Bifurcations and Transitions to Chaos in the Three-Dimensional Lotka- VolterraMap

L. Gardini; R. Lupini; C. Mammana; M. G. Messia

SIAM Journal on Applied Mathematics, Vol. 47, No. 3. (Jun., 1987), pp. 455-482.

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001

SlAM J. APPL. MA1-H. 01987 Society for Industrial and Applied Mathemat~cs Vol. 47, No. 3, Junc 1987

BIFURCATIONS AND TRANSITIONS TO CHAOS IN THE THREE-DIMENSIONAL LOTKA-VOLTERRA MAP*

Abstract. The three-dimensional cyclically invariant Lotka-Volterra map is studied, pointing out, by analytic and numeric techniques, the bifurcations of fixed points that have no counterpart in the corresponding system of ordinary differential equations. In particular, the Hopf-bifurcation of the three-population equi- librium point, and the related dynamics in a bounded invariant domain, are analysed. The analytic description of stable periodic orbits in a weak-resonant case is given, and the transitions to chaotic attractors via sequences of Hopf-bifurcations and period-doublings are discussed.

Key words. Lotka-Volterra map, bifurcations, chaotic attractors

AMS(M0S) subject classifications. 34C35, 32A15

1. Introduction. Most commonly, the dynamics of systems with N degrees of freedom is described by sets of ordinary differential equations, of the form x = f(x), x E R N, or by discrete-time iterations of the form x' = F(x), x E RN. Continuous dynamics is more common in physical systems, while in biological, social and economi- cal systems discrete dynamics is often more appropriate, because of the discontinuity in the accumulation and growth processes [I]-[3].

Within the latter class of systems, particularly in population dynamics, the most widely used model is the set of Lotka-Volterra equations [4], [5], L.V. henceforth, which can be written, in general, as

in the continuous-time formulation, and as

in the corresponding discrete-time version. Here the r,'s and R,'s represent the linear growth rates, while [a,] =A is the coefficient matrix of nonlinear interactions between the various components (species, or populations in the biological context). Note that for R,/R, = r,/r,, i, j = 1,2,3, (1.1D) represents the Euler discretization of (1.1C) with time step R,/r,, so that the orbits of (1.1D) converge to the orbits of (1.1C) in the limit R, +0, R,/ R, =const. = r,/r,, i,j = 1,2,3. Because of the meaning associated with the variables x, in biology, R," (which denotes the closure of R Y = {XER ~ I x , >0, i = 1, . . ,N)) is often chosen to represent the phase space of (1.1C) and (1.1D).

That greatly different dynamical behaviors may be predicted by the above two models can already be seen in the single case N = 1. In fact, for N = 1, (1.1C) becomes the trivial one-dimensional equation x= rx(1 -ax) which, for r >0 and apart from the unstable equilibrium, x = 0 , predicts asymptotic approach to the other stable equi- librium x = l l a . On the other hand, the discrete-time version x '= x + Rx(1 -ax) exhibits, in a suitable range of R-values, the well-known variety of asymptotic behaviors,

* Received by the editors October 18, 1985; accepted for publication (in revised form) August 18, 1986. IIstituto di Matematica, Facolta di Ingegneria, Universita di Ancona, 60100 Ancona, Italy. f Istituto di Matematica Applicata, Facolti di Ingegneria, Universiti d e L'Aquila, 67100 Aquila, Italy.

455

456 L. GARDINI, K. LUPINI, C. MAMMANA AND M. G. MESSIA

including periodic and chaotic attractors, typical of a large class of one-dimensional, unimodal maps [6], [7].

In the present paper we shall analyze a particular class of L.V. equations for N = 3, whose asymptotic dynamics in the continuous competitive case is well known and reiativeiy simple, while the corresponding discrete version will be found to exhibit great complexity in the structure of the attracting set. Such a class is obtained by requiring that the right member of ( l . lC) , f say, satisfies the symmetry relation f 0 P = P oJ; where P denotes the cyclic permutation in R':

(1.2) P(x1, x2r x3) = ( ~ 3 ,x2r XI).

In such a case [8], [9], after a suitable rescaling of time and of the variables, (1.1C) reads

while the corresponding discrete version, to be studied in the present paper, is given by

The chain of quadratic interactions involved in the two L.V. sets (1.3C, D ) may represent the completion or cooperation, depending on the signs of a and P, among three groups belonging to the same species, as illustrated in the following diagram (see also [8], [lo] for a discussion of the biological implications of such a model):

The relevant features of the orbit structure of the continuous L.V. set (1.3C), as reported in the literature [8], [9], [ l l ] , are the following. There are, at most, eight equilibrium points (eq. p. henceforth), that is, the origin 0, the so-called one-species equilibria E, and the two-species equilibria E,, i <j, i,j = l , 2 , 3 and the three-species eq. p. Q = (1, 1, l ) / ( l + a + P ) . Nontrivial dynamics are associated with a critical Hopf-bifurcation of Q, ( a+ P = 21, which belongs to R: for a + /3 > -1. For a + P = 2 all orbits in R: approach, asymptotically, the plane C, x, = I ; on such a plane there is an infinity of stable (not asymptotically) periodic orbits cycling around Q [8], 191. When a + P > 2, and a, P are both greater than 1, the El's are globally attracting in R:, apart from the stable manifold of Q, i.e., the straight line x, = x 2 = x ? , s say; otherwise, the El and Q are unstable and the w-limit set of all the orbits in R:, apart from s, consists of the three eq. ps. E, and the three heteroclinic orbits on the coordinate

BIFURCATIONS I N THE LOTKA-VOLTERRA MAP 457

planes connecting the El's (the E,. E in this case) [97. In contrast with such relatively simple dynamics, it will be shown, in the present paper, that the discrete version (1.3D) exhibits a much more complex bifurcation structure, giving rise to the generation of cyclic and quasi-periodic attractors, which in their turn may undergo transitions to chaotic attractors via typical sequences of flip- and Hopf-bifurcations.

The plan of the work is as follows. In O 2, after some general considerations on bifurcations in continuous and discrete models, we point out a few properties of aps with symmetry, and discuss the existence conditions of an invariant domain in R: for the map (1.3D). In O 3, we describe the bifurcation diagrams of fixed points for the two models (1.3C, D). Section 4 is devoted to the study of Hopf-bifurcations of the three-species eq. p. Q = (1,1,1)/(1+ a +p) , leading to the generation of quasi-periodic and periodic orbits, and transition to chaotic attractors via bifurcations of invariant curves. In particular, in a weak-resonant case of Hopf-bifurcation of Q, we show that the bifurcated invariant curve contains two 6-cycles; one of these cycles is given an analytic representation which allows for a detailed study of the cycle's subsequent bifurcations, in a neighborhood of the primary bifurcation point. The cycle's bifurcation sequence is then used to infer the occurrence of a global bifurcation of the invariant curve, via detaching of the 6-cycle, as confirmed by numerical computations. Sequences of primary flip-bifurcations and secondary bifurcations (flip- and/or Hopf-) in the one-species and two-species equilibria are discussed in O 5.

