数理解析研究所講究録
第 1786 巻 2012 年 85-92

85

Global asymptotic stability in a two-species
nonautonomous competition system
Hiroyuki USAMI, Gifu University

岐阜大学工学部数理デザイン工学科・ 宇佐美広介

This talk is based on ajoint work with Dr. Kunihiko Taniguchi, Hiroshima University.
We will consider the following nonautonomous competition system(S)$\{\begin{array}{l}x’=x(a_{1}(t)-b_{1}(t)F_{1}(x)-c_{1}(t)G_{1}(y))y’=y(a_{2}(t)-b_{2}(t)F_{2}(x)-c_{2}(t)G_{2}(y))\end{array}$

The following conditions are always assumed through this talk:$G_{i}\in C([0, \infty);[0, \infty)),$ $i=1,2$ , are strictly increasing functions $F_{i}(0)=$$G_{i}(O)=0$ , and(oo) $=G_{i}(\infty)=\infty$ ;$(A_{1})F_{i},$$F_{i}$$(A_{2})a_{i},$ $b_{i},$ $\mathfrak{g}\in C([0, \infty);(0, \infty)),$$( A_{3})\frac{a_{1}(t)}{b_{1}(t)},$$\frac{c_{1}(t)}{b_{1}(t)};\frac{a_{2}(t)}{c_{2}(t)},$$\frac{b_{2}(t)}{c_{2}(t)}$$i=1,2$ ;are all bounded and bounded away from$( A_{4})\int^{\infty}b_{1}(t)dt=\int^{\infty}c_{2}(t)dt=\infty$

$0$

near

$+\infty$;.

Note that (A3) andimply that $\int^{\infty}b_{i}(t)dt=\int^{\infty}c_{i}(t)dt=\infty,$ $i=1,2$ .
System (S) is a generalization of the classical Lotka-Volterra competition system with$(A_{4})$constant coefficients. We can show that if the initial values $x(O),$ $y(O)$ are both positive,then the corresponding solutions of (S) exist globally on, and remain positive there;
see for example [3, 4].
It is an important problem to find conditions for the global asymptotic stability of(S). When (S) is an autonomous system, there are many contributions to this problem bymeans of phase plane analysis; see for example [3, 4]. For nonautonomous cases of (S),Ahmad & Lazer [1], Ahmad & Montes de Oca [2], and Taniguchi [5, 6] have proposed
criterion for the global asymptotic stability of (S) by employing asymptotic behavior oftime averages of coefficients functions. In this talk, we will consider this problem from adifferent point of view.
Before going ahead we must describe well-known results concerning the classical LotkaVolterra competition systems$[0, \infty)$$\{\begin{array}{l}x’=x(\alpha_{1}-\beta_{1}x-\gamma_{1}y)y’=y(\alpha_{2}-\beta_{2}x-\gamma_{2}y)\end{array}$where$\alpha_{i},$$\beta_{i}$and$\gamma_{i},$(LV)$i=1,2$ , are positive constants. For (LV) we can introduce the isoclines$x+ \frac{\gamma_{1}}{\beta_{1}}y=\frac{\alpha_{1}}{\beta_{1}}$$\beta_{2}$$\overline{\gamma_{2}}^{X}+y=\frac{\alpha_{2}}{\gamma_{2}}$

,(1)

.

(2)

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Theorem A. If line (1) is located under line (2), then every positive solution $(x, y)$ of(LV) satisfies$\lim_{tarrow\infty}(x(t), y(t))=(0,0)$.

(Figure 1.)

Theorem B. Suppose that lines (1) and (2) are located as in Figure 2. (Therefore they
Then every positive solution $(x, y)$ of (LV) satisfies
intersect only at one point$(\xi, \eta).)$$\lim_{tarrow\infty}(x(t), y(t))=(\xi, \eta)$.

$y$

$y$

Figure 2

Figure 1

We will show that analogous results hold for our system (S). This is the aim of thetalk. When (S) reduces to classical Lotka-Volterra competition system (LV), our results
reduce to well-known classical ones.
To state our results, we introduce notation. For bounded function defined near oo
we put.
, and,$f$$f_{L}= \lim\inf f(t)tarrow\infty$$f_{M}= \lim_{tarrow}\sup_{\infty}f(t)$$+$$f_{\infty}= \lim_{tarrow\infty}f(t)$We introduce the following continuous curves in the first quadrant of the xy-plane:$F_{1}(x)+( \frac{c_{1}}{b_{1}})_{\infty}G_{1}(y)=(\frac{a_{1}}{b_{1}}I_{\infty},$$(S_{1})_{\infty}$$( \frac{b_{2}}{c_{2}})_{M}F_{2}(x)+G_{2}(y)=(\frac{a_{2}}{c_{2}})_{L}$,$(S_{2})_{ML}$$( \frac{b_{2}}{c_{2}})_{\infty}F_{2}(x)+G_{2}(y)=(\frac{a_{2}}{c_{2}})_{\infty}$.

