数理解析研究所講究録
第 827 巻 1993 年 1-11

1

PERMANENCE AND GLOBAL STABILITY
FOR DIFFUSION SYSTEMS
Zhengyi LU and Yasuhiro TAKEUCHI
Department of Applied Mathematics, Faculty of Engineering,
Shizuoka University, Hamamatsu 432, JAPANABSTRACT
Based on recently developed cooperative systems theory and an new result foressentially positive matrix, we consider two important ODE systems with discrete

diffusion. One is a cooperative Lotka-Volterra diffusion system and another is alogistic system with directed diffusion. Sufficient and necessary conditions are given
for the former to be permanent and for the latter to be globally stable.

1

INTRODUCTION

Recently, many authors consider the effect of spatial distributions of species over
the range of the habitat in population $dynamics[1- 8,11- 17,19- 23,26- 31]$ . It is shownthat spatial factors play a fundamental role in persistence and stability of popu-

lations, although a complete result has not yet been obtained even in the simplest
one-species case. If the population dynamics with the effects of the spatial heterogene-

ity is modeled by diffusion process, we have two typical equations. One is semilinear
parabolic equations, i.e., a reaction-diffusion system where the population is continuously spread out in $space[6,8,11,14,15,21- 23,26]$ . The other is a discrete diffusion sys-

tem where several species are distributed over an interconnected network of multiple

2
patches and there are population migrations among $patches[1- 5,7,8,12,13,19,20,27-$
$31]$

.

In this paper, we focus our attention to discrete diffusion systems, namely, a

cooperative Lotka-Volterra diffusion system and a logistic directed diffusion system.
For the cooperative Lotka-Volterra diffusion system, based on the homotopy function technique, Beretta[4] and Beretta and Takeuchi[5] provided some sufficient con-ditions for the existence of a positive globally asymptotically stable equilibrium point.
And in[l], Allen introduced a logistic system with directed diffusion. By using comparison theorem, Allen obtained a sufficient condition for the solutions of the system

to be bounded in 2-dimensional case.
In this paper, on the basis of the monotonicity for flows of cooperative systems,

for the cooperative Lotka-Volterra diffusion system, first, we prove sufficient andnecessary conditions ensuring permanence of the system and give a permanent systemwith two positive equilibrium points to show that permanence does not imply global
stability in general. Then, we give sufficient and necessary conditions for the directeddiffusion system to be globally stable. The fundamental tools to prove these results
are recently developed cooperative system theories[7,24,25,28] and an new result foran essentially positive matrix (Lemma 4).
Section 2 contains some background concepts and fundamental results for a co-operative system and an essentially positive matrix. Section 3 and 4 state our main

results: sufficient and necessary conditions for a cooperative Lotka-Volterra diffusion

system to be permanent and for a logistic directed diffusion system to be globallystable, respectively. We conclude the paper with some discussions.

In this paper, all matrices are supposed to be non-singular.

3

BACKGROUND CONCEPTS AND RESULTS

2

To begin with we state some concepts and results concerning a general n-dimensional

cooperative system:(1)$\dot{x}=F(x)$

where

$F$

is

$C^{1}$

on a domain $R_{+}^{n}=\{x\in R^{n}|x_{t}\geq 0, i=1, \ldots, n\}$ and has Jacobianmatrix $DF(x)$ with nonnegative off-diagonal elements, i.e. for all $i\neq j,$$(\partial F_{i}/\partial x_{j})\geq 0$, for all$x\in R_{+}^{n}$$i,$$j=1,\ldots,n$. Denote the solution to (1) as $x(t)$ whose initial value

is $x(0)$ .
We shall use a key result given by Kamke [10] and Selgrade $[24,25]$ for system (1),which in our case can be stated as follows :LEMMA 1. Let$y(O)$$R_{+}^{n}$be invariant for (1). If initial positions are ordered $x(O)\leq$, then $x(t)\leq y(t)$ for all $t\geq 0$ . In addition, if $0\leq F(x(O))$ then $x(t)$ is non-decreasing for $t\geq 0$ ; and if $F(x(O))\leq 0$ , then $x(t)$ is nonincreasing for $t\geq 0$ . In
either case, if the positive orbit of $x(O)$ is bounded then its$\omega$-limit set is precisely

one equilibrium point.