2. General considerations. We start this section by pointing out some simple and general relationships, to be used in our subsequent analysis of (1.3D), between bifurca- tions of fixed points in continuous models, represented by autonomous systems of ordinary differential equations,

(2.1c) X = f(x, P), X E R~

and the bifurcations in the related discrete models, or maps,

where F(x) =x+ Rf(x, P), R >0, and P = (p, ,. . .,p,) E Rm denotes the vector of parameters entering (2.1).

It is clear that the above two models have the same eq. ps., and that the correspond- ing stability and bifurcation properties may be different, but not independent, as the spectrum AF(x*) ={pi, i = 1, . . . ,N) of the Jacobian matrix JF(x*) of F at an equi- librium point x* is related to the spectrum Af (x*) ={Ai,i = 1, . . . ,N ) of the Jacobian matrix Jf(x*) by pi= 1+RA,, i = 1, . - .,N.

Let us briefly recall the well-known fundamental types of bifurcations occurring in families of continuous flows and in families of maps at an eq. p. x*. For simplicity, we shall assume here that p is a scalar parameter (m = 1).

Continuous model. We shall say that A(p) E Af(x*) is stable (unstable), when R,(A) <0 (R,(A) >0), while at a value p* such that R,(A(p*)) =0, we shall say that A(p) bifurcates. In particular, we shall denote by (i), (ii) the real-bifurcation and Hopf-bifurcation [12], respectively, that is,

(i) A (p) is real in a neighborhood of p*, and A (p*) =0, (ii) R,(A(p*)) =0, Im (A(P*)) # 0, d/dp(%(A(p)))I,~ f 0. Discrete model. We shall say that p(p)eAF(x*), (p(p)l< l(>) is a stable

(unstable) eigenvalue, and that p(p) bifurcates at p* when Ip(p*)l = 1. We shall

458 L. GARDINI, R. LUPINI, C . MAMMANA AND M. G. MESSIA

distinguish three cases as follows: (j) The real-bifurcation occurring for p (p ) real in a nieghborhood of p*, and

P(P*) = 1; (jj) The flip-bifurcation occurring for p(p) real in a neighborhood of p*, and

Y(P*)= -1; (jjj) The Hopf-bifurcation (for maps) when Im (p(p*)) Z 0, Ip(p*)J= 1,

dldp Jp(p)l,* # 0 and , u ~ ( ~ * ) 3,4.j.t 1 for k = For the change in the qualitative behavior of orbits (in a suitably small neighbor-

hood of x* and p*) associated with the bifurcations listed above, we refer to [12], [13]. We only note here that couples of periodic orbits of period k may exist in the so-called weak-resonant cases of a Hopf-bifurcation, that is, in case (jjj),when pk(p*)=

1, k Z 5. In fact, this kind of bifurcation is found to occur in map (1.3D) ( 5 4). Regarding model (2.1C), let us denote by SC(x*) the set of points in the parameter

space where the eq. p. x* is stable, and by B:(A), B:(A) the set of points where A bifurcates according to (i), (ii), respectively. The sets sD(x*), B:(~), Bf(p), B;(p ) are analogously defined for the discrete model (2.1D). Now, let A E Af(xs), and p = 1+RA E AF(x*). It is clear that Ipl< 1 implies R,(A) <0, so that the inclusion sC(x*)2 sD(x*) is obvious. Moreover, for any couple of such eigenvalues, BY =B:, B: has no counterpart in the continuous model while B$~B:= {PEMI R,(A) = Im (A) =0, A E Af(x*)), to which there correspond bifurcations of codimension at least two.

Finally, we state some simple, basic properties of N-dimensional maps that satisfy a symmetry relation of the form

where T is an invertible linear operator in R~ such that Tk =I, k >0. Property I. Let y be an orbit of F; then T(y) is also an orbit of F. Property 11. Let C, be an n-cycle of F; then either T(C,) f l C, =O (in which

case C,, T(C,), ,Tk-'(c,) are distinct cycles) or n = km, and

In particular, the latter alternative holds when F has a unique invariant curve y in some domain invariant with respect to T, C,, is stable and belongs to y.

Property 111. The stability conditions for cycles of the form (2.3) can be deduced from the analysis of the spectrum of the matrix

In fact, by (2.2), it follows that JF(Tx) = TJF(x)T-', so that

In particular, Properties 1-111 apply to the L.V. map (1.3D), with T =l? Moreover, if we denote by the L.V. map obtained from F (1.3D) after permutation

- -

BIFURCATIONS IN THE LOTKA-VOLTERRA MAP

and by PI , P2, the permutations defined by

(2.5b) P2(~1, ~ 3 ) X I , ~3)1~ 2 9 = ( ~ 2 , the next property follows.

Property IV. F(Plx) =p1F(x) and F(P2x) =p2F(x), that is,

In particular, as F is conjugate to F(Pyl =PI , P;' =P,), the permutation (a, P ) + (p, a ) does not introduce topological changes in the orbit structure. Moreover, for a =p, F commutes with P, and P2, and Properties 1-111 apply to F with T =Pl or T =P2.

As a conclusion to this section, we shall analyze the important question of the existence of invariant sets in R: for the map (1.3D), which we rewrite as

where li(x) = (a,, x), ai denotes the i - h row of the coefficient matrix

and (.,.) denotes the ordinary scalar product in R ~ . First, we note that, like in the continuous model (1.3C), the coordinate planes,

the coordinate axes and the straight line s are invariant under F. On s, F reduces to the unimodal map

(i.e. the logistic map [' = (1+R)[(l- e), after rescaling with [=

R(1+ a +p)x / ( l+ R)), which has two eq. ps., x =0 and x = 1/(1+ a +P), correspond- ing to the origin and to the eq. p. Q, respectively. For R E (0,2), the former is unstable and the latter attracts all points on s such that x = (0, (1 +R)/ R( l +a +P)), while all orbits starting at x <0 or x > (1+R)/ R (1 +a +P ) diverge to -m. By analogous arguments it can be seen that on each coordinate axis F reduces to the unimodal map

for x =xi, i = 1,2,3 (i.e. the logistic map, after rescaling). For R E (0,2) the origin is unstable and the eq. ps. Ei attract all points of the interval (0, (1 +R)/R).

