Asymptotic behavior of quasi-competitive species defined by Lotka-Volterra dynamics

Vol.11 No.4

ACTA MATHEMATICAE APPLICATAE SINICA

Oct., 1995

ASYMPTOTIC BEHAVIOR OF QUASI-COMPETITIVE SPECIES DEFINED BY LOTKA-VOLTERRA DYNAMICS* L U GANG ( ~ -

~])

(Department of Mathematics, Huazhong Normal University, Wuhan 430070, China) B . D . SLEEMAN

(Department of Mathematics and Computer Science, The University of Dundee, Dundee DD1 4HN, U.K.)

Abstract Competitive systems defined by Lotka-Volterra equations

j=l

where ri>0, aij :>0, have been extensively studied in the literature. Much attention has been drawn to, among other things, the non-periodic oscillation phenomenon, or May-type trajectory as it is called by some authors, since the discovery of that kind of trajectories in competitive LotkaVolterra systems made by May and Leonard [2]. Recently, the same phenomenon was reported to be existing in prey-predator systems. In this paper it is clear that one can expect the appearance of such phenomenon in a broader class of Lotka-Volterra systems, namely quasi-competitive systems (i.e. ri>0. (al)/ajj)+(aj:/a,i)>O in (I)), which cover both competitive and some prey-predator systems in addition to others. Conditions are established in terms of the parameters of the systems for the existence of stable equilibrium, periodic oscillation and non-periodic oscillation. Key words.

Non-periodic oscillation, competitive, prey-predator, LotkaoVolterra systems,

population biology

1.

Introduction

In population biology one of the most widely used mathematical models describing competitions among species is the Lotka-Volterra system

&i-~xi(ri4-fiaijxj),

i - - 1,2,-.. ,n,

(1.1)

j----1

Received April 11, 1992. *This work was supported by the Sino-British Friendship Scholarship Scheme and in part by the National Natural Science Foundation of China.

No.4

Q U A S I - C O M P E T I T I V E DYNAMICS

349

where x~ and r~ denote, respectively, the carrying capacity and the intrinsic growth rate of the species, aij denotes the interaction of the j - t h species on the i-th one. aii, the environmental constraint on the carrying capacity of the i-th species, is usually assumed to be negative. The system (1.1) is called a competitive system if aij < 0 (i # j ) , and a prey-predator system if a~jaj~ < 0 (i # j ) . It is well known t h a t the global asymptotic behavior of system (1.1) is easily determined in the two-species case. However for n >_ 3 the situation becomes extremely complicated. For example, Smale [5] pointed out that strange attractors may appear in competitive systems if n > 5. May and Leonard [2] studied a competitive system of three species (n = 3), and for certain ranges of parameters demonstrated the existence of (i) a globally stable positive equilibrium, (ii) periodic oscillations and most interestingly, (iii) non-periodic oscillations. Non-periodic oscillatory trajectories are now commonly referred to as May-type trajectories and have aroused considerable interest in recent years. In particular, Coste et al. [1] investigated general competitive systems using bifurcation methods and found similar phenomena for three species competitive Lotka-Volterra equations under certain conditions. In the case n = 4 they found that May-type trajectories are unlikely to be preserved since stable limit cycles may appear. Schuster et al. [41 presented another approach and proved t h a t for a class of unsymmetric three dimensional Lotka-Volterra systems there is an open set of May-type trajectories in R~_. Recently, Roy and Solimano [31 found May-type trajectories in a class of prey-predator Lotka-Volterra systems (or food chains) which suggests that for a broader class of three dimensional systems one may expect results similar to those May and Leonard [2] established for competitive systems. The results of this paper show that this conjecture is true for a class of Quasi-competitive Lotka-Volterra systems in a sense to be made precise below. 2.

Quasi-competitive

Systems

Assuming t h a t the three species have the same growth rate r > 0 and that the environmental constraint on the carrying capacity is negative, we propose the following system X l = X l ( r -- a l l X l + a12x2 ~- a13x3),

X2 = X2(T "~ a21xl -- a22x2 Jr a23x3),

(2.1)

~:3 : x 3 ( r A- a 3 1 x l + a32x2 -- a33x3),

where r > 0 , aij > 0, i = 1,2,3. If we scale x~ by the factor aii/r, i = 1,2,3, and the independent variable t by r, then (2.1) takes the form xl -- xl(1 - xl + a12x2 -4- a13x3), J:2 : X2(1 "~ a21xl -- x2 + a23x3),

(2.2)

X3 = X3(1 + a31xl + a32x2 -- X3). Note that the coefficients a~j in (2.2) should not be confused with those in (2.1). If aij < 0 in either (2.1) or (2.2) then the systems are referred to as "competitive systems". Here we propose a weaker condition and suppose t h a t aij + aji < 0,

i # j.