To prove global stability for systems in the paper, the following fact which is usedin [7] and [28] is very useful.

LEMMA 2. If system (1) possesses a positive equilibrium point$x^{*}$

satisfying$F(\lambda x^{*})\{\begin{array}{l}>0for\lambda\in(0,1)<0for\lambda\in(1,\infty)\end{array}$then$x^{*}$

is globally stable.

We also need the permanence concept (see, for example Hofbauer and Sigmund[9]).

4
De finition 1. System (1) is said to be permanent if there exists a compact set$K$in the interior of the state spacesuch that all solutions in the interior of$R_{+}^{n}$$R_{+}^{n}$enter ultimately $K$ .
Now consider the n-dimensional Lotka-Volterra cooperative system$\dot{x};=x_{i}(b_{1}+\sum_{j=1}^{n}a_{1j}x_{j})$Here$b=(b_{1}, \ldots, b_{n})^{T}$.

(2)

is a positive constant vector and $A=(a_{ij})_{nxn}$ a constant matrix

with $a_{ij}\geq 0(i\neq j, i,j=1, \ldots, n)$ and $a_{ii}<0(i=1, \ldots, n)$ (i.e.,$A$

is an essentially

positive matrix). For system (2), we can find the following important results quoted
in [9].

LEMMA 3. For system (2), the following statements are equivalent:(i) System (2) admits a positive equilibrium point;(ii) The matrix$A$

is VL stable (i.e. there exists a positive diagonal matrix

$C$

such

that $CA+A^{T}C$ is negative definite);(\"ui)System (2) is permanent;(iv) System (2) is globally stable in the sense that so is the positive equilibrium

point;For convenience, in the following discussions, we use a usual notation $A\in S_{w}$ todenote that matrix$A$

is VL-stable.

A key to prove the necessary conditions in the main theorems is as follows $[19,20]$ .

LEMMA 4. If an essentially positive matrix

a $K(\geq 2)$ -principal minor$A_{(i_{1},\ldots,i_{k})}$does not belong to$S_{w}$, thensuch that the system of linear equations$A_{(:_{1},\ldots,:_{k})}y=1$

has a solution $y>0$ .

$A$,$(1=(1, \ldots, 1)^{T})$

$A$

has

5

3

LOTKA-VOLTERRA SYSTEMS

We consider the following cooperative Lotka-Volterra diffusion systems with two dif-ferent patches:,$\dot{x}_{i}=x_{i}(b_{i}+\sum_{j=1}^{n}a_{ij}x_{j})+D_{t}(y_{i}-x_{i})$$\dot{y}_{i}=y_{i}(\overline{b}_{t}+\sum_{j=1}^{n}\overline{a}_{ij}yj)+\overline{D}_{i}(x_{i}-y_{i})$where$b_{i},\overline{b}_{i}(i=1, )n)$$\overline{A}=(a_{j})_{n\cross n}$constants and$Y$are positive constants,,$i=1,$$a_{ii},\overline{a}_{ii}(i=1,\ldots,n)$$\ldots,$

$n$

.

(3)

negative,$A=(a_{ij})_{n\cross n}$,essentially positive matrices, $D;,\overline{D};(i=1,\ldots,n)$ nonnegative diffusion$x_{i},$$y_{i}(i=1,\ldots,n)$describe the densities of species in the patch X and$i$

at time .
$t$

Based on Lemmas 1,3 and 4, we can prove our first main result as follows[19].
THEOREM 1. System (3) is permanent iff $A\in S_{w}$ and$\overline{A}\in S_{w}$.

From this theorem, we can obtain following corollary[19].

COROLLARY 1. System (3) is globally stable iff$A\in S_{w},\overline{A}\in S_{w}$and a positive

equilibrium point is unique.
A natural problem arising from above results is whether permanence implies global

stability, namely, permanence implies the uniqueness of a positive equilibrium point,in general.
The following example of a permanent system with two positive equilibrium points
shows that permanence does not imply global stability in general.

Example 1.
$\dot{x}_{1}=x_{1}(1.3-13x_{1}+3.1x_{2})+1.2(y_{1}-x_{1})$,$\dot{x}_{2}=x_{2}(1.3+53.1x_{1}-13x_{2})+23.1(y_{2}-x_{2})$,6$\dot{y}_{1}=y_{1}(1.3-13y_{1}+53.1y_{2})+23.1(x_{1}-y_{1})$$\dot{y}_{2}=y_{2}(1.3+3.1y_{1}-13y_{2})+1.2(x_{2}-y_{2})$

,

(4)

.