In the following we shall assume R E (0,2), so as to exclude those complex dynamical behaviors that are mere consequences of the chaoticity of the one-dimensional restrictions of the map (1.3D).

It is clear that the largest closed domain in R: which may be invariant under F is given by -(2.11) ~ = { x ~ ~ : l l ~ ( x ) 5 ( l + ~ ) / R , i = 1 , 2 , 3 )

because x E R: and F(x) E R: imply x E 3;accordingly, necessary and sufficient condi- tions for the invariance of 3 are

460 L. GARDINI, R. LUPINI, C. MAMMANA AND M. G. MESSIA

for any x E 3 , and i = 1,2,3. We note, however, that 3 is invariant under P while (A,, A,, A,), defined in (2.12), transforms into (A,, A, , A,). As a consequence, any one of the above inequalities implies the other two. Let us choose the first one, i.e.,

and denote by C the center of the family of quadrics A,(x) = const., and by a 3 the boundary of 3.By obvious geometric considerations we can state that

(J) If the quadratic form in Al(x) is definite (positive) and C E 3 , then A,(x) 2 0 in 3 iff A,(C) 2 0;

(JJ) If the quadratic form in A,(x) is definite but CE 3 , or indefinite, then A,(x) 2 0 in 3 iff Al(x) 2 0 in 83.

The "reduced" conditions for the invariance of 3 , however, are still too laborious to be worked out in full generality. We shall use, instead, another approach which leads to sufficient conditions on a , P ensuring the invariance of 3 , when a > 0 and P > 0.

Let us denote

We can easily ascertain that 3,U 3- is mapped by F into 3. Regarding points x E 3 := 3-{3+ U 3-), we note that max 3 ( x j -xi) = R/4, for i = 1,2,3, which implies, by obvious geometric considerations, that F(x) belongs to 3 for any x E 3 , when

(2.14) max {1,1/a, l / p ) + R/4<min (1 + R)/ R{1, 1/a , l/P}.

3. Stability analysis and bifurcations. In this section we analyse the various bifurca- tions of fixed points that take place as the parameters a , P vary in R' and R E (0,2). The analytic representation of the eight eq. ps. and of the relative spectra, that can be derived by simple algebraic manipulations, and by use of the results of the previous section, are reported in Table 1, while the stability domains and the bifurcation surfaces of E,, EC and Q are reported in Fig. 1. Details of the derivation of this figure and the definition of the various functions involved (+ and 9's) can be found in Appendix A.

TABLE1 Equilibrium points of the three-dimensional continuous and discrete L.V. models, and relative spectra.

Equilibrium point x* Eigenvalues of J f ( x * ) Eigenvalues of J,(x*)

E23=(0 , ( l - a ) l ( l - a p ) , ( l - P ) / ( l - a p ) ) ~ ~ = [ ( a ~ + ~ ~ - a - p - a p + l ) ] / for a p # 1 (1 - 4 )

BIFURCATIONS IN THE LOTKA-VOLTERRA MAP

FIG. 1. Sections of the stability regions and bifurcation surfaces of the equilibrium points of models (1.3C) and (1.3D), with a generic plane R = const., 0 < R < 2.

The only interesting change in dynamical behavior occurring in the continuous model (1.3C) is related to the critical Hopf-bifurcation of the eq. p. Q, cited in the introduction. In fact, for a +@ > 2, either the Ei's are stable and attracting, or they are unstable (because of a change of stability with the E,'s, at a = 1 or @ = 1, which takes the E, out of E ) , and the flow in { E - Q) converges to the o-limit set cited in the introduction. - For a +@5 -1, only the unstable points Ei's belong to R:and all the orbits in R: are divergent. In fact, let K = x, + x2+ x, and V = xlx2x3; the differential equations for V and K for a+@=-1-h , hZ-0, read

and imply that the orbits in R: are divergent. Regarding the discrete model (1.3D), complex dynamical behaviors are associated

not only with Hopf-bifurcations of Q through the surface q,(a, @, R) = 0, but also with flip-bifurcations both of the E,'s (through segments C and D of Fig. l(b) and of the E,'s (through the surface q3(a, @, R) = 0). These "primary" bifurcations are found to be followed by "secondary" bifurcations and will be analyzed and partly discussed in the following sections. On the other hand, transcritical bifurcations of the E,'s with the E,'s (through segments A and B of Fig. l(b), (d)) lead to a simple, asymptotic approach to stable equilibria.

To conclude this section we briefly discuss the bifurcations occurring in (1.3C, D) when a = @ (such a choice of the parameters would model species which are identical and interact identically). From Table 1 it can be seen that the equilibria E,, are unstable for a # 1 and that the spectrum of Q is real for any a. In the continuous model (1.3C)

462 L. GARDINI, R. LUPINI, C . MAMMANA AND M. G. MESSIA

the eq. p. Q is stable (unstable) for -112 < a< 1 ( a > I), while the Ei's are unstable (stable), respectively. Thus, bifurcation occurs only when a = 1, and it is of degenerate type: all the equilibria have the same spectrum, with one stable eigenvalue and two bifurcating ones. This is not a codimension-one bifurcation; in fact, in s s h a case, K =c:=, xi = 1 is a plane of fixed points, stable and globally attracting in R: because K = K ( ~ - K ) .

In the discrete model (1.3D), for -(2 -R)/(4+ R) < a < 1 (1 < a < (2 + R)/R) the eq. p. Q is stable (unstable) while the Ei's are unstable (stable), respectively. The bifurcation occurring at a = 1 is analogous to the one of (1.3C); in fact, as K t = K (1 + R -RK), the plane K = 1 consists of fixed points, it is stable for R E (0,2), and attracts points belonging to

The other two bifurcations are of degenerate type too, as, at a = -(2- R)/(4+ R) ( a = (2+ R)/R), the spectrum of Q (of the Ei's) contains two eigenvalues equal to -1. They give rise to "degenerate" flip-bifurcation sequences that will be discussed in detail in 99 4 and 5.

4. Bifurcations in the three-population equilibrium point Q. If we interpret the L.V. equations as a model of population dynamics, the orbit structure associated with bifurcations of Q is more interesting than the bifurcations related to the other equilibria. In fact, the flip-bifurcations of the Ei's and the Eq's (to be discussed in D 5), as well as their transcritical bifurcations, involve state vectors leaving R:. Accordingly, in this section, we shall restrict our considerations to the range a + p > -1 (for which Q E R:) and a S P , by Property IV. The bifurcation surface of Q apart from the line ( a , , R) = ( 1 1 R) is given by q4(a, p, R) = a 2 + p 2 - a -p+(a~-1)(2-R)/(1+R)=O,asectionofwhichwithaplaneR =const., 5(R) say, is shown in Fig. l(f). For a < p , 40, = 0 is the locus of the Hopf-bifurcation.