(2.3)

In this case (2.2) is referred to as a "quasi-competitive system". To be precise, we suppose that a~j + aji = - 2 ~ < 0, i # j (2.4)

350

ACTA MATHEMATICAE APPLICATAE SINICA

Vol.ll

a n d refer to (2.2) with (2.4) holding true as a uniform quasi-competitive system. Before discussing the b e h a v i o r of system (2.2) we mention two examples of uniform quasi-competitive systems which have been studied in the literature. E x a m p l e 2.1 [2]. This is an example of pure competition: Xl --~ Xl(1 - xl - c~x2 - ~x3), ~:2

----X2(1

- - j 3 X l - - X 2 - - O~X3) , "

Ot,~

>

0,

X3 ----X3(1 -- CeXl -- j3X2 -- X3). Example

2.2 [31. Here we have a p r e y - p r e d a t o r s y s t e m of the form ;~1 = X l ( 1 -- "T1 -- O~X2 -~ f~X3), :~2 = X 2 ( 1 3 t ' ~ X l

--Z2--O~X3),

0~,~ > O,

X3 = X3(1 -- (::~Xl -~ ~ X 2 -- X3)-

3.

The Main Results

In this section we formulate the main results of this paper, the proofs of which are contained in Section 4. Suppose condition (2.4) holds; then on writing a12 -- ~l, a23 = a2, a31 -- ~3 we have (1) W h e n 0 < ~ < t, the unique positive equilibrium point of (2.2) is globally a s y m p totically stable in R 3. (2) W h e n fl = 1 a n y t r a j e c t o r y in R~_ will t e n d (as t --* co) to a closed orbit contained in the plane xl + x2 + x 3 -- 1 and having the general form xlClx2C2x3Ca : v0, where cl -(I-}-O 1, the positive equilibrium point is unstable, and the system is bounded in the following sense. If ~i _< 1 - 2~ or c~{ _> - I (i --- 1,2,3) then any trajectory in R 3 (except for the positive equilibrium and possibly its one dimensional stable manifold) will tend to a singular closed orbit which is composed of three arcs lying on the three coordinate planes and connecting the three non-zero equilibrium points on the three coordinate axes respectively. If I - 2 ~ < ~i < - i (i -- 1,2,3) then any trajectory in R3, apart from the above exception, will tend to one of the six non-zero equilibrium points on the different coordinate planes and axes respectively. Biologically our results can be interpreted as follows (i) 0 < ;3 < I: Here we have stable competition with stable coexistence of the three species. (2) ~ -- I: In this case we have neutral competition leading to periodic oscillations of the three species. (3) ~ > i: If 1 - 28 < ~i < - 1 then there is eventual extinction of the two species, while if ~i _< 1 - 213 or cq >_ - 1 then there is eventual extinction of all three species through non-periodic oscillations. These are the so-called May type trajectories.

4.

Proofs

%Ve begin by showing t h a t for quasi-competitive systems ever)" t r a j e c t o r y in R3+ is b o u n d e d for all non-negative t. Letting U = ~ i =3 l xi, t h e n along trajectories in R ~ we have 3

3

3

3

3

i...~ l

i= l

2 i= l

i= l

i= l

i~j j>i

No.4

QUASI-COMPETITIVE

DYNAMICS

351

Consequently there exist constant C and e > 0 such t h a t 3

0 +

3

<

+

3 2

Z

i=1

i=l

=

x,(1

-

< c

i=l

Hence

u(t) < c + £

which implies t h a t every t r a j e c t o r y in R~_ is bounded. We now proceed to prove of the results (1), (2) and (3) given in Section 3. Firstly consider the case 0 < /3 < 1, or equivalently, - 2 < aij + aji < 0 in (2.4) and assume t h a t s y s t e m (2.2) admits a unique positive equilibrium X0 = C - (cl, c2, c3) T > O. It is easy to see t h a t the function 3 W(xi,x2,x3)

=

c i

-

_

i=1

is well-defined and non-negative in R 3, with W = 0 only if X = X0 = C. By taking the derivative of W with respect to t along the trajectories of (2.2) in R 3, we obtain 3

