数理解析研究所講究録
第 1917 巻 2014 年 55-62

55

Global stability and influence of feedback controls of
delayed Lotka-Volterra systems with patch structure
室谷義昭 (早稲田大学・理工学術院)
Yoshiaki Muroya (Department of Mathematics, Waseda University)

1

Introduction

Motivated by our attention to recent works of Chen [1], Li et al. [5] and Faria and
Muroya [3]) for Lotka-Voltera systems with feedback controls and Takeuchi et al. [8]and Faria [2] for Lotka-Voltera systems with patch structure, we investigate the global
dynamics for the following -species Lotka-Volterra system with infinite delays, feedback
controls and patch structure.
$n$$\{\begin{array}{l}x_{i}’(t)=x_{i}(t)(b_{i}-\mu_{i^{X}i}(t)-\sum_{j=1}^{n}a_{ij}\int_{0}^{+\infty}K_{ij}(s)x_{j}(t-s)ds-c_{i}u_{i}(t))+\sigma_{i}u_{i}(t)+\sum_{j=1}^{n}(\alpha_{ij}\int_{0}^{+_{(}\infty}K_{ij}(\mathcal{S})x_{j}(t-s)ds-\alpha_{ji}x_{i}(t)) ,u_{i}’(t)=-e_{i}u_{i}(t)+d_{i}x_{i}(t) , i=1, 2, .

.

.

(1.1)

, n,\end{array}$with initial conditions of system (1.1):$\{\begin{array}{l}x_{i}(\theta)=\varphi_{i}(\theta) , u_{i}(\theta)=\psi_{i}(\theta) , \theta\in(-\infty, 0],(1.2)\varphi_{i}(0)>0, \psi_{i}(0)>0, i=1, 2, .

.

.

, n,\end{array}$$e_{i}>0,$
where$a_{ij}\in R$ , andand$j=1$ , 2, . . . , arenon-negative and bounded continuous functions on $(-\infty, 0$ ].
Hear, $x_{i}(t)(i=1,2, \ldots,n)J$ denotes the number of species in the patch
denotes the per capita death rate for the species during dispersion from patch tois the intrinsic rate for the species in patch
represents the regulation andis thedispersal coeﬃcient of the species from patch to patch
denotes the feedback$[0,+\infty$$arrow[0,+\infty$control variable and the kernels
:
)
) are
functions, normalized sothat $\int_{0}^{+\infty}K_{ij}(s)ds=1$ , for $j=1$ , 2, . . . , . For the species to disperse from patch toin the model, for simplicity, we neglect the per capita death rate for the species during
dispersion from patch to (cf. Takeuchi et al. [8]).
$\mu_{i},$$c_{i}d_{i},$$\sigma_{i}\geq 0,$$\alpha_{ij}\geq 0$$b_{i},$$\varphi_{i},$$\psi_{i},$

$x$

$i,$

$n$

$i,$

$\gamma_{ij}\geq 0$

$j$

$i,$

$i,$

$\mu_{i}$$\alpha_{ij}$

$j$

$i,$

$i$

$j$

$i$

$i,$

$u_{i}(t)$$L^{1}$$K_{ij}$$n$$j$$b_{i}$

56
The unique solution of (1.1) with initial conditions (1.2) is expressed by $(x(t), u(t))=$with $x(t)=(x_{1}(t), x_{2}(t), \ldots, x_{n}(t))$ and $u(t)=(u_{1}(t), u_{2}(t), \ldots, u_{n}(t))$ .
Moreover, we suppose that for all , the linear operators defined by, for: $(-\infty, 0$ ] $arrow R$ bounded, are non-atomic at zero,
which amounts to have $K_{ii}(0)=K_{ii}(0^{+})$ , and$(x(t;\varphi), u(t;\psi))$$i$$L_{ii}( \varphi_{i})=\int_{0}^{+\infty}K_{ii}(s)\varphi_{i}(-s)ds$

an$n\cross n$$\varphi_{i}$matrix$[\alpha_{ij}]$

is irreducible.