$(S_{2})_{\infty}$

and

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By our assumption
these three curves are all downward-sloping continuous curves.
When (S) reduces to classical Lotka-Volterra system (LV), these three curves reduce toisoclines of the system, namely (1) and (2). These three curves can be regarded as isoclinesof limiting systems of (S), in some sense. We will show that asymptotic stability of system(S) can be determined from the relative positions of these curves. Theorems 1 and 2 below
are, respectively, generalizations of Theorems A and B.
$(A_{1})$

Theorem 1. Letexist. If curve
and, then every solution $(x, y)$ of (S) satisfies$(a_{1}/b_{1})_{\infty}$$(c_{1}/b_{1})_{\infty}$$(S_{1})_{\infty}$

is located under curve$(S_{2})_{ML}$$\lim_{tarrow\infty}x(t)=0$

and$0<G_{2}^{-1}(( \frac{a_{2}}{c_{2}})_{L})\leq\lim\inf y(t)tarrow\infty\leq\lim_{tarrow}\sup_{\infty}y(t)\leq G_{2}^{-1}((\frac{a_{2}}{c_{2}})_{M})\cdot$(Figure 3.)

Theorem 2. Let
exist. Suppose that curves
andandare located as in Figure 4. (Therefore they intersect only at one point
Then every solution $(x, y)$ of (S) satisfies$(a_{1}/b_{1})_{\infty},$$(S_{1})_{\infty}$$(c_{1}/b_{1})_{\infty},$$(b_{2}/c_{2})_{\infty}$$(a_{2}/c_{2})_{\infty}$$(S_{2})_{\infty}$$(\xi, \eta).)$$\lim_{tarrow\infty}(x(t), y(t))=(\xi, \eta)$

$y$

.

$y$

Figure 3

Figure 4

We need a simple lemma concerning ordinary differential inequalities, on which theproof of Theorems 1 and 2 are essentially based. Let us consider the following differential
inequalities near :$z’\leq z(p(t)-q(t)H(z))$(3)$\infty$

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and$w’\geq w(p(t)-q(t)H(z))$.

(4)

For them we assume the following:$(L_{1})H\in C([0, \infty);[0, \infty))$ is a strictly increasing function, $H(O)=0$ , and $H(\infty)=\infty$ ;$(L_{2})p,$ $q\in C([0, \infty);(0, \infty))$;(L3) $0<(p/q)_{L}\leq(p/q)_{M}<\infty$ ;$( L_{4})\int^{\infty}q(t)dt=\infty$

Lemma 3. (i) Let$z$.

be a positive function satisfying inequality (3) near$w$

be a positive function satisfying inequality (4) near$\lim\inf w(t)tarrow\infty\geq H^{-1}((p/q)_{L})$. Then,.

$\lim_{tarrow}\sup_{\infty}z(t)\leq H^{-1}((p/q)_{M})$(ii) Let$\infty$$\infty$. Then,.

Proof of Lemma 3. (i) The proof is divided into three cases.
Suppose that $z(t)\geq H^{-1}(p(t)/q(t))$ near . Since $p(t)-q(t)H(z(t))\leq 0$ , we find that$z(t)$ decreases,
exists and $\lim_{tarrow\infty}z(t)\geq H^{-1}((p/q)_{M})$ . To prove$H^{-1}((p/q)_{M})$ , suppose to the contrary that $\lim_{tarrow\infty}z(t)>H^{-1}((p/q)_{M})$ . Then there are
two positive constants and satisfying$\infty$$\lim_{tarrow\infty}z(t)=$$\lim_{tarrow\infty}z(t)$$\delta_{1}$$\delta_{2}$for all sufficiently large .

$H(z(t))>H( \delta_{1})>H(\delta_{2})>\frac{p(t)}{q(t)}$$t$Then inequality (3) implies that$z’(t) \leq q(t)(\frac{p(t)}{q(t)}-H(z(t)))z(t)<q(t)(H(\delta_{2})-H(\delta_{1}))z(t)$$=-q(t)(H(\delta_{1})-H(\delta_{2}))z(t)$.