IV(t) = ~

3

y ~ aii(xi - ci)(x i - cj) = _ I x T (A + A T ) x ,

i=1 j=l

where A = ( - a i j ) with a~i --- - 1 , X = (xl - c l , x 2 - c2,xa - c 3 ) T . If A + A T is positive definite, t h e n l)d(t) < 0 and consequently X0 = C is globally asymptotically stable in R 3. T h e principal minor d e t e r m i n a n t s of the m a t r i x A + A T are det J1 = 2, det J2 = 4(1 - / 3 ) > 0 and d e t J 3 -- 2(/3 - 2)2(/3 -t- 1) > 0, which ensures t h a t A + A T is positive definite in the range - 1 < / 3 < 1, and includes the case 0 < / 3 < 1. Next, we consider the case/3 -- 1 or aij q- aji = --2 in (2.2) by introducing two auxiliary a xlc~, where ci (i = 1, 2, 3) are the same as those defined functions U --- ~ i =3 1 xi and V = Yli=l in (2) of Section 3. After p e r f o r m i n g differentiations along the t r a j e c t o r y of (2.2) in R~_, we obtain U(t) = U(1 - U) and l/(t) -- V(1 - U), from which we conclude t h a t every t r a j e c t o r y of (2.2) will eventually a p p r o a c h the plane U -- 1 and t h a t there is a family of closed orbits of (2.2) on t h a t plane, which surround the equilibrium X0 -- C and fill up the p a r t in R 3 o f the plane. For the last case /3 > 1, or aij q- aji < --2 in (2.2), we need to know the qualitative properties, such as stability and invariant manifolds, of the equilibria of (2.2), before we investigate the a s y m p t o t i c behavior of any t r a j e c t o r y in R~_ T h e s y s t e m (2.2) has at most eight equilibria in R 3 w h i c h are O(0,0, 0)i P(Cl, c2,ca) (ci > 0), Q1 (1, O, 0), Q2(o, 1, 0), Qa(O, o, 1), $1 (811,812,0), S2(o , 822 , 823), a n d S 3 ( s 3 1 , O, 833 ) > o). T h e following four l e m m a s accomplish our first goal a b o u t the equilibria and their properties. T h e proofs are ~eft to the appendices. L e m m a 1. & e R 3 ( i -- 1,2,3) if and only if 1 - 2/3 < ai < - 1 or equivalently aij < - 1 (i ¢ j ) . If P ( c l , c~, c3) e R 3, then Si is unstable with a stable manifold in R 3, i -- 1,2,3. L e m m a 2. Qis are a s y m p t o t i c a l l y stable if 1 - 2/3 < a~ < - 1 , i.e. aij < - 1 (i ¢ j ) , and unstable either if a i < 1 - 2/3, i.e. a21,aa2,ala > - 1 , or if c~i > - 1 , i.e.

352

ACTA MATHEMATICAE APPLICATAE SINICA

Vol.ll

a12, a23, a31 > --1. W h e n Qis are unstable, there exist saddle connections Q1Q2, Q2Q3 and Q3Q1 on the planes x l - x2, x2 - x3 and x3 - x l , respectively, which implies t h a t there is a singular loop Q1 ~ Q2 --* Q3 --~ Q1 on the coordinate planes i f a ~ < 1 - 2 f l (or Q1 --* Q3 ~ Q2 --+ Q1 if a i > - 1 ) . L e m m a 3. Let A = (-a~j) with all = - 1 , and I = (1, 1, 1) T. If ~ > 1, a i _> - 1 (i = 1 , 2 , 3 ) or a i < - 1 (i = 1,2,3), then the e q u a t i o n A X = I has a positive solution Xo = C = (cl,c2,c3) T if a n d only if the equation A T y = I has a positive solution Y0 = 3 ci = ~'~=t 3 d ~ < 1. • D = ( d l , d2, d3) T, 0 < ~-,~=1 L e m m a 4. If ~ > 1, then P(Cl, c2, c3) > 0 is an equilibrium point with one negative eigenvalue A1 = - 1 and two positive (real part) eigenvalues; hence P is unstable with a one dimensional stable manifold. H a v i n g o b t a i n e d a qualitative information a b o u t the equilibria, we can move on to achieve our second goal for the case ~ > 1, i.e. to e x a m i n e t h e a s y m p t o t i c behavior of any t r a j e c t o r y of (2.2) in R 3 b y means of a set of auxiliary functions as follows: 3 U -~Exi, i~1 3 Y ----H xd~ i , i=l R - - C l X 2 -- C2Xl

C3X 1 -- ClX 3

A T D = f,

D = (dl, d2, d3) T > O.