System (4) has at least two positive equilibrium points $(x_{*}; y_{*})=(1,3;3,1)$ and$(x^{*}; y^{*})=(2,7;7,2)$. Note that

$A,\overline{A}\in S_{w}$.

Comparing Theorem 1, Corollary 1 and Example 1 for diffusion system (3) with
Lemma 3 for isolated patch (2), we know that, since global stability is one hnd of

permanence, the diffusions will not change the dynamical behaviour of the system inthe sense of permanence, but will change it in the sense of global stability.

4

LOGISTIC SYSTEMSIn the preceding section, we have shown sufficient and necessary conditions for acooperative Lotka-Volterra diffusion system to be permanent. In this section, we

consider the following logistic system with directed diffusion terms$\dot{x}_{t}=x_{i}(a_{i}-b_{i}x_{t})+\sum_{j=1,j\neq t}^{n}D_{ij}(x_{j}^{2}-\alpha_{ij}x_{i}^{2})$Denote $A=(a_{J}\cdot)_{nXn}$ , where $a_{ij}=D_{t}j$ for $j\neq i,$

suppose that$a_{i}$

and$b_{i}$

condition[l] constants.

(5)$a_{ii}=-b_{i}-\Sigma_{j=1,j\neq i}^{n}D_{ij}\alpha_{ij}$are positive constants, the diffusion constants$\alpha_{ij}$are nonnegative. Obviously, matrix$A$$D_{ij}$. We

and boundary

defined as above isan essentially positive one. In Allen[l], for system (5) as $n=2$ , the strong persistence

result is shown and some sufficient conditions for the existence of unbounded solutions
are also given. In the present section, we obtain the sufficient and necessary conditions

for the system to have a globally stable positive equilibrium point, and we show thatevery solution of the system is unbounded if the conditions are failed to be satisfied.

This extends the known result for 2-dimensional system[l] to general n-dimensionalone.

7
THEOREM 2[20]. Consider system (5).
i) The System possesses a globally stable positive equilibrium point$x^{*}$, if $A\in S_{w}$ ;ii) every solution of the system is unbounded, i.e., $\lim_{tarrow T_{x}}x(t)=\infty$ , if

$A\not\in S_{w}$.

Here, $(0, T_{x})$ is the maximal interval of existence for $x(t)$ .

In the following, we assume, without loss of generality, that the K-th principal

minor given in Lemma 4 is the K-th leading one of$A$, that is, $i_{l}=l$ for $l=1,$$K$$\ldots,$.

To prove Theorem 2, we need the following lemma.
LEMMA 5. If$A\not\in S_{w}$, for any positive parameter , the following system of linear$\mu$equations$a_{11}x_{1}^{2}+a_{12}x_{2}^{2}+\cdots+a_{1K}x_{K}^{2}=\mu$,$a_{21}x_{1}^{2}+a_{22}x_{2}^{2}+\cdots+a_{2K}x_{K}^{2}=\mu$

,

.

(6)

$a_{K1}x_{1}^{2}+a_{K2}x_{2}^{2}+\cdots+a_{KK}x_{K}^{2}=\mu$

has a positive solution$x_{K}^{2}= \frac{|\begin{array}{llll}a_{11} a_{12} \cdots 1a_{21} a_{22} \cdots 1\vdots \vdots \ddots \vdots a_{K1} a_{K2} \cdots 1\end{array}|\mu}{det(a_{ij})_{K\cross K}}$$x_{1}^{2}= \frac{|\begin{array}{llll}1 a_{12} \cdots a_{1K}1 a_{22} \cdots a_{2K}\vdots \vdots \ddots \vdots 1 a_{K2} \cdots a_{KK}\end{array}|\mu}{det(a_{ij})_{K\cross K}},$$\cdots,$

Proof. This lemma is a direct consequence of Lemma 4.
Proof of Theorem 2.
Now we write system (5) in the vector form
th $=diag(a_{1}, \ldots, a_{n})x+A(x^{2})=G(x)$ ,

.