The results of numerical investigations in such a range are briefly reported here. Along paths in the parameter space crossing the curve 5(R) through points for which kth roots of unity do not occur in the spectrum of Q, for k = l ,2,6, an invariant attractive curve, y say, is detected (as predicted by Hopfs theorem [13] in the nonresonant case) with quasi-periodic orbits on it (supercritical bifurcation) (Fig. 2(a)).

FIG. 2. Projections on the plane no,Pixi= 0,of the invariant curve y bifurcated from Qfor R = 1 , P = 0, and a = 1.37 (a); a = 1.5 (b).

463 BIFURCATIONS IN THE LOTKA-VOLTERRA MAP

For positive values of LY and P, as we move away from the bifurcation surface, y increases in size and tends to the w-limit set of the continuous model (Fig. 2(b)). We note that, in order to emphasize the difference of topological properties, the point set of Fig. 2(b) has been reduced to the same scale of the point set of Fig. 2(a); the same scale reduction has been applied to all the plots of point sets reported in figures without explicit numerical scaling on the axes.

When P is negative, more complex sequences of bifurcations of y are detected some of which are "typical" for n-dimensional maps with n 2 2 (see [14]-[16]), that is

-Cascades of Hopf-bifurcations leading to chaotic attractors (Fig. 3, Fig. 4); -Period doubling cascades of a periodic orbit leading to a chaotic attractor

(Fig. 13).

FIG. 3 . Projections on noof attracting sets for R = 1, a =0.5. (a) Quasi-periodic orbit on the invariant curve y Hopf-bifurcated from Q (P = -0.61). ( b ) Primary Hopf-bifurcation of y (P = -0.665). (c) Chaotic attractor generated by cascade of Hopf-bifurcations of y (P = -0.67).

Moreover, invariant curves and cycles may coexist, as well as chaotic attractors and invariant curves (Fig. 5).

We are trying to understand a complex phenomenology. At least the occurrence of the cycle C6of Figs. 5(a), 6(a), its bifurcations, and the role it plays in the bifurcations of y can be partly studied by analytical tools.

L. GARDINI, R. LUPINI, C. MAMMANA AND M. G. MESSIA

FIG. 4. Projections on the plane noof attracting sets for: R = 1 , a = 1 (weak-resonant case). ( a ) Stable 6-cycle C6 (on the invariant curve y ) Hopf-bifurcated from Q ( P = -0.51). ( b ) Quasi-periodic orbit on the y6-attractor Hopf-bifurcated from C , ( P = -1) . ( c ) Chaotic attractor generpted by cascade of Hopfbifurcations of C6 ( p = -1.1).

We shall show (Theorem 1) that C6 is generated via a weak-resonant Hopf-bifurcation of Q occurring on the surface a = 1, at P = ( R - 2 ) / ( 1 + R ) , as p ( 1 , ( R -2 ) / (1+ R ) , R )= ei"I3, a sixth root of unity.

The stability of C 6 , near the bifurcation value, will be demonstrated in Theorem 2.

THEOREM1. Let a = 1, 2 > R >0. Then, for any /3 <( R-2 ) / ( R+ 1) there exists a period-6 cycle, given by

c6= {x*, Pix*, Px*, PPlx*, p2x*, P ~ P ~ x * }

where

x* = {[3a-1+&(a - 1)(1+3a)]/2b, [3a -1-J 3 ( a -1)(1+3a)]/2b,lib)

and

a = ( l + R ) ( l - p ) / 3 , b = R ( l - P ) .

BIFURCATIONS IN THE LOTKA-VOLTERRA MAP

FIG. 5. Projections on the plane 11, of attracting sets for R = 1.4, a = 1 (weak-resonant case). ( a ) Stable 6-cycle C,, bgurcated via Hopf at p = -0.25, for p = -0.51, after detaching from the y-attractor.

Magnification of a small spherical neighborhood of point S is reported in Fig. 11. (b ) y-attractor coexisting with y6-attractor ( p = -0.58). ( c ) Chaotic y-attractor coexisting with y, ( P = -0.607). As the projections of the two attractors overlap,

we have reported y,, separately, in ( d ) ,for the same values of the parameters. ( d ) y6-attractor ( p= -0.607). ( e ) ,(f) Merging of 7 and y6 into a single chaotic attractor; j3= -0.63 and P = -0.66, respectively.

466 L. GARDINI, R. LUPINI, C . MAMMANA AND M. G . MESSIA

Prooj Let us denote by IIS the plane Z:-, xi=35, and by 1the vector (1,1,1). To the orthogonal decomposition of vectors in R ~ :x =51+ u , u E n o ,5:E R, there corre- sponds the following equivalent representation of the map (1.3D):

(4.lb) u l = R(Au, u ) /3 -R5(ai, u ) +ui ( l+R -R(1+ a +/3)5)

where (Au, a ) can be written as (1 - ( a + /3) /2) l~1~. Let us denote by TS the circle on n, centered at 51 of radius p, = [35(1- ((1 +a +/3))/(1 - ( a +/3)/2)1"~, and by f the projection of r, on IIo. Clearly, X E T, implies F ( x )E nS , by (4.la). Now, let a = 1 and 5" = (1+R)/3R. The restriction of the map (4.lb) to fScan be rewritten as

The following symmetry properties of G, with respect to permutations of a :

can now be used to show that if u * satisfies the equation G ( u ) = p , u then Cz = {u*, Plu*, Pu*, PPluh , p2u*, P~P,U*} is a 6-cycle of G on fS* .

In fact, by (4.3), we deduce:

In conclusion, the proof will be complete if we can show that the equation Plu= G(u) admits a solution in f e e , for, in such a case, C, =5*1+ CZ is a 6-cycle of (4.1).

In fact, let us write such equations in component form

An equivalent system is obtained by taking the sum and the difference of (4.5a) and (4.5c):

BIFURCATIONS IN THE LOTKA-VOLTERRA MAP

It has to be u1 # u2, since otherwise, we would obtain u3 = -2u, and u: = a ( a - l)/b2, which contradicts (4.5a) and (4.5b). As a consequence, (4.6~) reduces to u3 = (1- a ) /b which also implies that (4.6a) and (4.6d) are identically satisfied.