A C = I,

C = (cl, c2, c3) T > O.

T h e differentiations of the functions along t h e trajectories of (2.2) lead to the following expressions

= V i E3 di + ( - d l + a21d2 + a31d3)xl + (a12dl - d2 + a32d3)x2 i=1 + (a13dl + a23d2 - d3)x3] 3

3

i=l 3

i=1

/

'~

U);

(4.1)

i=l

2 i=1 3

1)

i=1

=u(1

-

u);

3 ~r = u - Z x~ -

(4.2)

2~(~lX~ +

~1x3 + ~ x 3 ) = v - X ~ B X ,

i=l

where the m a t r i x 1

!)

(4.3)

No.4

QUASI-COMPETITIVE

DYNAMICS

353

has the eigenvalues At = 1 + 213 > O, Cl

A2'3 = 1 - t3 < O;

(4.4)

= ( C ~ l - - - ~ 3 ) ~ { - ( ~ + 213 - 1)c3Xg~l - (~2 + 213 - 1 ) ~ g ~ -- (oz3 + 213 -- 1)c2x~xl - (Cel + 1 ) c 3 x l x g -- (oz2 + 1)ClX2X 2 -- (0~3 + 1)C2XaX~ +

+ C~I q- 213)Cl "}- (O~1 Jr- O~2 + 213)C2 q- ((~2 + ~3 -~" 213)C3]XlX2;g3} •

(4.5)

T h e following conclusions m a y be drawn from (4.1)-(4.5), which will complete our proof for the case/3 > 1. We observe, firstly, from (4.4) t h a t U -- X T B X is a two-sheeted hyperboloid with one sheet in R~_and the positive equilibrium X0 -- C on t h a t sheet. Secondly, we conclude, from (4.1)-(4.3), t h a t every t r a j e c t o r y in R3will eventually enter the region { ( x l , x 2 , x 3 ) : 0 < Y < 1, u - - X T B X < 0} and in this region when U > ~i3__1 c~, If < 0 (from (4.1)) which shows that the trajectories will go towards the b o u n d a r y of R~_, i.e. the c o o r d i n a t e planes, or in other words, the w-limit sets of the trajectories in R~.(except Xo = C and its stable manifold) lie on the c o o r d i n a t e planes. Therefore, we focus our attention on the coordinate planes and discuss two situations: (a) 1 - 213 < a l < - 1 (13 > 1), i.e. a~j < - 1 (i # j ) in (2.2): from L e m m a s 1 a n d 2, the a t t r a c t o r s on the planes are Sis and Q~s, where Sis are unstable b u t each has a stable ~3 manifold in R~_, therefore, almost all the trajectories in R + h a v e one of the Qis as their w-limit sets; (b) a i < 1 - 2 / 3 (/3 > 1), (i.ea21,a32,a13 > - 1 in (2.2)), o r a ~ > - 1 , (i.e. a12,a23, a31 > - 1 in (2.2)): on the one hand, since the w-limit sets of the trajectories are on the coordinate planes, x l x 2 x 3 ~ 0 as t --* + o e , which implies t h a t the cubic terms in (4.5) are negligible for t sufficiently large. Under the current conditions, the remaining t e r m s in (4.5) are either all positive (a~ < 1 - 213) or all negative (a~ > - 1 ) , m e a n i n g t h a t the' trajectories r o t a t e when they are near the coordinate planes. On the other hand, L e m m a 2 shows t h a t on the planes, there is only one singular loop Q1 --~ Q2 --* Q3 --~ Q1 (a~ < 1-213) or Q1 --* Q3 --~ Q2 --* Q1 (hi > - 1 ) , which has the p r o p e r t y of a b s o r p t i o n to the trajectories in R 3+- Hence every t r a j e c t o r y in R a ( e x e e p t Xo = C and its stable manifold) is a M a y t y p e t r a j e c t o r y which displays non-periodic oscillations.

5.