(1.3)

Put
$\delta_{ij}=\{$

1,
,

$0$

if $i=j,$if $i\neq j,$$\tilde{\alpha}_{ii}=\sum_{j=1}^{n}(1-\delta_{ji})\alpha_{ji},$$i=1$ , 2,

...,

(1.4)

$n,$

and

$M(0)=\{\begin{array}{lllllll}b_{1}+ \lrcorner\sigma d\lrcorner e_{1} -\tilde{\alpha}_{11} \alpha_{12} \cdots \alpha_{1n} \alpha_{21} b_{2}+ -A2_{-\tilde{\alpha}_{22}}e_{2} \cdots \alpha_{2n} \vdots \vdots \ddots \vdots \alpha_{n1} \alpha_{n2} \cdots b_{n}+\frac{\sigma d}{e_{n}}-\tilde{\alpha}_{nn}\end{array}\}$

.

(1.5)

Let the stability modulus of an $n\cross n$ matrix $M$ , denoted by $s(M)$ , be defined by $s(M)$ $:=$$\max$ {: is an eigenvalue of $M$ }. If $M$ has nonnegative oﬀ-diagonal elements andis irreducible, then $s(M)$ is a simple eigenvalue of $M$ with (component-wise) positive
eigenvector. A positive equilibrium $E^{*}=(x^{*}, u^{*})$ of (1.1) withand, satisfies the following equations:$Re\lambda$$\lambda$$a$$x^{*}=(x_{1}^{*}, x_{2}^{*}, \ldots, x_{n}^{*})$$u^{*}=(u_{1}^{*}, u_{2}^{*}, \ldots, u_{n}^{*})$$\{\begin{array}{l}x_{i}^{*}((b_{i}-\tilde{\alpha}_{ii})-\mu_{i}x_{i}^{*}-\sum_{j=1}^{n}a_{ij}x_{j}^{*}-c_{i}u_{i}^{*})+\sigma_{i}u_{i}^{*}+\sum_{j=1}^{n}(1-\delta_{ij})\alpha_{ij}x_{j}^{*}=0,-e_{i}u_{i}^{*}+d_{i}x_{i}^{*}=0, i=1, 2, .

Since

$u_{i}^{*}= \frac{d_{i}}{e_{i}}x_{i}^{*},$$(x_{1}, x_{2}, \ldots, x_{n})$.

.

, n.\end{array}$(1.6)$i=1$ , 2, . . . , $n$ , the positive equilibrium of (1.1) is the solution $x=$of the system $F(x)=0$ in$R^{n}$, where$\{\begin{array}{l}F(x)=(f_{1}(x), f_{2}(x), \ldots, f_{n}(x))^{T}, x=(x_{1}, x_{2}, \ldots, x_{n})^{T},f_{i}(x_{1}, x_{2}, \ldots, x_{n})\equiv-[x_{i}\{(b_{i}+\frac{\sigma_{i}d_{i}}{e_{i}}-\tilde{\alpha}_{ii})-(\mu_{i}+a_{ii}+\frac{c_{i}d_{i}}{e_{i}})x_{i}-\sum_{j=1}^{n}(1-\delta_{ij})a_{ij}x_{j}\}+\sum_{j=1}^{n}(1-\delta_{ij})\alpha_{ij}x_{j}]=0, i=1, 2, .

.

.

, n,\frac{\partial f_{i}(x_{1},x_{2},\ldots,x_{n})}{\partial x_{j}}=\{-(\alpha_{ij}-ij),forj\neq i,i=1,2-(b_{i}+\frac{\sigma_{i}d_{i}}{x_{i}ae_{i}}-\tilde{\alpha}_{ii})+2(\mu_{i}+a_{ii}+.\frac{c_{i}d_{i}}{e_{i_{n}}})x_{i}, for j=i, i=1, 2, .

.

.