Accordingly, (4.6) is equivalent to

(4.7a) u3= (1- a)/b,

(4.7b) u:+ u; = ( a - 1)(5a+ l ) / b2,

(4 .7~ ) u 1 + u 2 = ( a - l ) / b ,

(4.7d) u lu2= (1-a)(1 +2a)/b2.

Finally, as (4.7b) is identically satisfied by any solution of (4 .7~) and (4.7d), system (4.7) admits two solutions given by

(4.8) u f = (1 -a) /b, UT,~=[(a -1)*d3(a- 1)(1+3a)]/2b

and the proof is complete. We point out that the choice of the sign in the last equation is immaterial, as the orbit generated by a*,one solution of (4.5), is identical to the orbit generated by P2u*, the second solution of (4.5).

Remark. The 6-cycle C6 of Theorem 1 belongs to the plane II(l+R)/3R and tends to the eq. p. Q as E = [(R -2) -@(R+ 1)]/2 + O+. For E in a small right neighborhood of zero, the invariant curve y predicted by Hopf's Theorem must contain C6, but it cannot belong to the plane II(,+,,/,, and then reduce to r(l+R)/3R, as r ( l + R ) / 3 R is not invariant with respect to G (see (4.2)). In fact C6 = y fl n(]+R)/3R.

THEOREM2. For a = 1, P < 0, the 6-cycle C6 of Theorem 1 is stable when

is positive and suitably small. Proof: By Property I11 of 9 2, C6 is stable iff the spectrum of

(4.10) H := P~J(P,X*)J(X*)

lies within the unit disk of the complex plane. Clearly as E +O+, x* + Q and H + Ho:= p2 (1-R / ( ~ + P ) A ) ~ =[ P ( I -R / ( ~ + P ) A ) ] ~ ,as A commutes with P. The spectrum of P ( I -R/(2+P)A), a symmetric matrix, is easily found to be given by ((1 +2P)/ (1-p) , 1,1}; as 0 < (1+ 2/3)/(1- P ) < 1, we have only to study the perturbations of the double eigenvalue h = 1, for small values of E. The elements of H with respect to the orthonormal basis of eigenvectors of Ho, ((1, 1, I)/&, (1, -1,0)/&, (1,1, -2)/&}, are reported in Appendix B. The coefficients of the corresponding characteristic polynomial,

are analytic functions of

Note that l - T o + ~ o - A o = O , as p(1 ,0)=0 and that p(Ao,O)=O, as Ao= ((1+ 2P)/(1- P ) ) ~ . Moreover, p"(1, 0) = 6-2(2+ A,) > 0. Therefore the double root

L. GARDINI, R. LUPINI, C. MAMMANA AND M. G. MESSIA

FIG. 6 . Qualitative behavior of the characteristicpolynomialp(A, E ) , (see (4.1 I ) ) , with coe@cients truncated at various orders: ( a ) order &I/'; ( b ) order E'; ( c ) order E'/'.

h = 1 will split into two real, stable roots, if, at some higher order in &'I2, 1 - T + 6 - A = p ( l , s ) > O a n d 3 - 2 T + S = p ( l , s ) > O . Now, as T,=A1=6,=Oand -T2+6,-A2=0, it follows that A = 1 is still a root of p at order E. But, to the same order, 3 -2T+ 6 = -2T2+ 6, = (-12/3)/(1- /3), >0, so that, to order E, one stable real root separates from 1 (see Figs. 6(a), (b)). As a consequence, it is sufficient to show that 1 - T + 6 - A > 0 at some order higher than E. As -T3+ 63-A3 = -T4+ 64 -A4 = 0, h = 1 is still a root at order E,, but, at last, we find -TS+ SS-As = 12(2-P)/ (1 - >0. In conclusion, P ) ~ the double root h = 1 splits into two stable real roots of the form A, = 1 -a2& + o ( E ~ / ~ ) , h3= 1- with~ , E ' / ~ + o ( E ~ ) , 0 (see Fig. 6(c)). By Theorems 1 and 2 we can then assert that the weak resonant Hopf-bifurcation at a = 1, /3 = (R -2)/(R + 1) (0 <R <2) is characterized by the occurrence on the bifurcated invariant curve, y say, of a couple of 6-cycles, one stable and the other unstable, C, and C, say. Moreover, as the points of C, are stable fixed points of F, , the implicit function theorem guarantees that C, can be continued, as a stable 6-cycle, in a (suitably small) interval of values around a = 1, although, for a # 1, C, does not admit a simple analytic representation. The continuation of C, to the right and to the left of the surface a = 1 stops at some surface of parameter values where the spectrum of J+, on C,, contains the value 1. On such a surface C, and its unstable counterpart e6annihilate via a saddle-node (s.n.) bifurcation. The s.n. bifurcation surface of C, and the Hopf-bifurcation surface of Q, p4= 0, must meet along the curve of singular points a = 1, /3 = (R -2)/(R + I), as sketched in Fig. 7. Typically, as we move away from the Hopf-bifurcation surface of Q, within the shaded region of Fig. 7, the dimensions of C, increase until C, it undergoes a Hopf-bifurcation, with generation of an attractor consisting of six invariant curves, and coexisting with y. Along the surface a = 1, the analytic representation of C, allows one to determine the spectrum of the matrix H (4.10), by simple numerical computa- tions. The typical (qualitative) behavior of the spectrum of H, as a function of E, along a = 1, is reported in Fig. 8; that is, we have Hopf-bifurcation of C, at some value, E,

say, of E, but no flip or s.n. bifurcation. The values of E, can be obtained by use of the analytic, necessary and sufficient condition that a 3 x3 matrix has to satisfy in order to have a couple of complex eigenvalues of modulus one (see (4.11)):

BIFURCATIONS IN THE LOTKA-VOLTERRA MAP

FIG.7. Details of the Hopf-bifurcation sutface for Q and of the s.n. bifurcation surface for C, on a plane R =const. C, exists in the shaded region ("Amol'd tongue" for 116 frequency locking).

FIG.8. Typical qualitative behavior of the spectrum of H (4.10) when a = 1, as a function of E : one stable real eigenvalue A , tends to become negative, while the two real eigenvalues bifurcating from 1, A, and A, become complex and eventually exit from the unit circle (Hopf-bifurcation).

coupled with stability condition of the real root, that is, / A [ < 1. The graph of the curve (4.12) in the (P,R) plane is reported in Fig. 9.