Discussion

In this section, we first present several e x a m p l e s to illustrate the results in Section 3, and then briefly discuss their biological significance. Examples (1) P r e y - p r e d a t o r : /3 = 1/2, a l = 2, ~2 = 3, a3 = 5 or a12 = 2, a~l = - 3 , a13 = - 4 , a31 -- 5, a23 --- 3, a32 --- - 4 in (2.2). Since 13 = 1/2 < 1, P ( c l , c 2 , c 3 ) is globally stable in (2) C o m p e t i t i o n : /3 = 1, ~1 -- - 1 / 2 , a2 -- - 1 / 3 , a3 = - 1 / 5 or hi2 = - 1 / 2 , a21 ---- 3 / 2 , a23 ---- - 1 / 3 , a32 ------ 5 / 3 , hi3 ---- - 5 / 9 , a31 ---- - - 1 / 5 in (2.2). Since 13 =- 1, the s y s t e m is oscillatory periodically on the plane Xl + x2 + x3 -- 1. (3) P r e y - p r e d a t o r : 13 = 1, a l = 2, a2 = 3, a3 = 5 or a12 = 2, a21 = - 4 , a13 = - 7 , a31 = 5, a23 = 3, a32 --- - 5 . Since/3 = 1, the situation is the same as in (2). (4) P r e y - p r e d a t o r - c o m p e t i t i o n : 13 = 2, 0~1 = - 1 / 2 , a2 = 2, a3 = 3 or a12 = - 1 / 2 , a21 = - 7 / 2 , a13 = - 7 , a31 -- 3, a23 = 2, a32 -~-- --6. Since 13 = 2 > 1 and c~{ > - t , the

354

ACTA MATHEMATICAE A P P L I C A T A E SINICA

Vo1.11

system admits non-periodic oscillation towards the singular loop Q1 --+ Q3 --+ Q2 ---+ Qt on the coordinate planes. (5) Competition: 13 -- 2, a l = - 7 / 2 , a2 -- - 1 0 / 2 , a3 -- - 1 1 / 4 or a12 = - 7 / 4 , a21 = - 9 / 4 , a13 = - 5 / 4 , a31 = - 1 1 / 4 , a2a = - 5 / 2 , a32 = - 3 / 2 . S i n c e ~ = 2 > t and 1 - 2/3 < a~ < - 1 , almost all the trajectories in R3will go towards one of (1, 0, 0), (0,1, 0) and (0, 0, 1). We have shown in the above sections that there are four possibilities of the species evolution for the three species uniform quasi-competition Lotka-Volterra systems, namely, stable coexistence of the three species; periodic coexistence of the three species; eventual extinction of the two species and extinction of all three species in a form of non-periodic oscillation. Numerically, the non-periodic oscillation phenomenon may not easily be observed if t is reasonably large because as the trajectory approaches the coordinate planes, the precision requirement for continuing the oscillation may be beyond the computer capacity. W h a t one actually observes is that the trajectory will settle down at one of (1,0,0), (0, 1, 0) and (0, 0, 1), depending on the pre-chosen precision of integration of the system. Biologically, May and Leonard [r] pointed out that the non-periodic oscillation phenomenon has no significance in reality, the reason being that the Lotka-Volterra models may not be good simulations of the real ecological system. Indeed the L-V equations are linear expressions for the relative growth rates (1/xi)(dxi/dt) of the species. Mathematically, however, it is of interest to observe in broader classes of the L-V equations the non-periodic oscillation phenomenon. At a first glance, we can easily derive the conditions for the existence of singular loop on the coordinate planes for the case of quasi-competitiofi systems, i.e. if aij + aji < - 2 (i ~ j ) and a12,a23,a31 > --1 (or a21,a32,a13 :> - - 1 ) , then there is a singular loop Q 1 - + Q3 --+ Q2 --* Q1 (or Q1 --* Q2 --+ Q3 -+ Q1) on the coordinate planes, where Q1 -- (1, 0,0, ), Q2 = (0,1, 0), Q3 = (0, 0,1). For the more general case when ri, the growth rates of the species, are not necessarily equal to each other, it seems that the singular orbit on the coordinate planes may still exist. Nevertheless, it is not clear whether such conditions are sufficient for the singular orbit to be the unique attractor of all the trajectories in R~_(except the positive equilibrium point). Appendix

1.