, n,\end{array}$

(1.7)

57
where the R\’echet derivative of $F(x)$ is
the ordering of vectors and matrices in

Consider a solution
with$F’(x)=[\frac{\partial f_{i}(x_{1},x_{2},\ldots,x_{n})}{\partial x_{j}}]$$R^{n}$

as the usual component-wise one in$(\overline{x}(t),\overline{u}(t))=(\overline{x}(t;\varphi),\overline{u}(t;\psi))$$\overline{x}(t)=(\overline{x}_{1}(t),\overline{x}_{2}(t), \ldots,\overline{x}_{n}(t))$

and. Hereafter, we use$R^{n}.$of the auxiliary cooperative system, given by$\overline{u}(t)=(\overline{u}_{1}(t),\overline{u}_{2}(t), \ldots,\overline{u}_{n}(t))$$\{\begin{array}{l}\overline{x}_{i}’(t)=\overline{x}_{i}(t)((b_{i}-\tilde{\alpha}_{ii}-\alpha_{ii})-\mu_{i}\overline{x}_{i}(t)+\sum_{j=1}^{n}|a_{ij}^{-}|\int_{0}^{+\infty}K_{ij}(s)\overline{x}_{j}(t-s)ds)+\sigma_{i}\overline{u}_{i}(t)+\sum_{j=1}^{n}\alpha_{ij}\int_{0}^{+\infty}K_{ij}(s)\overline{x}_{j}(t-s)ds,\overline{u}_{i}’(t)=-e_{i}\overline{u}_{i}(t)+d_{i}\overline{x}_{i}(t) , i=1, 2, .

.

.

, n.\end{array}$(1.8)

with the same initial conditions$\{\begin{array}{l}\overline{x}_{i}(\theta)=\varphi_{i}(\theta) , \overline{u}_{i}(\theta)=\psi_{i}(\theta) , \theta\in(-\infty, 0],\varphi_{i}(0)>0, \psi_{i}(0)>0, i=1, 2, .

where we use the notations
For $n\cross n$ matrices$(x_{1}, x_{2}, \ldots, x_{n})$$a_{ij}^{+} \equiv\frac{|a_{i}|+a_{i}}{2}\geq 0$$\tilde{A}^{0}=[\tilde{a}_{ij}^{0}],$$A^{0}=[a_{ij}^{0}]$, put $n\cross n$ matrices$[\delta_{ij}x_{i}\mu_{i}-(x_{i}|a_{ij}^{0}|+x_{j}|a_{ji}^{0}|)/2]$

and

.

.

(1.9)

, n,\end{array}$$|a_{ij}^{-}|= \frac{|a_{i}|-a_{i}}{2}\geq 0,$$i,$$j=1$ , 2, . . . , $n.$

and $A=[a_{ij}]$ and a positive vector $x=$$\tilde{M}^{0}(x)\wedge=[\delta_{ij}x_{i}\mu_{i}-(x_{i}|\tilde{a}_{ij}^{0}|+x_{j}|\tilde{a}_{ji}^{0}|)/2],$$\hat{M}^{0}(x)=$and $\hat{M}(x)=[\delta_{ij}x_{i}\mu_{i}-(x_{i}|a_{ij}|+x_{j}|a_{ji})/2]$ , respectively.

In this paper, we obtain the following result.
Theorem 1.1 Assume that $s(\dot{M}(0))\leq$ O. If there exists an $n\cross n$ matrix $\tilde{A}^{0}=[aa]$such that$\tilde{a}_{ij}^{0}\leq a_{ij},$and for the positive left eigenvector
constants
with$(\tilde{\theta}_{i1},\tilde{\theta}_{i2}, \ldots,\tilde{\theta}_{in})$$i,j=1$ , 2, . . . , ,$\tilde{\omega}=(\tilde{\omega}_{1},\tilde{\omega}_{2}, \ldots,\tilde{\omega}_{n})$$\tilde{\theta}_{ii}=1,$(1.10)$n$of $M(O)$ , there exist positive$i=1$ , 2, . . . , $n$ such that$\tilde{\omega}_{i}(\mu_{i}-|aa)\geq\sum_{j=1}^{n}(1-\delta_{ij})\frac{1}{2}(\tilde{\theta}_{ij}\tilde{\omega}_{i}|\tilde{a}_{ij}^{0}|+\frac{1}{\tilde{\theta}_{ji}}\tilde{\omega}_{j}|\tilde{a}_{ji}^{0}|)$, $i=1$ , 2, . . . , ,$n$(1.11)

then the trivial solution $E^{0}=(0,0)$ is globally asymptotically stable.
In particular, if $a_{ij}\geq 0,$ $j=1$ , 2, . . . , , then for $s(M(O))\leq 0$ , the trivial solution$E^{0}=(0,0)$ is globally asymptotically stable.
Note that if $a_{ij}\geq 0,$ $i,j=1$ , 2, . . . , , then for $s(M(O))\leq 0$ , the trivial solution$E^{0}=(O, 0)$ is globally asymptotically stable (seeLemma 2). If anmatrixis diagonally dominant, then for
,
$i,$