On the other hand, numerical computations suggest that, on a = 1, the invariant curve y (to which C6 belongs for E small enough) persists as a stable curve diffeomorphic to a circle, for some range of &-values (0, E ~ ) > EH (see Figs. 5(a), (b)). Therefore, with as C6 cannot belong to y for E 2 E,, a detaching of C6 from y must take place at some intermediate value of E, E* E (0, EH). This new kind of nonlocal bifurcation, however, is incompatible with the behavior of y as a function of E suggested above. In fact, let us denote by T, the restriction of F~ on y, for a given value of E. Clearly, if we assume that y is diffeomorphic to a circle for all E E (0, E ~ ) , then T, is conjugate to a diffeomorphism of the circle, which (by Hopf theorem) for values of E small enough, has twelve fixed point (C6UC,), while for E > EH has no fixed point (see Fig. 10).

L. GARDINI, R. LUPINI, C. MAMMANA A N D M. G. MESSIA

F I G . 9 Graph on the (P , R ) plane of the Hopfbifurcation curve of the 6-cycle C, at a = 1 (4.12).

FIG. 10. ( a ) Qualit_ative graph of the invertible map a, on the invariant curve y when the stable cycle C6 and the unstable cycle C6 belong to it. ( b ) Numerical computation of the graph of a, on the y-attractor at R = 1.4, a = 1 , P = -0.52.

Now, continuity of IT;with respect to E would imply the occurrence of a saddle-node bifurcation of C6at some intermediate value of E in the range (0, E ~ ) , which, however, is incompatible with the behavior of the spectrum of C6as a function of E cited above (Fig. 8).

This contradiction can be explained if we assume that, for some subset of (0, E ~ ) , the y-attractor is not diffeomorphic to a circle. In fact, more accurate analysis after detaching of C6reveals a complex substructure of y in the neighborhood of the points of the unstable 66.In Fig. 1 1 we report a magnification of the spherical neighborhood of point S of Fig. 5(a) which represents one of the six zones on the y-attractor where the orbit is "more dense" suggesting closeness of the unstable 6-cycle 66.Such a figure confirms that the detaching of C6 is accompanied with an increase of the topological dimension of y [16]. The attractors discussed so far, C6 and y, undergo various

BIFURCATIONS IN THE LOTKA-VOLTERRA MAP

FIG. 1 1 . Magnification of the spherical neighborhood of radius of thepoint S = (0.595,0.8171,0.1937) of Fig. 5(a).

transitions to chaotic attractors when the parameter vectors (a, p, R) take values sufficiently far from the Hopf-bifurcation surface of Q.

-If we take R = 1, a = 1 and increase E, the one-dimensional attractor bifurcated from C6 at E = eH, Y6 say, undergoes a cascade of Hopf-bifurcations, leading to a chaotic attractor as reported in Figs. 4(b), (c).

-For larger values of R, on decrease of P, the transverse substructure of the y-attractor "diffuses," and eventually englobes y6 (R = 1.4 in Figs. 5(c), (e), (f)).

-If a is decreased or increased from unity, while R and p are kept fixed at 1 and -1.1, respectively, the same sequence of bifurcations in the attracting set is detected: collapse of the six branches of y6 (Figs. 12(a), (b)), followed by twelve invariant curves bifurcating via reverse Hopf-bifurcation into a 12-cycle (Figs. 12(c), (d)), which even- tually undergoes a sequence of flip-bifurcations (Figs. 13(a), (b), (c)) leading to a chaotic attractor (Fig. 13(d)).

As a conclusion to this section we shall discuss the results of numerical computa- tions referring to the case of degenerate bifurcation of Q in the model (1.3D) which occurs where a = p. The spectrum of JF(Q), for R E (0,2), consists of a stable eigenvalue and a couple of real eigenvalues that cross -1 when a crosses the critical value a * := -(2 -R)/(4+ R). We recall that, by Property IV, in case a = p, the attracting set is symmetric with respect to the three planes xi = xj.

The observed asymptotic dynamics for values of a less than a*, for any R E (0,2) is characterized by transitions to three distinct attractors via sequences of flip-bifurca- tions taking place on the planes xi = xj, followed by a global bifurcation with generation of a single space-filling chaotic attractor. As an example, let us fix R = 1. As we decrease a in the range (-0.24, -0.2), the three 2-cycles bifurcated from Q (one on each plane xi =xi), undergo a sequence of flip-bifurcations ending in chaotic attractors (Figs. 14(a), (b)). On further decrease of a , the three branches coalesce and merge into the single space-filling attractor of Fig. 14(c).

472 L. GARDINI, R. LUPINI, C . MAMMANA AND M. G. MESSIA

FIG. 12. Projections on the plane Il, of attracting sets for R = 1 , P = - 1 . 1 . (a), ( b ) Chaotic attractors for a = 0.9995 and a = 0.995 respectively, exhibiting the collapse of the six branches of the attractor of Fig. 4(c). (c) Quasi-periodic orbit on a one-dimensional attractor consisting of 12 closed curves (a= 0.9905). (d) 12-cycle ( a = 0.99).

5. Bifurcations in the equilibrium points Ei and Eij. One-species eq. ps. E,. On crossing the segment denoted as D in Fig. l(b) the El's

undergo flip-bifurcation in the L.V. map (1.3D). The same bifurcation occurs in the restriction of (1.3D) to the coordinate planes (see Appendix C); therefore the three stable 2-cycles bifurcated from the E,'s belong to the coordinate planes.

Let us briefly discuss the results of numerical investigations referring to the crossing of D along the straight line R = 1, P = 2 (different values of P, R do not lead to qualitative changes in the attracting set).

For 1 < a < 3, the computed orbits are found to be either divergent or convergent to one of the (locally stable) E,'s, depending on the initial conditions. On crossing a = 3, the El's become unstable and a stable 2-cycle appears on each coordinate plane which persists in the range 3.562 a >3. At a ~ 3 . 5 6 0 8 the two 2-cycles, in turn, flip-bifurcate into Ccycles, not belonging to the coordinate planes (which are stable

BIFURCATIONS IN THE LOTKA-VOLTERRA MAP

FIG. 13. Projections on the plane noof attracting sets for R = 1, P = -1.1. ( a )Stable 12-cycle ( a =0.98). (b ) Stable (12x 4)-cycle Pip-bifurcated from the 12-cycle ( a =0.9705). ( c ) Periodic orbit of high period flp-bifurcated from the 12-cycle ( a =0.969).( d ) Chaotic attractor generated via cascade offlip-bifurcations of the 12-cycle ( a =0.96).

manifolds of the unstable two-cycles). On further increase of a the three 4-cycles undergo a cascade of flip bifurcations (in R:), while the three unstable 2-cycles Hopf-bifurcate and give rise to invariant curves, on each coordinate plane (Figs. 15(a), (b), (c)). The basin of attraction in R: of the attractor generated by the flip-sequence is quite small, as the major part of the orbits, after a chaotic transient spent on this attractor, eventually converge to one of the three stable invariant curves (Fig. 15(d)).