Proof

of Lemma

1

We consider the system (2.2) with the condition of aij + aji < - 2 (i ~ j). Let A(X) be the variational matrix of (2.2) at the point X and Sz(sll,s12,0), S2(O, s22,s~3) and S3(s31,0, s33) be the equilibrium points of (2.2) on the coordinate planes but not on the axes, where A12 Sll

---~ A 1

822 ~

1 -{- a 1 2 --

/~23 __ -/~2

A31 833

~,,

A3

1 -- a 1 2 a 2 1

A21 ,

8 1 2

,

A32 823 ~--A 2

,

831 = i 3

1 + a23 1 -- a 2 3 a 3 2

1 -}-a31 =

1 -- a13a31

~

A 1

1 J r a21 --

1 -- a12a21

,

1 -}- a 3 2 --

A13

1 -- a23a32

,

1 -{- a31 --

1 -- a13a31

--3

are all positive so that Si E R+. We see that Aij and Aji cannot be both positive since then we would have 1 -t- aij > 0 and 1 + aji > 0 (i :fi j), which would imply that 2 -t- aij -t- aji ~ 0 --3

(i ¢ j), contradicting our assumptions. Thus sij > 0, or equivalently Si E R+(i = 1, 2, 3), hold if and only if A~i < 0, Aj~ < 0 and A~ < 0 (i---- 1,2,3, j ~ i), i.e. a~j < --1 (i ¢ j).

No.4

QUASI-COMPETITIVE DYNAMICS

355

We assume that the positive equilibrium point P(cl,c2, c3) E R z+, and examine the stability of $1; the same method applies to $2 and $3. The variational matrix of (2.2) at $1 is a12311

A(S ) = ( a12812 --Sll

--812 0

0

\

a13811

a23812 ~ ' 1 -{-a31811 -}-a32812

which has the eigenpolynomial det (AI - A(S1)) = (A - 1 - a31sl, - a32s12)[A 2 + (sll + s12)A + (1 - at2a21)slls12]. Since A1 = 1 -- a12a21 < O, $1 = -$1 is a saddle on the (xl,x2) plane with A1 > 0, A2 < 0. The third eigenvalue of A(S1) is

A3 = l +aalSll +a32s12 = ~---~(Al +a31A12+a32A21)=

~-~c3detA,

where det A c3 = det

-a21

1

\ --an1

.

--a32

Now we have A1 ---- 1 -- ai2a21 < 0 (since a l i < --1), ca > 0 (since P(ci, c2, ca) E R~) and det A > 0 (we can prove that det A(cl + c2 + c3) > 0), so A3 < 0 and $1 is a saddle with two negative and one positive eigenvalues and has a stabte manifold in R 3+.

A p p e n d i x 2.

P r o o f of L e m m a 2

The variational matrix of (2.2) at the equilibrium point Q~(1, 0, 0, ) is

A(Q1) =

(-1 0

a~2 1 -}- a12

0

0

a~3) 0 1 2a a31

with eigenvalues A(Q1) = - 1 , 1 + a21 and 1 + a31. If aij < - 1 (i # j), then A(Q1) < 0. By arguing as above we obtain that aij < - 1 (i • j) imply A(Q1), A(Q2) < 0 and A(Q3) < 0. In other words, Q1,Q2 and Q3 are stable nodes if $1,$2 and $3 E R~. We turn to consider another case, i.e. the conditions

aij + aji < --2

(i # j);

a12, a23,a31 > - - 1

aij+aji - 1 .

or

This time we see that Q1, Q2 and Q3 are saddles with two negative and one positive eigenvalues. On the (xl,x2) plane, the original three dimensional system (2.2) reduces to a two dimensional one ;~1 --'=Xl(1

--

Xl "]- a12X2),

5:2

-----

X2(1 -]- a21xl -- x2),

and the equilibria Q1(1,0,0) and Q2(0,1,0) degenerate into two dimensional equilibria QI(1, 0) and (~2(0,1) with the eigenvalues A(Q1) -- - 1 , 1 + a21 and A((~2) = - 1 , 1 -{- a12. Consequently, Q1 is a saddle and Q)2 is a stable node on the (xl, x2) plane if a21 > - 1 and the unstable manifold of Q1 in R~. goes to Q2 to form the Q1 -* Q2 part of the whole singular loop Q1 --* Q2 -'~ Q3 --~ Q1, which is easily accomplished by taking the same consideration on (x2,x3) and (x3,xi) planes. The case a12,a23, a31 > - 1 can be dealt with

356

ACTA MATHEMATICAE APPLICATAE SINICA

Vol.ll

similarly. Appendix

3.

