# cellular systems—ii. stability of compartmental systems

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Cellular Systems-II. Stability of Compartmental Systems*

G. S. LADDE

Department of Mathematics, The Slate University of New York at Potsdam,

Potsdam, New York 13676

Communicated by R. Rosen

ABSTRACT

In this work, by employing the concept of vector Lyapunov functions, and the theory

of differential inequalities, the stability analysis of compartmental systems is initiated in a

systematic and unified way. Furthermore, an attempt is made to formulate and partially

resolve the complexity vs. stability problem in open compartmental systems. The recent

stability results obtained for open chemical systems are applied to open compartmental

systems in a natural way. As a byproduct of our analysis, we obtain an estimate for

washout functions. Finally, it has been demonstrated that the stability analysis of

compartmental systems provides a common conceptual framework for innumerable,

apparently unrelated systems in biological, medical, and physical sciences.

1. INTRODUCTION

The concept of compartmental processes provides suitable means of description for several wide ranges of dynamic processes in biological, physical, and social sciences. A similar fact has been stated by Rosen [22] with respect to morphogenetic and two-factor processes in a systematic and unified form. Furthermore, the scope of compartmental analysis has been well documented in the recent monographs by Jacquez [7] and by Rosen [23], in an excellent paper by Hearon [4], and also in the monograph by Rescigno and Segre [18]. However, a systematic stability analysis of com- partmental systems awaits development. An attempt in this direction has been made by Hearon [4] in connection with the stability of the steady state of linear compartmental systems. For nonlinear compartmental systems, Bellman [2] was the first to formulate the concentration-dependent trans-

*The research reported herein was supported by SUNY Research Foundation Faculty Fellowship No. 0454-O l-03 1-74-O.

MATHEMATICAL BIOSCIENCES 30, l-21 (1976)

0 American Else&x Publishing Company, Inc., 1976

1

2 G. S. LADDE

portation rates and also study the stability of the steady state of closed compartmental systems. In a recent work, Jacquez [7] has used linearized approximations to show the stability of the steady state of nonlinear open compartmental systems. However, these results lack the realism and struc- tural insight of the stability analysis. Lyapunovs second method has been utilized successfully for the stability study of chemical systems [22, 25, 331, controlled biological systems [31, 321, ecosystems [ 121, etc.; however, this method has remained unused for the stability study of compartmental systems, even though it has been recognized that chemical systems [4, 7, 221 and ecosystems [4, 13, 221 can be viewed as compartmental systems.

Because of the fundamental importance of open compartmental systems for demonstrating analogies between systems at different levels of biological and physical organization, we propose to extend our stability analysis of chemical systems [8] to compartmental systems. This fact will support a very fundamental and important attempt made by Rosen [21, 221 to unify innumerable, structurally diverse dynamic systems in biological and physi- cal sciences. As remarked by Rosen [22], such a unification, with the resultant economy of thought, seems absolutely necessary in these days of explosive growth in scientific knowledge, if the student is not to be hope- lessly submerged under the torrent of specific but specialized investigations relevant to his interest. Indeed, unless expanding knowledge is accompanied by corresponding conceptual economies, a comprehensive understanding of even a small corner of science may become impossible before much longer..

Our present stability analysis is the continuation of our stability analysis of chemical systems [8], and it is strongly influenced by the remarkable work of Hearon [4], Jacquez [7], Rescigno and Segre [ 181, and Rosen [21, 221. Furthermore, two of the important classes of dynamical analogies, namely, morphogenetic systems [22, 231 and two-factor systems [19, 221, have been realized as hierarchic compartmental systems. It is important to note that our method of study does not demand either linearization ap- proximations or reduction techniques for analyzing the stability properties of open compartmental systems.

For linear, nonlinear, and hierarchical open compartmental systems, our stability analysis is analogous to the corresponding stability analysis of linear, nonlinear, and hierarchical open chemical systems [8]. In fact, it is a generalization of the first part of our work [8], so as to include wide varieties of dynamic systems in life and physical sciences.

The paper is organized as follows: In Sec. 2, we recognize the fact that the transport rate matrix of the

linear compartmental system is the Metzler matrix 1141 that is used by several economists to obtain strong stability results for economic systems.