$n$

$n$

$n\cross n$$[\delta_{ij}\tilde{\omega}_{i}\mu_{i}-(\tilde{\omega}_{i}|\tilde{a}_{ij}^{0}|+\tilde{\omega}_{j}|\tilde{a}_{ij}^{0}|)/2]$$\tilde{M}^{0}(\tilde{\omega})\wedge=$$(\tilde{\theta}_{i1},\tilde{\theta}_{i2}, \ldots,\tilde{\theta}_{in})=(1,1, \ldots, 1)$58$i=1$ , 2, . . . , $n$ , (1.11) holds.

Theorem 1.2 Assume that $s(M(O))>0$ and suppose that$\omega_{i}(\mu_{i}\omega_{i}-\sum_{j=1}^{n}|a_{ij}^{-}|\omega_{j})>0,$

Then, there exists a positive equilibrium(1.8) withandstable and satisfy$\overline{E}^{*}=(\overline{x}^{*},\overline{u}^{*})$, and(1.12)of the auxiliary cooperative system$\overline{u}^{*}=(\overline{u}_{1}^{*},\overline{u}_{2}^{*}, \ldots,\overline{u}_{n}^{*})$$\overline{x}^{*}=(\overline{x}_{1}^{*},\overline{x}_{2}^{*}, \ldots,\overline{x}_{n}^{*})$$\lim_{tarrow+}\sup_{\infty}x_{i}(t)\leq\overline{x}_{i}^{*}$$i=1$ , 2, . . . , $n$ .

which is globally asymptotically$\lim_{tarrow+}\sup_{\infty}u_{i}(t)\leq\overline{u}_{i}^{*},$$i=1$ , 2, . . . , $n$ ,(1.13).

(1.14)

and$F(\overline{x}^{*})=[\overline{x}_{i}^{*}\{(a_{ii}^{+}+\frac{c_{i}d_{i}}{e_{i}})\overline{x}_{i}^{*}+\sum_{j=1}^{n}(1-\delta_{ij})a_{ij}^{+}\overline{x}_{j}^{*}\}]\geq 0$(i)If$\{\begin{array}{l}\alpha_{ij}>0, for any i, j=1, 2, .

.

.

, n such that a_{ij}^{+}>0,\sigma_{i}>0, for any i=1, 2, .

.

.

, n such that c_{i}>0, and(\mu_{i}+\frac{c_{i}d_{i}}{e_{i}})\omega_{i}+\sum_{j=1}^{n}a_{ij}\omega_{j}>0, for any i=1, 2, .

.

.

, n,\end{array}$(1.15)

then the system (1.1) is permanent and$\min_{1\leq i\leq n}\lim_{tarrow+}\inf_{\infty}(x_{i}(t)/\omega_{i})$$\geq\underline{\hat{x}}\equiv\min\{(\min_{a_{ij}^{+}>0,i,j\in\{1,2,\ldots,n\}}\frac{\alpha_{ij}}{\omega_{i}a_{ij}^{+}}) , (\min_{c_{i}>0,i\in\{1,2,\rangle n\}}\ldots\frac{\sigma_{i}}{\omega_{i}c_{i}})$,(1.16)$( \min_{1\leq i\leq n}\frac{(b_{i}+_{e_{i}}-L\Delta-\tilde{\alpha}_{ii})\omega_{i}+\sum_{j=1}^{n}(1-\delta_{ij})\alpha_{ij}\omega_{j}}{\omega_{i(1A}(\mu_{i}+e_{e_{i}}^{d})\omega_{i}+\sum_{j=1}^{n}a_{ij}\omega_{j})})\},$

is a positive eigenvector corresponding to the spectral radius$\rho(M(O))=s(M(0))>0$ which satisfies

where$\omega=(\omega_{1}, \omega_{2}, \ldots, \omega_{n})$$(b_{i}+ \frac{\sigma_{i}d_{i}}{e_{i}}-\tilde{\alpha}_{ii})\omega_{i}+\sum_{j=1}^{n}(1-\delta_{ij})\alpha_{ij}\omega_{j}>0,$$i=1$ , 2, . . . , $n$ .