Proof

of Lemma

3

For convenience, we set a12 = or1, a 2 3 - - a 2 , a 3 1 ---- oz3 a n d aij + aji = - 2 ~ (i # j) in (2.2), and then the conditions aij + aji < - 2 (i # j) b e c o m e f~ > 1 under the new notations and the coefficient m a t r i x of (2.2) now has the form of A =

Ctl

+2¢~

1

-~3

o

-

12

~ 2 + 2¢~

If det A # 0, t h e n b o t h A X = I and A T y = I ( I = (1, 1, 1) T) have, respectively, unique solutions X = C a n d Y = D. Noticing t h a t D T A C = ( D T A C ) T, we have 3

3

ci = c T I =

C T A T D = (D T AC) T = D T A C + D T I = ~

i=1

di.

i=1 3

3

F r o m (4.2) we deduce t h a t 0 < ~ = 1 ci < 1, and hence 0 < Y~=~ d~ < 1. Let U0 = 3 ~-2~i=1ci = Y'~i=l d~, 2/3 = 2 + 6, c~i = ui - 1, and rewrite A C = I and A T D = I as 3

ul + 6 -u3

0

-u2

u2 + 5

0

~3 ]

=

-ul

0

u2 + 5

u3 + 6

-u2

0

~2 773

1

,

1 =

,

OF

AL~ = I ,

AT~ = I,

w h e r e ~ i = c i / ( 1 - U o ) and rh = d ~ / ( 1 - U o ) (i = 1,2,3). Since ci~i > 0 a n d dir/i > 0, we need only to prove t h a t AI~ = I has a positive solution ~ > 0 if and only if A ~ = I has a positive solution r! > O. By C r a m e r ' s rule we set ~.~ = A i / A and ~h = A ~ / A (i = 1, 2, 3), where A = d e t Al = 5152 + 5(ul + u2 + u3) + (utu2 + u2u3 + u3ul)]; 3

a~ =6: + 6(2~: + ~ ) + ~: ~ ~ ;

3 ! ~62

~,~

+ 5(2~: + ~,) + ~ Z ~';

i=1

i=l

3

zx~ :62 + 6(2~3 + ~1) + ~3 Z ~;

3

zx; = 62 + 6(2u3 + ~2) + ~z ~

u~;

i=1

i=1

3

3

a~ =62 + 6(2~1 + u~) + ~1 ~ u~;

~

= 62 + 6(2ul + ~31 + ~ 1 E u~.

i=1

i=1

Since 3

3

3

~Ai=~-2A~=362+36~ui+ i=1

i=1

3

(i~=lul)

2

>0,

i=1

we proceed to show t h a t Ai > 0 (i = 1, 2, 3) and A~ > 0, (i = 1,2, 3) are equivalent.

No.4

QUASI-COMPETITIVE

DYNAMICS

357

I f f l > 1 a n d a~ > - l , i . e . 6 > 0 and ui >_ 0, b o t h Ais and A~s are positive, so the result holds in this case. If fl > 1 a n d a~ < - 1 (i = 1, 2,3), i.e. 6 > 0 and u~ < 0 (i = 1,2,3), noticing t h a t the q u a d r a t i c functions &is and A~s of 5 have the m i n i m a l points

&,~ = -

&,: = -~(2u~+~ + ud

(2ug+l + u~+2),

on 0 < 5 < + e c , we know t h a t Ai > 0 (i = 1, 2, 3) and A~ > 0 (i = 1, 2, .3) correspond, respectiveIy, to min { A i ( & ~ ) } > 0 and min {A~(6A:)} > 0. B u t 1,2,3

1,2,3

{ A i ( S A ~ ) } = ~{4U2U 3 _ U2, 4UlU 3 _ U2, 4 U l U 2 _ u 2} = { / k : ( S A : ) } '

so the m i n i m a of the two sets are equal, and consequently Ai > 0 (i = 1, 2, 3) implies t h a t A~ > 0 (i = 1, 2, 3) and vice versa. Hence the result holds for the case ui < 0 (i = 1,2, 3), and this completes the proof. Note. B y the same a r g u m e n t as used in the proof of L e m m a 3, one can prove t h a t det A > 0 if P(c~, c2, c3) ~ R~_.

Appendix

4.