. This is a contradiction.
, this implies that $z(t)arrow-$ oo as
Since
Next suppose that $z(t)\leq H^{-1}(p(t)/q(t))$ near . Obviously in this case there isnothing to prove.
Finally suppose that the function $z(t)-H^{-1}(p(t)/q(t))$ changes its sign in any neighborhood of . Suppose that $\lim suptarrow\infty^{z(t)}>H^{-1}((p/q)_{M})$ . Then, there are three sufficiently large numbers $t_{1}<\tau<t_{2}$ satisfying $z’(\tau)=0,$ $z(t_{i})=H^{-1}(p(t_{i})/q(t_{i})),$ $i=1,2$ ,.
. It follows therefore that $z’(t)<0$ for
and $z(t)>H^{-1}(p(t)/q(t))$ for
This is a contradiction. The proof is complete.
(ii) As in the proof of (i), the proof is divided into several cases.
Suppose that $w(t)\leq H^{-1}(p(t)/q(t))$ near . Since $p(t)-q(t)H(w(t))\geq 0$ , we findthat $w(t)$ increases,
exists and $\lim_{tarrow\infty}w(t)\geq H^{-1}((p/q)_{L})$ . To prove$tarrow\infty$$\int^{\infty}q(t)dt=\infty$$\infty$$\infty$$t\in(t_{1}, t_{2})$$t\in(t_{1}, t_{2})$$\infty$$\lim_{tarrow\infty}w(t)\in(0, \infty)$

89$\lim_{tarrow\infty}w(t)=H^{-1}((p/q)_{L})$, suppose to the contrary that $\lim_{tarrow\infty}w(t)<H^{-1}((p/q)_{L})$ .

Then there are two positive constants$\delta_{1}$and$\delta_{2}$$H(w(t))<H( \delta_{1})<H(\delta_{2})<\frac{p(t)}{q(t)}$satisfying

for all sufficiently large .
$t$Then inequality (4) implies that$w’(t) \geq q(t)(\frac{p(t)}{q(t)}-H(w(t)))w(t)>q(t)(H(\delta_{2})-H(\delta_{1}))w(t)$.

. This is a contradiction.
, this implies that $w(t)arrow\infty$ as
Since
Other cases can be treated similarly; so we omit them. The proof is complete.
$tarrow\infty$$\int^{\infty}q(t)dt=\infty$$\square$Sketch of the proof of Theorem 1. By Lemma 3, it suffices to show that $\lim_{tarrow\infty}x(t)=$. From the first equation of (S) we get$0$$x’\leq x(a_{1}(t)-b_{1}(t)F_{1}(x))$

near$\infty$. Lemma 3-(i) shows that$\lim_{tarrow}\sup_{\infty}x(t)\leq F_{1}^{-1}((a_{1}/b_{1})_{\infty})^{p}=^{ut}X_{1}$Then, by the second equation of (S) for every$\epsilon>0$.

we have$y’\geq y[(a_{2}(t)-b_{2}(t)F_{2}(X_{1}+\epsilon))-c_{2}(t)G_{2}(y)]$near$\infty$. By Lemma 3-(ii) and our assumptions$\lim\inf y(t)tarrow\infty\geq G_{2}^{-1}((\frac{a_{2}}{c_{2}})_{L}-(\frac{b_{2}}{c_{2}})_{M}F_{2}(X_{1}))^{p}=^{ut}Y_{1}$Again returning to the first equation of (S), we have for every.

$\epsilon>0$$x’ \leq b_{1}(t)[(\frac{a_{1}(t)}{b_{1}(t)}-\frac{c_{1}(t)}{b_{1}(t)}G_{1}(Y_{1}-\epsilon))-F_{1}(x)]$near. So, if, then assumptions (A3) andshow that$\lim_{tarrow\infty}x(t)=0$ ; accordingly the proof is complete. So we may suppose that$\infty$$(A_{4})$$(a_{1}/b_{1})_{\infty}-(c_{1}/b_{1})_{\infty}G_{1}(Y_{1})\leq 0$$(a_{1}/b_{1})_{\infty}-$$(c_{1}/b_{1})_{\infty}G_{1}(Y_{1})\geq 0$.

Repeating the above consideration, we find that,$\lim_{tarrow}\sup_{\infty}x(t)\leq F_{1}^{-1}((\frac{a_{1}}{b_{1}})_{\infty}-(\frac{c_{1}}{b_{1}})_{\infty}G_{1}(Y_{1}))^{p}=^{ut}X_{2}$and$\lim\inf y(t)tarrow\infty\geq G_{2}^{-1}((\frac{a_{2}}{c_{2}})_{\infty}-(\frac{b_{2}}{c_{2}})_{\infty}F_{2}(X_{2}))^{p}=^{ut}Y_{2}$.