(1.17)(ii) In addition to (i), if$\alpha_{ij}-\overline{x}_{i}^{*}a_{ij}^{+}\geq 0,$$i,$$j=1$ , 2, . . . , $n$ ,(1.18)

59
then there exists a positive equilibrium $E^{*}=(x^{*}, u^{*})$ of (1.1) such that (1.6) holds.
(iii) Moreover, if there exists anmatrixsuch that$n\cross n$$a_{ij}^{0}\leq a_{ij},$$\alpha_{ij}-x_{i}^{*}(a_{ij}-a_{ij}^{0})\geq 0,$$i,$$A^{0}=[a_{ij}^{0}]$$j=1$ , 2, . . . , $n$ ,and$[\alpha_{ij}-x_{i}^{*}(a_{ij}-a_{ij}^{0})]$is irreducible,(1.19)

and for the positive vector $v=(v_{1}, v_{2}, \ldots, v_{n})$ defined by$\sum_{j=1}^{n}v_{j}(1-\delta_{ji})\{\alpha_{ji}-x_{j}^{*}(a_{ji}-a_{ji}^{0})\}x_{i}^{*}=v_{i}\sum_{j=1}^{n}(1-\delta_{ij})\{\alpha_{ij}-x_{i}^{*}(a_{ij}-a_{ij}^{0})\}x_{j}^{*},$$i=1$ , 2, . . . , $n,$

(1.20)

there exist positive constants$(\theta_{i1}, \theta_{i2}, \ldots, \theta_{in})$with$\theta_{ii}=1,$$v_{i}( \mu_{i}-|a_{ii}^{0}|)\geq\sum_{j=1}^{n}(1-\delta_{ij})\frac{1}{2}(\theta_{i}v_{i}|a_{ij}^{0}|+\frac{1}{\theta_{j}}v_{j}|a_{ji}^{0}|)$then the positive equilibrium$E^{*}$$i=1$ , 2, . . . , $n$ such that

, $i=1$ , 2, . . . , ,(1.21)$n$of (1.1) is globally asymptotically stable.

Note that if an $n\cross n$ matrix
dominant, then foris diagonally, $i=1$ , 2, . . . , , (1.21) holds.

$\hat{M}^{0}(v)=[\delta_{ij}v_{i}\mu_{i}-(v_{i}|a_{ij}^{0}|+v_{j}|a_{ji}^{0}|)/2]$$(\theta_{i1}, \theta_{i2}, \ldots, \theta_{in})=(1,1, \ldots, 1)$$n$Theorem 1 implies that concerning the global stability of the positive equilibrium of(1.1), there is no influence of the feedback controls.
If we choose the $n\cross n$ matrixin (iii) of Theorem 1, then we obtain thefollowing corollaries.
$A^{0}=[a_{ij}^{0}]$(a) First, we choose$a_{ij}^{0}=a_{ij}^{-},$$i,$$j=1$ , 2, . . . , $n.$

Corollary 1.1 Assume that $s(M(O))>0$ and the conditions of $(i)-(ii)$ of Theorem
1 hold. If an $n\cross n$ matrix
is irreducible and for a positive vector $v=$
such that$[\alpha_{ij}-x_{i}^{*}a_{ij}^{+}]$$(v_{1}, v_{2}, \ldots, v_{n})$$\sum_{j=1}^{n}v_{j}(\alpha_{ji}-x_{j}^{*}a_{ji}^{+})=v_{i}\sum_{j=1}^{n}(\alpha_{ij}-x_{i}^{*}a_{ij}^{+})$

there exist positive constants$(\theta_{i1}, \theta_{i2}, \ldots, \theta_{in})$

with

,

$i=1$ , 2, . . . , $n$ ,$\theta_{ii}=1,$$v_{i}( \mu_{i}-|a_{ii}^{-}|)\geq\sum_{j=1}^{n}(1-\delta_{ij})\frac{1}{2}(\theta_{ij}v_{i}|a_{ij}^{-}|+\frac{1}{\theta_{j\iota’}}v_{j}|a_{ji}^{-}|)$(1.22)$i=1$ , 2, . . . , $n$ such that