Proof

of Lemma

4

T h e variational m a t r i x of (2.2) at P is _C 1 A(P) =

-(~z + 2Z)cl)

-(al +23)c2

~1Cl --C 2

C~3C3

- { ~ 2 + 2fl)cz

OL2C2

-c3

and the eigenpolynomial is [ A I - A(P)] =

A+cl (~1 + 2 9 ) c ~ --O~3C3

-alcl

(a3 + 2fl)ct

A + c2 (a2 + 23)cz

-c~2c2 ,\ + c3

(A4.1)

Since cl, c2, c3 satisfy the linear system -- C1°+ otlC2 -- (O~3 -4- 2fl)c3 = --1,

cl - aic2 + (a3 + 23)cz = 1,

( a l + 23)ct + c2 - a 2 c z = 1, -

-- (aX + 2fl)Cl -- C2 + a2c3 = --1,

or

C~3Cl -- (aS + 2#)C2 -- C3 = --1,

a3cl + (as + 2 3 ) c 2 + c3 = 1,

hence if we take A = - 1 in (A4.1), then ] -

I -

A(P)!

( a l + 2fl)c2

- ( a l + 2Z)cl + a~c3

(a3 + 2fl)ct --r~2C2

--OL3C3

( a 2 -~- 2fl)C 3

c~zcl - (a2 + 2fl)c~

a l c 2 - (a3 + 2fl)c3

=

--

1 CLC2C3

-alcl

0 0

-alclc2 - ( a l + 2fl)clc2 + O~2C2C3

(O~3 + 2fl)ClC3 ~{:~2C2C3

0

(a2 "+"2fl)C2C3

--(a2 -t- 2fl)C2c3 + ~3ClC3

= 0,

which suggests t h a t the eigenequation IM-A(P)I=O has a root A = - 1 . If we write the equation (A4.2) as A3 + ~ A2 + ~?A + 7 = 0,

(A4.2)

358

ACTA MATHEMATICAE A P P L I C A T A E SINICA

Vol.11

A3 + ~A2 + ~TA+ ' y ~ (A + 1)[A2 + ( ~ _ 1) + (77- ~ + 1)],

(A4.3)

we have where ~? - ~ + 1 = 7- After simple calculations we find ~=C

1 "~- C2 -]- C3,

~/ = ClC2Cadet A ,

wherect=

detA~ c 2 = d~tA~ and c a = ~ w ith det A ' det A ' det A A =

(al + 2~) -aa

1 (a2 + 2/~)

12

and Ai is the matrix obtained by replacing the ith column of the matrix A with the vector (1, 1, 1) r . We assert that det A > 0 and so 7 > 0, since ci > 0 (i ---- 1, 2, 3). In fact, 3

E

det A3 =3 + 12~32 + 2/~(3al + 3a2 + 3a3 - 3) + (al + (~2 + a3) 2

i=1

=3 + 12/~2 + 6 ~ ( ~ - 1) + 42 =12D 2 + 6~(¢ - 1) + (3 + 42)

(al + a2 + 53 = 4)

is a quadratic polynomial of Z and has its minimum at fl0 = - ( 1 / 4 ) ( ¢ - 1) with the minimal 3 value q(~0, 4) ---- (1/4)(4 + 3) 2 > 0, hence for each Z > 1 and a t , c~2,a3, ~ i = l det A~ > 0. Consequently, det A = (~-~3=1det A ~ ) / ( ~ i =3 1 c~) > 0 which confirms our assertion. Now we consider the roots of the quadratic polynomial in (A4.3) A2 ~ - ( ~ - 1 ) A + ( ~ / - ~ + 1 ) = 0 , we have A± = (1 - ~) 4- X/(~ - 1) 2 - 4(~} - ~ +"'i"i'' 2

where 1 - ~ = 1 - ( C l a r C 2 + C 3 ) > O (~ :> 1 implies 0 < cl + c 2 + c 3 < 1) and ~ / - ~ + 1 = 7 > 0, so either A+ and A_ are both complex with positive real part or they both positive real roots, hence P ( c l , c2, ca) is unstable with a one dimensional stable manifold if ~ > 0.

References [1] J. Coste, J. Peyraud and P. Coullet. Asymptotic Behaviours in the Dynamics of Competing Species. SIAM J. Appl. Math., 1979, 36: 516-543. [2] R. May and W. Leonard. Nonlinear Aspects of Competition Between Three Species. SIAM J. AppL Math., 1975, 29: 243-253. [3] A. Roy and F. Solimano. Global Stability and Oscillations in Classical Lotka-Volterra Loops. J. Math. Biol., 1987, 24: 603--616. [4] P. Schuster, K. Sigmund and R. W'olf. On u~-limit for Competition Between Three Species. SIAM ,I. AppL Math., 1979, 37: 49-54. [5] S. Smale. On the Differential Equations of Species in Competition. J. Math. Biol., 1976, 3: 5-7.