90Continuing this procedure, we can construct inductively two infinite sequences
satisfying
and,$\{X_{n}\}_{n=1}^{\infty}$$\{Y_{n}\}_{n=1}^{\infty}$$X_{1}=F_{1}^{-1}((a_{1}/b_{1})_{\infty})$$F_{1}(X_{n})+( \frac{c_{1}}{b_{1}})_{\infty}G_{1}(Y_{n-1})=(\frac{a_{1}}{b_{1}})_{\infty}$$( \frac{b_{2}}{c_{2}})_{M}F_{2}(X_{n})+G_{2}(Y_{n})=(\frac{a_{2}}{c_{2}}I_{L}$,,$n=2,3,4,$$n=1,2,3,$$\ldots$$\ldots$

,

(5)
(6)

,

and
$G_{1}(Y_{n})< \frac{(a_{1}/b_{1})_{\infty}}{(c_{1}/b_{1})_{\infty}}$,$n=1,2,3,$$\ldots$,

unless$( \frac{a_{1}}{b_{1}})_{\infty}-(\frac{c_{1}}{b_{1}})_{\infty}G_{1}(Y_{m})\leq 0$

for some $m\in$ N.

By the assumption of Theorem 1, we can show inductively that
$Y_{1}<Y_{2}< \cdots<Y_{n}<Y_{n+1}<\cdots<G_{1}^{-1}(\frac{(a_{1}/b_{1})_{\infty}}{(c_{1}/b_{1})_{\infty}})$Soturn. Let.

exists, which means that $\lim_{narrow\infty}X_{n}put=\tilde{X}\geq 0$ also exists in$narrow\infty$ in (5) and (6). We then find that$\lim_{narrow\infty}Y_{n}put=\tilde{Y}>0$$F_{1}( \tilde{X})+(\frac{c_{1}}{b_{1}})_{\infty}G_{1}(\tilde{Y})=(\frac{a_{1}}{b_{1}})_{\infty}$,

and$( \frac{b_{2}}{c_{2}})_{M}F_{2}(\tilde{X})+G_{2}(\tilde{Y})=(\frac{a_{2}}{c_{2}}I_{L}$.

These means that two curves
andintersect atcontradiction to the assumption of Theorem 1. Thus it must hold that$(S_{1})_{\infty}$$(S_{2})_{ML}$$( \frac{a_{1}}{b_{1}})_{\infty}-(\frac{c_{1}}{b_{1}})_{\infty}G_{1}(Y_{m})\leq 0$

Since for every$(X, \tilde{Y})\in[0, \infty)\cross[0, \infty)$, afor some $m\in$ N.

$\epsilon>0$$x’ \leq b_{1}(t)[(\frac{a_{1}(t)}{b_{1}(t)}-\frac{c_{1}(t)}{b_{1}(t)}G_{1}(Y_{m}-\epsilon))-F_{1}(x)]$near, we can show that $\lim_{tarrow\infty}x(t)=0$ by assumptions (A3) andthe proof.
$\infty$$(A_{4})$. This completes
口Sketch of the proof of Theorem 2. Arguing as in the sketch of the proof of Theorem 1,
we obtain sequences, andsuch that$\{X_{n}\}_{n=1}^{\infty},$$\{\underline{X}_{n}\}_{n=1}^{\infty},$$\{Y_{n}\}_{n=1}^{\infty}$$\overline{X}_{1}=F_{1}^{-1}((a_{1}/b_{1})_{\infty})$,$\{\underline{Y}_{n}\}_{n=1}^{\infty}$$\overline{Y}_{1}=G_{2}^{-1}((a_{2}/c_{2})_{\infty})$

;

91
,

$0< \underline{X}_{n}\leq\lim\inf x(t)tarrow\infty\leq\lim_{tarrow}\sup_{\infty}x(t)\leq\overline{X}_{n}$$0< \underline{Y}_{n}\leq\lim\inf y(t)tarrow\infty\leq\lim_{tarrow}\sup_{\infty}y(t)\leq\overline{Y}_{n}$$n=1,2,3,$,$n=1,2,3,$;$\ldots$$\ldots$

;

(7)

(S)

and
$F_{1}( \overline{X}_{n+1})+(\frac{c_{1}}{b_{1}})_{\infty}G_{1}(\underline{Y}_{n})=(\frac{a_{1}}{b_{1}})_{\infty}$;(9)$( \frac{b_{2}}{c_{2}})_{\infty}F_{2}(\underline{X}_{n})+G_{2}(\overline{Y}_{n+1})=(\frac{a_{2}}{c_{2}})_{\infty}$

;

(10)

$F_{1}( \underline{X}_{n})+(\frac{c_{1}}{b_{1}})_{\infty}G_{1}(\overline{Y}_{n})=(\frac{a_{1}}{b_{1}})_{\infty}$$( \frac{b_{2}}{c_{2}})_{\infty}F_{2}(\overline{X}_{n})+G_{2}(\underline{Y}_{n})=(\frac{a_{2}}{c_{2}})_{\infty}$

These four formulas show that the points
andand the pointsand
exist on curve
we can find inductively that$(\overline{X}_{n+1}, \underline{Y}_{n})$$(\underline{X}_{n}, \overline{Y}_{n+1})$$(X_{n}, \underline{Y}_{n})$

;

(11)

.