Regarding the degenerate bifurcation of the Ei's in the case a =p, we note that, as in the previous case, the same bifurcation occurs in the restrictions of (1.3D) to the coordinate planes, so that, again, the invariant set bifurcating from the Ei's on crossing the critical value a" = (2 +R)/R must belong to the coordinate planes.

The attracting set detected at R = 1, a > 3 is formed by two 2-cycles on each coordinate plane, which, in turn, Hopf-bifurcate at a =3.63 (Figs. 16(a), (b)). Finally,

474 L. GARDINI, R. LUPINI, C . MAMMANA AND M. G. MESSIA

FIG. 14. Projections on the plane II, of attracting sets for R = 1 and a =P : degeneratejlip-bifurcating of the eq. p. Q.( a ) Three distinct 2-cycles belonging to the three planes xi = xi ( a =P = -0.24). ( b ) Three chaotic attractors jlip-bifurcated from the 2-cycle on the three planes xi = x, ( a =P = -0.26). ( c ) Single space-Jilling chaotic attractor ( a =p = -0.27).

on further increase of a the bifurcated invariant curves on the coordinate planes merge into a single chaotic attractor (Figs. 16(c), (d)).

Two-species eq.ps. Et. The eq. ps. Eij exist when ap Z 1 and belong to E: if a and p are both larger or less than one. The relevant piece of bifurcation curve in (1.3D) related to these points, with no counterpart in the continuous model, is the surface of the equation rp,(a, p, R ) = a2+p2-a -p +(1-ap)(2+R)/ R =0 (except for a =p = 1) (see Appendix A), a section of which is reported in Fig. l (d) . Moreover, as these bifurcations do not occur in the restriction of the map to the coordinate planes (see Appendix C), the 2-cycles, flip-bifurcated from the Eij's, cannot belong to the coordinate planes, which represent the stable manifolds of the EV's as long as the parameter vector does not intersect the surface q2(a, p, R) =0. Let us discuss the results

BIFURCATIONS IN THE LOTKA-VOLTERRA MAP

ALFA= 3.5 BETA =2 ALFA =3.61 BETA =2

ALFA =3.63 BETA =2 ALFA =3.63 BETA =2

FIG. 15. Attracting sets for R = 1, P =2. ( a ) 2-cycle Pip-bifurcated from the eq. p. E, belonging to the coordinate plane (x,, x,) ( a =3.5). ( b ) Projection on the plane (x,, x,) of the 4-cycle (not belonging to that plane) flip-bifurcated from the 2-cycle ( a =3.61). ( c ) 10-cycle detected on the invariant curves bifurcated via Hopf from the 2-cycle on the coordinate plane (x, , x,) ( a =3.63). ( d ) Projection on the (x, , x,) plane of an orbit which, after a chaotic transient, tends to the 10-cycle on that plane.

of numerical experiments referring to the case R = 1, P =2.9 while a crosses the bifurcation value, a: say, (p3(af, 2.9,l) =0. First, we observe three 2-cycles belonging to R: flip-bifurcated from the EV's. On further decrease of a , each 2-cycle undergoes Hopf-bifurcation (a=0.833). On the bifurcated invariant curves, quasi-periodic and periodic orbits are detected (see Figs. 17(a), (b)). Finally, at some value a$ such that tpz(a$, 2.9, 1) =0, the EV's flip-bifurcate into period-2 orbits which Hopf-bifurcate in turn, in the restriction of the map to the coordinate planes, giving rise to three invariant curves, one for each coordinate plane (see Figs. 18(a), (b)), which eventually merge into single chaotic attractors (Fig. 18(c)).

6. Conclusions. A description of the bifurcation curves of the fixed points of the 3-dimensional L.V. system of O.D.E. and of the associated map has been worked out in the case of cyclic invariance of the coefficient matrix. We have identified those branches of the bifurcation curves that are loci of types of degenerate and nondegenerate bifurcation in the L.V. map, different from the continuous model. By analytic and numerical techniques, we have analyzed the changes taking place in the attracting sets when the parameters are varied across such branches of the bifurcation curves. It is found that the loss of stability of the three-species eq. p. Q is accompanied, in the case of positive values of a and j3,by the generation, via Hopf-bifurcation, of an invariant curve similar in shape to the w-limit set of the continuous model, while, for negative

L. GARDINI, R. LUPINI, C. MAMMANA A N D M. G. MESSIA

1

FIG. 16. Attracting sets on the plane (x,, x-,) for R = 1 and a =P : degeneratejlip-bifurcation of the Ei's. (a) Closed curves Hopf-bifurcated (secondary bifurcation) from the two 2-cycles Pip-bifurcated (primary bifurcation) from the equilibria E,, E3 ( a =P =3.65). (b )Magnification of the invariant curves bifurcated from E,. ( c ) Two chaotic attractors bifurcated from E2 and E3 ( a =P =3.668). ( d ) MagniJication of the chaotic attractor bifurcated from E, .

p, the original Hopf-bifurcation is followed by more complex bifurcation sequences. Making use of symmetry properties of the map in the case a = 1, we have given, analytically, an explicit characterization of a family of 6-cycles generated in a weak resonant Hopf-bifurcation of Q, pointing out their saddle-node and Hopf-bifurcations as well as their relationships with the invariant curve generated in the same Hopf- bifurcation of Q.

By numerical experiments, we have shown the occurrence in the L.V. map of a large variety of transitions from elementary to chaotic attractors (via typical sequences of Hopf and flip-bifurcations), that contrast with the simple asymptotic dynamics of the continuous L.V. system.

Such a drastic difference in asymptotic behaviors exhibited by the continuous-time formulation and by the discrete-time version of the same dynamical model confirms

BIFURCATIONS IN THE LOTKA-VOLTERRA MAP

ALFA = .83 BETA =2.9

0.5'

0

u 0.9

-0.4 2 0.2 ( a )

ALFA = .82 BETA =2.9

-0.6 0.3

( b ) FIG. 17. Projections on the (x,, x,) plane of attracting sets (not belonging to that plane) at R = 1 , P =2.9.