, $i=1$ , 2, . . . , ,(1.23)$n$then the positive equilibriumof (1.1) is globally asymptotically stable.
In particular, if for an
matrixis diagonally dominant, then for, $i=1$ , 2, . . . , , (1.23)
holds.
$E^{*}$$n\cross n$$A^{-}=[a_{ij}^{-}],$$\hat{M}^{-}(v)=[\delta_{ij}v_{i}\mu_{i}-(v_{i}|a_{ij}^{-}|+v_{j}|a_{ji}^{-}|)/2]$$(\theta_{i1}, \theta_{i2}, \ldots, \theta_{in})=(1,1, \ldots, 1)$

$n$

60
$a_{ij}=0,$ $j\neq i,$ $i=1$ , 2, . . . ,, then for
Corollary 1.2 If $c_{j}=\sigma_{i}=0,$$s(M(O))>0$ , there exists a unique positive equilibrium of (1.1) which is globally asymptotically stable.
The models of Takeuchi et al. [8, Theorem 2.1] and Faria [2, Theorem 3.5] satisfies thiscondition.
$j=1$ , 2, . . . ,(b) Second, we choose
Corollary 1.3 Assume that $s(M(O))>0$ and (1.15) hold. If $a_{ij}\geq 0,$
is iroeducible, then there exists a
matrix$j=1$ , 2, . . . , and an$E^{*}=(x^{*},u^{*})$
positive equilibriumof (1.1) which is globally asymptotically stable.
$\mu_{i}-|a_{ii}^{-}|\geq 0,$$a_{ij}^{0}=0,$

$i,$

$n$

$n.$

$\alpha_{ij}-\overline{x}_{i}^{*}a_{ij}\geq$

$0,$

$i,$

$n\cross n$$n$$[\alpha_{ij}-\overline{x}_{i}^{*}a_{ij}]$(c) Third, we choose$i,j=1$ , 2, . . . , $n.$
Corollary 1.4 Assume that (1.12) and (1.15) hold. Then, if $s(M(O))\leq 0$ and for the
positive left eigenvector of $M(O)$ , anmatrixis diagonally dominant, then thetrivial solution $E^{0}=(0,0)$ of (1.1) is globally asymptotically stable, and if $s(M(0))>0$
andmatricesandfor the positive eigenvector of $M(O)$ and the positive
vector $v=(v_{1}, v_{2}, \ldots, v_{n})$ defined by (1.20), are diagonally dominant, then there exists a
positive equilibrium $E^{*}=(x^{*}, u^{*})$ of (1.1) which is globally asymptotically stable.
Next, consider the case thatand $a_{ij}\geq 0,$ $j=1$ , 2, . . . , of (1.1).
$a_{ij}^{0}=a_{ij},$$\hat{M}(\omega)$$n\cross n$$\hat{M}(\tilde{\omega})$$n\cross n$$\tilde{\omega}$$\hat{M}(v)$$\omega$$\mu_{i}=c_{\dot{\eta}}=\sigma_{i}=0$

$i,$

$n$

Then, (1.1) becomes$\{\begin{array}{l}x_{i}’(t)=x_{i}(t)(b_{i}-\sum_{j=1}^{n}a_{ij}\int_{0}^{+\infty}K_{ij}(s)x_{j}(t-s)ds)\sum_{j=1}^{n}(\alpha_{ij}\int_{0}^{+\infty}K_{ij}(s)x_{j}(t-s)ds-\alpha_{ji}x_{i}(t)) .\end{array}$Corollary 1.5 For (1.24), assume that there exists a positive vector(1.24)$\overline{x}^{0}=(\overline{x}_{1}^{0},\overline{x}_{2}^{0}, \ldots,\overline{x}_{n}^{0})$such that$M(0)(\overline{x}^{0})^{T}\leq 0,$$\alpha_{ij}-\overline{x}_{i}^{0}a_{ij}\geq 0,$$i,j=1$ , 2, . . . , ,$n$(1.25)

and$-b_{i}+ \sum_{j=1}^{n}\overline{x}_{j}^{0}a_{ji}>0,$

If $s(M(O))\leq 0$ , then the trivial equilibrium$i=1$ , 2, . . . , $n$ .