STABILITY OF COMPARTMENTAL SYSTEMS 3

In the present work, it is shown that for the linear compartmental system, the quasidominant diagonal property of the corresponding transport rate matrix is necessary and sufficient condition for stability of the steady state of the linear compartmental system. Based on our study, we conclude that the dominant diagonal property enables us to draw conclusions about the tolerance for increased complexity of stable open compartmental systems. Furthermore, we also recognize a complexity vs. stability problem similar to that in open chemical systems [8] and show that the dominant diagonal property is a suitable mechanism for partially resolving this problem. However, for more details, we refer to [8]. In Sec. 3, nonlinear compartmen- tal systems, and in Sec. 4, hierarchic compartmental systems are briefly formulated, and further to avoid tedium, detailed mathematical analysis, remarks, and conclusions are summarized by reference to the first part of this work [8]. Furthermore. our analysis suggests a notion of quasi-washout function which includes the usual notion of washout function [5], and it gives us an estimate of it in a natural way. In Sec. 5, the scope of our stability analysis of hierarchical compartmental systems is exhibited by identifying classes of dynamical analogies-for example, morphogenetic systems [22, 231 and abstract two-factor systems [19, 22]-as hierarchic compartmental systems. In addition to this, the stability study of several important network problems [19, 20, 221, such as the epigenetic network, simple series network, discrimination network, and learning network, is given as a special case of the stability study of compartmental networks or hierarchical compartmental systems.

2. LINEAR COMPARTMENTAL SYSTEMS

Let us consider a compartmental system consisting of n compartments described by a linear differential equation

i=Ax+a.

where x E R is a state variable n-vector x =(x1,x2,. .,x,), and T stands for transpose of a vector. For i~1. X, represents the concentration of a labeled observable quantity (e.g., a drug), or the strength of a labeled observable quality (excitatory, or inhibitory, etc.) in the ith compartment, where 1={1,2,..., n}. A = (ati) is the n X n constant observable quantity (or quality) transport rate matrix, and a E R is the constant input rate vector from external sources. In view of the physical significance, the elements of the matrix A must satisfy the conditions:

ai, < 0, iEI (2) -

4 G. S. LADDE

and

ay > 0, i#j, iJ E I. (3)

From (2) and (3) we immediately observe that the constant observable quantity (or quality) transport rate matrix A is a Metzler matrix [14]. Throughout our discussion, we assume that the linear compartmental sys- tem (1) is open. See [7, 18, 231.

It has been observed [ 1, 21 that the solution x(t, to,x,,) of (1) is non- negative for all tat,, (t,,x,)ER+XR;, where R+=[O,cs), R;={xE R :x, > 0 Vi E I}. Furthermore, the steady state x* of (1) is equal to -A -]a whenever A is nonsingular. Therefore, the stability of x* = -A -a of (1) is equivalent to the stability of x* = 0 of

.k=Ax. (4)

For more details, see [8]. By following the argument used in [8], we conclude that the steady state

x* = 0 of (4) is stable [22] if and only if the Metzler matrix A is a Hicks matrix [ 141.

The main objective of the present work is to study the effects of the strength of the interactions among the compartments on the stability of the steady state of an open compartmental system under structural perturba- tions. In order to study the effects of the strength of interactions among compartments on the stability of the steady state x* in (1) let us assume that the elements av of the matrix A have the following form:

(5)

In (5) eG are elements of the n x n interconnection matrix E =(e,), and they can take on values between zero and one. By eV we represent the strength of connection of the compartment x, with the compartment xi, ranging from disconnection (eV = 0) to full connection (eU = 1). Furthermore, the interconnection matrix E represents the structural changes in the interactions among the compartments in (1). In particular, the shutdown of a pathway from ith compartment to the jth compartment in the compart- mental system is represented by e,, = 0. The disappearance of the pathways between the ith and jth compartments in the system is shown by eU = eji = 0.

STABILITY OF COMPARTMENTAL SYSTEMS 5

The disconnection or disappearance of the pth compartment in the com- partmental system (1) is represented by eiP = ePj = 0 for all ij E I. If ePi = 0 for all

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