(7)

8
where

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

positive vector$w$. Since all $a_{i}(i=1, \ldots, n)$ are positive, for sufficiently small, we have $G(w)>0$ . Hence, according to Lemma 1, the region$R_{+}^{n}+w=\{x\in R_{+}^{n}|x_{i}\geq w;, i=1, \ldots, n\}$is positively invariant, and furthermore, weknow that all solutions enter ultimately this invariant region. If at least one solution
is bounded, then again by Lemma 1, we know that the system possesses a positive
equilibrium point$x^{*}$. It is easy to check that
$G_{t}(\lambda x^{*})=a_{i}x^{\dot{*}}\lambda(1-\lambda)$

then by Lemma 2,$x^{*}$,

is globally stable.

i) When $A\in S_{w}$ , we take a Liapunov function as follows$V(x)= \frac{1}{3}\dot{\sum_{=1}^{n}}c;x_{i}^{3}$

where $c_{i}(i=1, \ldots, n)$ are diagonal elements of a diagonal matrix$A^{T}C$$C$

such that $CA+$

is negative definite. Then,$\dot{V}(x)=(x^{2})^{T}(CA+A^{T}C)(x^{2})+\sum_{1=1}^{n}c_{i}a;x_{t}^{3}<0$

for large enough . Therefore all solutions are bounded, namely, the system possesses$x$

a globally stable positive equilibrium pointii) Suppose that

$A\not\in S_{w}$$x^{*}$.

. Since the boundedness of at least one solution implies

global stability of the system, we only need to check that under condition$A\not\in S_{w}$, thesystem is not globally stable. Therefore, it is sufficient to show that for any compact
set $E$ in$R_{+}^{n}$, there exists an initial $x(0)\not\in E$ such that $\Omega(x(0))\cap E=\emptyset$ . Clearly, wecan, without loss of generality, suppose that $E$ is the intersection of$R_{+}^{n}$and a given

ball with center at the origin .
$0$

Since

$A\not\in S_{w}$, by Lemma 5, we know that there is a minimum $K\geq 2$ such thatfor any given positive$\mu$, the linear equations (6) have a positive solution (7). Now

we take an initial value $x^{0}=x(\mu)=(x_{1}(\mu), \ldots, x_{n}(\mu))$ . Here $x_{i}(\mu)(i=1, \ldots, K)$ are9given by (7) for sufficiently large$\mu$and the remaining $x_{j}(\mu)(j=K+1, \ldots, n)$ aresufficiently small such that $G(x(\mu))>0$ . By Lemma 1, the solution $x(t)$ with theinitial value$\omega$$x^{0}$

is increasing for $t\geq 0$ . Therefore, either $x(t)$ is unbounded or has an-limit set disconnected to$E$.

This completes the proof of Theorem 2.

5

DISCUSSION

In this paper, based on the specific property of cooperative systems and some results

for monotone flow of solutions given by Kamke[10] and Selgrade $[24,25]$ , we have obtained the sufficient and necessary conditions for Lotka-Volterra cooperative systems
with diffusion to be permanent. Theorem 1 and Example 1 show that if each isolated
patch is permanent, then diffusion between patches cannot destroy the permanence,

although the diffusion system can have two or more positive equilibrium points.
The global stabihty of the system is considered and a corollary to guarantee theglobal stability is obtained. Under the condition of both

$A$

and

$\overline{A}$belonging to$S_{w}$,

the uniqueness of positive equilibrium points ensures global stability.

In Section 4, global asymptotic behavior of a single species dispersing among
multiple patches is discussed. Sufficient and necessary conditions for the directeddiffusion system to be globally stable are obtained. It is shown that every solutionof the system is unbounded if the conditions are failed to be satisfied. This extends

a known result for 2-dimensional system[l] to general n-dimensional one.
The key to prove the necessities of both main Theorems 1 and 2 is a result for an
essentially positive matrix (Lemma 4) which seems a new one.
It needs to be stated that for a concrete system, the conditions $A\in S_{w}$ and$\overline{A}\in S_{w}$are not difficult to be checked according to Lemma 3. And on the basis of recently

developed theory[18] of numerically determining solutions of systems of polynomial

10
equations, it is also possible to find all positive equilibrium points of system (3) whose
number of positive equilibrium points will decide whether it is globally stable or not.

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