$\tilde{E}^{0}=(0,0, \ldots, 0)$(1.26)of (1.24) is globally asymp-

totically stable, and if $s(M(O))>0$ , then there exists a positive equilibrium $\tilde{E}^{*}=x^{*}=$of (1.24) which is globally asymptotically stable. Moreover, (1.24) is equivalent to a multi-group SI epidemic model (see Kuniya and Muroya [4]).
is equivalent to $s(M(O))\leq 0.$
Note thatis equivalent to $s(M(0))>0$ and$(x_{1}^{*}, x_{2}^{*}, \ldots, x_{n}^{*})$$\tilde{R}_{0}>1$$\tilde{R}_{0}\leq 1$

61
2

Global stability for $s(M(O))\leq 0$
We first give a basic result on the positiveness and the auxiliary cooperative system(1.8).
Lemma 2.1 For system (1.1) with initial conditions (1.2), there exists a unique solution $(x(t), u(t))=(x(t;\varphi), u(t;\psi))$ with $x(t)=(x_{1}(t), x_{2}(t), \ldots, x_{n}(t))$ and $u(t)=$$(u_{1}(t), u_{2}(t), \ldots, u_{n}(t))$ which
satisfies $x_{i}(t)>0$ , for any $i=1,$ $2\ldots,$ , and $t>$ O. For
the solutionof the auxiliary cooperative system (1.8) withsame initial conditions (1.2),
and, it holds,, for any $i=1$ , 2, . . . , $n,$ $t\geq 0.$
Lemma 2.2 For $s(M(O))\leq 0$ , if there exists anmatrixsuch that (1.10)$n$$(\overline{x}(t),\overline{u}(t))=(\overline{x}(t;\varphi),\overline{u}(t;\psi))$$\overline{x}(t)=(\overline{x}_{1}(t),\overline{x}_{2}(t), \ldots,\overline{x}_{n}(t))$$(\overline{u}_{1}(t),\overline{u}_{2}(t), \ldots,\overline{u}_{n}(t))$$x_{i}(t)\leq\overline{x}_{i}(t)$$\overline{u}(t)=$$u_{i}(t)\leq\overline{u}_{i}(t)$$n\cross n$$\tilde{A}^{0}=[\tilde{a}_{ij}^{0}]$and (1.11) hold, then the trivial solution $E^{0}=(0,0)$ is globally asymptotically stable.
In particular, if $a_{ij}\geq 0,$ $j=1$ , 2, . . . , , then for $s(M(0))\leq 0$ , the trivial solution$E^{0}=(0,0)$ is globally asymptotically stable.
Proof of Theorem 1.2 By Lemma 2.2, we obtain Theorem 1.1.
$i,$

