two- or three-dimensional region R for i = 1, 2 . . . . . S where the Xt are the components of the S-dimensional vector X of relative concentrations, D is the (positive) diffusion coefficient, ~ is an S x S matrix, E is a constant parameter, and l(X) is a nonlinear scalar function of the Xl. We will impose on 1 the conditions s__ Ol(X)

X~ = nl(X)

~=~ 0Xt

(2a)

for some n and ~ 0l(X) ~jkX~ = O. j,~=l aXj

(2b)

Note that (2a) says that l is a homogeneous function of the X, of order n. Equation (1) with (2a, b) implies that I(X) will satisfy the equation ~l

s

- -t = 3

D

L

01

OX, V2X* + n(E - 1)l.

(3)

i=l

We will consider the case where I(X) is a homogeneous quadratic polynomial. Then I(X) may be written in the form 8

l(x) =

(4) i,i=1

where all = a i i . Thus, (3) becomes

2D ~ allXiV2Xl+2(E-l)l.

--= 0t

(5)

~,j=l

Now consider the S • S orthogonal matrix U that diagonalizes the aij matrix: 8

(UT)t~a/c~UII = aiSIJ.

(6)

k,l=l

Then with the vector Y defined b y

(7)

=

J we have l

BOUNDS F O R G E N E R A L I Z E D D R E I T L E I N - S M O E S MODELS

741

and so (5) may be put in the form O1 - 2D ~. ~ Y,V 2 Y, + 2(E - l)l. 0t

(9)

Defining the quantities

N(m)(t) - fR/re(X) dx,

N(t) - N(1)(t)

(10)

gives by (9) dt

- 2D fR ~t ~Y~V2Y~dx+ 2EN-2N(2).

(11)

We are now in a position to determine bounds on N(t) by the methods used previously (Fizell and Rubin, 1976).

3. Dirichlet Boundary Conditions. Consider now (11) supplemented with homogeneous Dirichlet boundary conditions Xt = 0 on OR

(12)

l(X) = 0 on 0R.

(13)

for all i. This also gives

Now by Gauss' theorem, we obtain __dN =< - 2 D dt

2 ~IvY~I 2 d x + 2 E N - 2 N ( 2 )

(14)

t=1

from (11). Using the inequality (Fizell and Rubin, 1976; Rosen, 1975)

fR lVOl2dx> g fRO2dx,

(15)

with n a ~ -1,

2 - dim.

g = -~/

,

3 -dim.

(16)

leads to dN N 2 ~ -1, where ~ is the volume (area) of R gives d2V

< 2(E-Dg)N-2~-IN dt =

2,

(18)

which integrates to yield N(t) < N(O) [exp (--[JDt)T2N(O)(flD~)-I(1--exp (--flDt)] -1

(19)

flD -- 2 ( E - D g ) .

(20)

with

The asymptotic behavior of N(t) then for t --> oo is governed by the sign of riD. We have/V(oo) = 0 for flD < 0 and N(oo)
O. We note that our equation (19) and the bounds (21) are of the same form as equation (10) and bounds (11) in Fizell and Rubin (1976). This verifies the expectations of Dreitlein and Smoes (1974) that the generalized model should display behavior similar to that of the more specialized model. 4. Robin Boundary Conditions. conditions of the Robin type

We now supplement (11) with boundary

3X~ --+KXI ~v

= 0

on ~R

(22)

for all i and ~ a constant parameter. Using now the inequality (Fizell and Rubin, 1976; Rosen, 1974)

fR

_

Bounds for generalized Dreitlein-Smoes models.

BULLETIN OF MATHEMATICAL BIOLOGY VOLV~E 38, 1976 NOTE B O U N D S F O R G E N E R A L I Z E D DREITLEIN-SMOES MODELS 9 PAV-LE. RUBr~ Systems Analys...
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