数理解析研究所講究録
第 828 巻 1993 年 171-178

171

GLOBAL DYNAMICAL BEHAVIOR FOR
LOTKA-VOLTERRA SYSTEMS
Zhengyi LU (陸征一) and Yasuhiro TAKEUCHI (竹内康博)
Department of Applied Mathematics, Faculty of Engineering,
Shizuoka University, Hamamatsu 432, Japan

ABSTRACT
Recently, Redheffer[13] obtained a global stability result of a positive equilibrium
point for a class of Lotka-Volterra systems with reducible interaction matrices.
The present paper, following his proving method, extends his result in some senseand shows that qualitative stability of an interaction matrix implies global stabilityof the system.

INTRODUCTION

1

It is known that global asymptotic behavior of the solutions of the general n-dimensional Lotka-Volterra system$\dot{x};=x:\sum_{j=1}^{n}a_{lj}(x_{j}-x_{j}^{*}),$

where$x^{*}=(x_{1}^{*}, x_{2}^{*}, \ldots, x_{n}^{*})^{T}$$i=1,2,$$\ldots,$

$n$

,

(1)

is a positive equilibrium point of the system, largely

depends upon the structure of the Jacobian or the interaction matrix$A=(a_{i_{J}}\cdot)_{n\cross n}$of the system. A lot of results about the relationship between the structure of theinteraction matrix and global asymptotic behavior of solutions of system (1) arequoted by Hofbauer and Sigmund[4].

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In this paper, we consider a basic and interesting question in the study of LotkaVolterra systems, i.e., what structure of the interaction matrix can imply global
stability of a positive equilibrium point of the system. Hereafter, we call system (1)globally stable if and only if so is the positive equilibrium point. We define similarly
global stability for subsystems of the system.
In Section 2, we discuss so called D-stability of a matrix and some related conjectures. We will prove that a special kind of D-stability, i.e., qualitative stability of

the interaction matrix, implies global stabihty of system (1) in Section 3. Finally, abrief dicussion is given.

2SOME CONJECTURES
The following is a well-known concept for a matrix:
De finition 1. A matrix is (i) semistable if the real parts of its eigenvalues are all

nonpositive; (ii) quasistable if it is semistable and no eigenvalue with zero real partis repeated in its minimal polynomial; (iii)stable if the real parts of its eigenvalues

are

$al1$

negative.

De finition 2.

$A$

is D-stable if for every positive vector$x^{*},$$diag(x^{*})A$is stable.

In [4], Hofbauer and Sigmund proposed the following

Hofbauer-Sigmund Conjecture: If the interaction matrix$A$

is D-stable, then sys-tem (1) is globally stable.
They[4] have also indicated that their conjecture is a special case of the well-known

Jacobian conjecture stated below.

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Let’s now consider a general n-dimensional system of ordinary differential equations in$R^{n}$:$\dot{x}=f(x)$(2),where $f(x)$ has continuous first-order partial derivatives. Then we have the following[10]:Jacobian Conjecture: Suppose that(i)$0$is a fixed point, i.e., $f(O)=0$ ,(ii) the Jacobian is a stable matrix at every point $x\in R^{n}$ . Then$0$is globallystable.
Jacobian conjecture has not been completely solved even in two-dimensional caseyet. Meisters and Olech[ll] proved that in two-dimensional case the conjecture is truefor the class of polynomial$f_{2}$$f$:$R^{2}arrow R^{2}$, i.e., for maps $f=(f_{1}, f_{2})$ such that$f_{1}$

andare real polynomials in two variables. Recently, Gasull, Llibre and Sotomayor[2]extended Meisters and Olech’s result to a larger class of analytic vector fields.
In the next section, we shall consider system (1) with an interaction matrix satis-fying different stability concept, motivated from qualitative economics[15], i.e., qualitative stability, which implies D-stability[4]. We will prove that qualitative stabilityof the matrix implies global stability of Lotka-Volterra system. This can serve as anaffirmative answer to a special case of Hofbauer-Sigmund conjecture.