(12)

,
exist on curve$n=1,2,3,$. Therefore,$(\underline{X}_{n}, \overline{Y}_{n})$$(S_{2})_{\infty},$$(S_{1})_{\infty}$$\ldots$,$\underline{X}_{n}\leq\underline{X}_{n+1}\leq\xi\leq\overline{X}_{n+1}\leq\overline{X}_{n}$and$\underline{Y}_{n}\leq\underline{Y}_{n+1}\leq\eta\leq\overline{Y}_{n+1}\leq\overline{Y}_{n}$.

So, these four sequences all have positive limits as $narrow\infty$ :$\lim_{narrow\infty}\underline{X}_{n}=\underline{\xi},\lim_{narrow\infty}\overline{X}_{n}=\overline{\xi},\lim_{narrow\infty}\underline{Y}_{n}=\underline{\eta}$

Letting$narrow\infty$

and$\lim_{narrow\infty}\overline{Y}_{n}=\overline{\eta}$.

in (9), (10), (11) and (12), we have(13)

$F_{1}( \overline{\xi})+(\frac{c_{1}}{b_{1}})_{\infty}G_{1}(\underline{\eta})=(\frac{a_{1}}{b_{1}}I_{\infty}\cdot$$( \frac{b_{2}}{c_{2}})_{\infty}F_{2}(\underline{\xi})+G_{2}(\overline{\eta})=(\frac{a_{2}}{c_{2}})_{\infty}$$F_{1}( \underline{\xi})+(\frac{c_{1}}{b_{1}})_{\infty}G_{1}(\overline{\eta})=(\frac{a_{1}}{b_{1}})_{\infty}$$( \frac{b_{2}}{c_{2}})_{\infty}F_{2}(\overline{\xi})+G_{2}(\underline{\eta})=(\frac{a_{2}}{c_{2}}I_{\infty}$

.

(14)

;

(15)

.

(16)

The formulas (13) and (16) imply that two curves;intersect atandsimilarly curvesand. By the assumption of Theorem 2, itintersect at
must hold that, that is,$(S_{1})_{\infty}$$(S_{1})_{\infty}$$(S_{2})_{\infty}$$(S_{2})_{\infty}$$(\overline{\xi}, \underline{\eta})$$(\underline{\xi}, \overline{\eta})$$(\overline{\xi}, \underline{\eta})=(\underline{\xi}, \overline{\eta})=(\xi, \eta)$$\lim_{narrow\infty}\overline{X}_{n}=\lim_{narrow\infty}X_{n}=\xi$, and$\lim_{narrow\infty}\overline{Y}_{n}=\lim_{narrow\infty}\underline{Y}_{n}=\eta$.

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Lettingproof.

$narrow\infty$in (7) and (8), we can get $\lim_{tarrow\infty}(x(t), y(t))=(\xi, \eta)$ . This completes the$\square$

References[1] S. Ahmad & A. C. Lazer, Average conditions for global asymptotic stability in anonautonomous Lotka-Volterra system, Nonlinear Anal., 40 (2000) 37-49.
[2] S. Ahmad & F. Montes de Oca, Average growth and extinction in a two dimensional
Lotka-Volterra system, $Dyn$ . Conti. Discrete Impuls. Syst. Ser. A Math. Anal., 9(2002) 177-186.
[3] H. I. Freedman, Deterministic Mathematical Models in Population Ecology, M.
Dekker, 1980, New York.

[4] J. Hofbauer & K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998, Cambridge.

[5] K. Taniguchi, Asymptotic property of solutions of nonautonomous Lotka-Volterra
model for N-competition species, Differ. $Equ$ . Appl., 2 (2010), 447-464.
[6] K. Taniguchi, Permanence and global asymptotic stability for a generalized nonautonomous Lotka-Volterra competition system, Hiroshima Math. J., to appear.
Hiroyuki Usami

Department of Mathematical and Design Engineering
Faculty of Engineering
Gifu University
Gifu City, 501-1193
Japanhusami@gifu-u.ac.jp