3

$n$

Basic results on the global stability for $s(M(O))>0$
Lemma 3.1

If $s(M(O))>0$ and (1.12) holds, then there exists a unique positive

equilibriumandof (1.8) withwhich is globally asymptotically stable and (1.13) and (1.14) hold.
Lemma 3.2 If $s(M(0))>0$ and (1.15) are satisfied, then the system (1.1) is permanent$\overline{E}^{*}=(\overline{x}^{*},\overline{u}^{*})$$\overline{x}^{*}=(\overline{x}_{1}^{*},\overline{x}_{2}^{*}, \ldots,\overline{x}_{n}^{*})$$\overline{u}^{*}=(\overline{u}_{1}^{*},\overline{u}_{2}^{*}, \ldots,\overline{u}_{n}^{*})$and (1.16) holds.
Lemma 3.3 Assume that $s(M(O))>0$ and (1.12) hold, then there exists a positive
equilibriumandof (1.8) withwhich satisfy (1.13). Moreover, if (1.18) hold. Then, the system $F(x)=0$ has a positive$i=1$ , 2, . . . , which is equivalent to thatsolutionin(1.1) has “at least”’ one positive equilibrium $E^{*}=(x^{*}, u^{*})$ .
Lemma 3.4 If anmatrixis irreducible, then the system (1.20)
has a positive solution
defined by $(v_{1}, v_{2}, \cdots, v_{n})=(C_{11}, C_{22}, \ldots, C_{nn})$ ,$1\leq i,j\leq n$ , andwhere$\overline{E}^{*}=(\overline{x}^{*},\overline{u}^{*})$$x^{*}=(x_{1}^{*}, x_{2}^{*}, \ldots, x_{n}^{*})$$n\cross n$$\overline{x}^{*}=(\overline{x}_{1}^{*},\overline{x}_{2}^{*}, \ldots,\overline{x}_{n}^{*})$$\overline{u}^{*}=(\overline{u}_{1}^{*},\overline{u}_{2}^{*}, \ldots,\overline{u}_{n}^{*})$$n$$0<x_{i}\leq\overline{x}_{i}^{*},$$[\alpha_{ij}-x_{i}^{*}(a_{ij}-a_{ij}^{0})]$$(v_{1}, v_{2}, \ldots, v_{n})$$\tilde{\beta}_{ij}=\{\alpha_{ij}-x_{i}^{*}(a_{ij}-a_{ij}^{0})\}x_{j}^{*},$$\tilde{B}=\{\begin{array}{lllll} -\tilde{\beta}_{21} \cdots \cdots -\tilde{\beta}_{n1}\sum_{j\neq 1,-\tilde{\beta}_{12}}\tilde{\beta}_{1j} \sum_{j\neq 2}\tilde{\beta}_{2j} \cdots \cdots -\tilde{\beta}_{n2}\cdots \cdots \cdots \cdots \cdots-\tilde{\beta}_{1n} -\tilde{\beta}_{2n} \cdots \sum_{j\neq n} \tilde{\beta}_{nj}\end{array}\},$

and$C_{ii}$denotes the cofactor of the i-th diagonal entry of$\tilde{B},$$1\leq i\leq n.$624

Global stability of the positive equilibrium for $s(M(O))>0$

Proof of Theorem 1.2 Assume that $s(M(O))>0$ and suppose that (1.12) holds.
of theThen, by Lemmas 2.1 and 3.1, there exists a positive equilibriumandauxiliary cooperative system (1.8) withwhich is globally asymptotically stable and satisfy (1.13) and (1.14).
(i) Suppose that (1.15) holds. Then, by Lemma 3.2, system (1.1) is permanent and (1.16)
holds.
(ii) Suppose that in addition to (i) and (ii), (1.18) holds. Then, by Lemma 3.3 there
and $u^{*}=$
exists a positive equilibrium $E^{*}=(x^{*}, u^{*})$ of (1.1) withsuch that (1.6) holds.
such that (1.19)(iii) Moreover assume that there exists an $n\cross n$ matrix
holds and for the positive vector $v=(v_{1}, v_{2}, \ldots, v_{n})$ defined by (1.20), there exist positive$i=1$ , 2, . . . , such that (1.21) holds. Then, wewithconstants
can prove Theorem 1.2 by applying Lemma 3.4. The detail proof will be shown in the fort
coming paper Muroya [6] or Muroya$\overline{E}^{*}=(\overline{x}^{*},\overline{u}^{*})$$\overline{x}^{*}=(\overline{x}_{1}^{*},\overline{x}_{2}^{*}, \ldots,\overline{x}_{n}^{*})$$\overline{u}^{*}=(\overline{u}_{1}^{*},\overline{u}_{2}^{*}, \ldots,\overline{u}_{n}^{*})$$x^{*}=(x_{1}^{*}, x_{2}^{*}, \ldots, x_{n}^{*})$$(u_{1}^{*}, u\S, .

.

.

, u_{n}^{*})$$A^{0}=[a_{ij}^{0}]$$(\theta_{i1}, \theta_{i2}, \ldots, \theta_{in})$$\theta_{ii}=1,$$n$$[7|.$$ae\vee’ き$Xﬄ[1] F. Chen, The permanence and global attractivity of Lotka-Volterra competition system with feedback controls. Nonlinear Anal. $RWA7$ (2006) 133-143.
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