3

MAIN RESULTS
Let $Q(A)$ denote the convex cone consisting of all $n\cross n$ matrices

have the same sign pattern $(+, -, 0)$ as
we have$A$

so that sgn$\tilde{a}_{lj}=sgna_{ij}$$\tilde{A}=(\tilde{a}_{ij})$thatfor all and . Then
$i$

$j$

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Definition 3.

A matrix$A$

is qualitative semistable (qualitative quasistable,qualitative stable) if each member of $Q(A)$ is semistable (quasistable, stable).

The digraph $G(A)$ of an $n\cross n$ matrix$A$

consists of$n$

vertices representing$n$populations together with edges representing the interrelations between populations.
Black dot$\bullet$(when $a:i<0$ ) or white dot $o(a_{il}=0)$ will be put on each vertex. If,

in general discussions, we will not be concerned in the concrete sign ofbe used. An edge between$a_{ji}\neq 0$

$i$

and

$j$

$i_{1},$$i_{2},$$i_{p}$$\ldots,$$\oplus will$is called a stronge-edge whenever $a_{ij}\neq 0$ and. A p-cycle of $G(A)$ is a nonvanishing product ofof distinct indices$a_{ii},$$a_{i_{1^{t}2}}a_{i_{2^{t}3}}\ldots a_{i_{p}i_{1}}$for a sequence.

To explain these, we use a system quoted from Jeffries[5].

Example 1. Consider system (1) with the interaction matrix$A_{f}=(\begin{array}{lllll}a_{11} a_{12} a_{13} 0 0a_{21} 0 0 a_{24} 0a_{31} 0 0 0 a_{35}0 a_{42} 0 0 00 0 a_{53} 0 0\end{array})$$A_{f}$

as follows,(3)where $a_{11}<0,$ $a_{ij}a_{ji}<0$ for $i\neq j$ . Then the $G(A_{f})$ is expressed as$0—0—\bullet---0---0$ .
The following result was established by Quirk and Ruppert[12] on the basis of
Hurwitz Theorem and Liapunov function technique.

LEMMA 1. A matrix$A$is qualitative semistable if and only if it satisfies:i) each l-cycle in $G(A)$ is nonpositive;ii) each 2-cycle in $G(A)$ is nonpositive;iii) $G(A)$ has no p-cycle for $p\geq 3$ .

When

$A$

is irreducible these conditions imply qualitative quasistability.

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It is easy to check, based on Lemma 1, that matrix$A_{f}$in (3) is qualitative qua-sistable.
The following stronge community notation are quoted from Jeffries[5] and Jeffries
et $a1[6]$ , in their cases, they called it as predation community and strong component
respectively.
De finition 4. Associate with a fixed vertex all the other vertices, if any, to which

it is connected by strong edges. Then associate with these vertices all additionalvertices connected by strong edges, and so on. The maximal set of all such vertices

so connected to the first vertex is called the strong community containing the firstvertex.
By introducing the color test notation, Jeffries gave a complete solution for amatrix to be qualitative stable[5].

De finition 5. A strong community $G(A)$ can pass the color test if each vertex init may be colored black and white with the result thati) each vertex with $a_{ii}<0$ is black;ii) there is at least one white dot;iii) each white dot is connected by a strong edge to at least one other white dot;iv) each black dot connected by a strong edge to one white dot, is connected bya strong edge to at least one other white dot.

The following theorem is due to Quirk and Ruppert[12] in the case where $a_{ii}<0$
for all$i$

and to Jeffries[5] in general case.

JQR THEOREM. A matrix $A=(a_{j})_{n\cross n}$ is qualitative stable if and only if itsatisfies:i) each l-cycle in $G(A)$ is nonpositive;ii) each 2-cycle in $G(A)$ is nonpositive;

176iii) $G(A)$ has no p-cycle for $p\geq 3$ ;iv) each strong community in $G(A)$ can not pass the color test;v) $\det A\neq 0$ .

It is easy to check that matrix $A_{f}(3)$ is not qualitatively stable, since $G(A_{j})$ canpass the color test.
In [8], we have considered system (1) with the following lower-triangle interaction

matrix$A=(a_{ij})_{n\cross n}=(A_{\cross}\cross^{1}$$A_{\cross^{2}}^{0}$$..$.

$A_{k}^{0}0)$,(4)where each submatrix $A_{1}(i=1, \ldots, k)$ is irreducible, all elements in the upper-rightblocks are zero and all matrices$\cross$in the left-lower have any elements and obtained:

THEOREM 1[8]. If system (1) withi) each

$A_{1}\in\overline{S}_{w}(i=2, \ldots, k)$ii) each diag$(x_{i}^{*})A_{i}$$A$in (4) satisfies;

is stable $(i=1, \ldots, k)$ ;iii) each subsystem th $i=diag(x;)A_{i}(x_{i}-x_{i}^{*})$ is globally stable $(i=1, \ldots, k-1)$ , then$\Omega(x)\subseteq E=\{(x_{1}^{*}; \ldots;x_{k-1}^{*}; x_{k})\in R_{+}^{n}|x_{k}\in M\}$, where$\Omega(x)$is the -limit set of system$\omega$(1) and $M$ is the LaSalle’s invariant set of the subsystem $\dot{x}_{k}=diag(x_{k})A_{k}(x_{k}-$$x_{k}^{*})$$(x_{1}^{*}$. Here;$x_{2}^{*}$; ,..;$x^{*}=$$x=(x_{1}; x_{2};\ldots;x_{k})^{T}=(x_{11}, \ldots, x_{11_{1}} ; x_{21}, \ldots, x_{2};_{2} ; \ldots;x_{k1}, \ldots, x_{ki_{k}})^{T},$$x_{k}^{*})^{T}=(x_{11}^{*}, \ldots x_{1i_{1}}^{*} ; x_{21}^{*}, \ldots, x_{21_{2}}^{*} ; \ldots;x_{k1}^{*}, \ldots, x_{kt_{k}}^{*})^{T}$

and $i_{1}+i_{2}+...$ $+i_{k}=n$and a matrix $B\in S_{w}$ if and only if there exists a positive diagonal matrix$C$suchthat $CB+B^{T}C$ is negative semi-definite.

Combining Theorem 1 and JQR Theorem, we have
THEOREM 2[9]. If the interaction matrix$A$of system (1) is qualitative stable,

then the system is globally stable for any given positive equilibrium point$x^{*}$.

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Remark.

By using Theorem 1 in [13] and JQR Theorem, we can also prove

Theorem 2.

4

DISCUSSION
We have proved that qualitative stability of the interaction matrix of a Lotka-Volterra system implies global stability of the system. Since a qualitative stablematrix must be D-stable[4], our Theorem is an affirmtive answer to a special case of
Hofbauer-Sigmund conjecture.
A number of results [1,3,7,14,16,17] given before in the literature are special casesof Theorem 2 in Section 3, since in these cases, the interaction matrix of system (1)is qualitative stable.

The remaining question is the relationship between the weaker qualitative prop-erties such as quahtative semistability or qualitative quasistability of matrix$A$andglobal stability of the corresponding system (1). But semistability is not so interesting in the sense that it cannot guarantee even the boundedness of the solutions of acorresponding linear system[6].
It seems that there is no general known condition ensuring a matrix to be qual-itatively quasistable. From Lemma 1, we know that a qualitative semistable matrix

is qualitative quasistable whenever it is irreducible. Furthermore a qualitative qua-sistable matrix ensures the boundedness for the solutions of a corresponding linearsystem. Hence, from this subclass of qualitative quasistable (qualitative semistable

and irreducible) matrices we can expect to find out some interesting structures of a
LaSalle’s invariant set which is not identical with just a unique positive equilibrium
point.

REFERENCES

178
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