( a ) Quasi-periodic orbit at a =0.83. ( b ) 8-cycle at a =0.82, detected on the invariant curoe Hopfbifurcated (secondary bifurcation) from the 2-cycle, Pip-bifurcated (primary bifurcation) from the eq. p. E,.

that, when modeling biological or economical systems, the choice between continuous- time and discrete-time dynamics is crucial.

This conclusion should, a fortiori, hold for classes of L.V. systems more general than the one examined in the present paper. For example, inclusion of cubic non- linearities already, in the case of a single species, introduces substantial complications in the structure of the attracting set (May [17]).

For the same reason, it is quite reasonable to expect that the asymptotic dynamics presented above will not be persistent under similar substantial changes in the model, like elimination of the cyclic invariance of the coefficient matrix or increase in the

L. GARDINI, R. LUPINI, C. MAMMANA AND M. G. MESSIA

FIG. 18. Attracting sets belonging to the (x,, x,) plane at R = 1 , P =2.9. (a) Invariant closed curves Hopf-bifurcated from the unstable 2-cycle at a =0.4755. (b) MagniJication of one of the two curves reported in (a). (c) Chaotic attractor at a =0.4745.

number of interacting species. However, the cyclically invariant, 3-dimensional L.V. map studied in the present paper may provide a useful and simple example of transitions to chaotic behavior in low-order systems, induced by multi-component, nonlinear interactions, which raises the important question of how to discriminate between noise in experimental data due to internal dynamics and noise due to random fluctuations of the environment [21].

A more detailed characterization of the bifurcation sequences and of the attractors described in the present paper, as well as a comparison between discrete dynamics and continuous dynamics in more general forms of the L.V. equations [18], 1191, [20], is in progress.

Appendix A. Equilibrium points and relative spectra of model (1.3C) and (1.3D) are reported in Table 1. Clearly, the origin is an unstable node for both models and

BIFURCATIONS IN THE LOTKA-VOLTERRA MAP 479

for any (a, p ) E R ~ .As regards to the remaining points, a separate discussion of the stability properties is needed.

One-species eq. ps. Ei. Continuous model (1.3C). For a > 1 and P > 1 the Ei's are stable nodes. On the

half-lines of equations ( a = 1, /3 > 1) and ( a > 1,p = I), denoted as A and B, respec- tively, in Fig. 1 (a), any Ei coincide with some EV and vice versa, and one real eigenvalue is zero. As we shall show below, A and B are loci of transcritical bifurcation, where the Ei's exchange their stability with the E,'s

Discrete model (1.3D). For any R E (0,2) and 1 <a , p <(2 +R)/ R the Ei's are stable node, see Fig. l(b). On the segments A ={ a = 1, 1 <P <(2+ R)/R} and B = {p= 1, 1 <a <(2+R)/ R}, one eigenvalue takes the value 1, while on the segments C = { p = 3 , 1 < a < ( 2 + R ) / R ) a n d D = { a =(2+R)/R, l<P<(2+R)/R},oneeigen- value takes the value -1. As we shall see below, on A and B transcritical bifurcations occur between the Ei's and E,'s, while a flip-bifurcation of the Ei's may occur on segments C and D.

Two-species eq. ps. E,. Continuous model (1.3C). The spectrum corresponding to the E,'s is real, and A,

is always stable. The stability of A, is the region bounded by the couple of straight lines of equation p,(a, p ) = (P - 1)(1 - a ) =0 and by the hyperbola of equation +(a, p ) = 1-ap =0. The stability domain of A, is the region above the curve +(a, p ) =

0. As a consequence, sC(E,) is the domain of Fig. l(c). We note that for (a, P ) E sC(Ev), the E,'s do not belong to R:. On the half-lines denoted as A and B in Fig. l(c), A, =0 and it is easy to verify that these sets are loci of transcritical bifurcation of the E,'s with respect to the Ei's.

Discrete model (1.3D). For any R E (0,2) the eigenvalue p1is clearly stable. The stability domain for p2on the generic plane R =cost. is the region bounded by the two curves of equation p2(a, p, R) = a +P -ap(2+ R)/ R +(2 -R)/ R =0 and pl(a, P ) =0. In particular, we have p2= -1 on the hyperbola of equation 9, =0, and p2= 1 on the two straight lines of equation p, =0. The stability domain of p, is the region bounded by the two hyperbolas of equation p,(a, p, R) = a2+p2-a -p +(1-ap)(2+ R)/R =0 and $(a, P) =0. In particular, we have p, = -1 on the hyperbola p, =0. The resulting sD(E,) is reported in Fig. l(a); in such domain, the E,'s do not belong to z.The segments A and B in this figure, belonging to the curve p1 =0 where p2= 1, are loci of transcritical bifurcation of the E,'s with the Ei's, while on the piece of curve of equation p, =0, flip-bifurcation of the Eg7s may occur.

Three-species eq. p. Q. The eq. p. Q exists for any value of (a , p ) except on the line a +P = -1. In particular, for a +p > -1 (<-1) Q belongs to R: (R,- R:). Regarding the stability properties, the following results are obtained.

Continuous model (1.3C). s ~ ( Q ) is the domain bounded by the straight lines a +/3 = -1 and a +p =2 (see Fig. l(e)); the latter is the locus of Hopf-bifurcation, because for a +p =2, besides Re (A2,,) =0, Im (A2,,) Z 0 and d / d ~(Re (h2,,))Is-2 =

116# 0, where 7=a +P. As stated in the introduction, such Hopf-bifurcation is critical. Discrete model (1.3D). For any R E (0,2) p, is stable and ( / ~ 2 , , ( ~5 1 iff p4(a, p, R) =

a2++p2-a-p + ( a p - 1)(2- R) / ( l+ R ) SO; it follows that s ~ ( Q ) is the domain bounded by the ellipsis, ((R), of equation p4(a, P, R ) =0 (see Fig. l(f)). On 5, except for the two points on the line a =p, p2,3are complex. At a =p = 1, p2=p3=+1, while at =p = -(2- R)l(4+ R), p2=p, = -1. For any R E (0,2) the stable manifold of Q is the straight line x, =x2=x, (see (2.10)).

480 L. GARDINI, R. LUPINI, C. MAMMANA AND M. G. MESSIA

Appendix B. The elements of H = with respect to the basis P ~ J ( P ~ X * ) J ( X * ) , el = ( 1 , 1 , I ) /&, e2= ( 1 , - 1 , 0 ) / f i , e3= ( 1 , 1 , -2)/& can be determined by the for- mulae Hik= (JF(P1x*)JF(x*)ek,Pei),where

1-y1 -Y1 -PY1 JF(x*)= -PYZ Y ~ / Y Z - Y Z -YZ 9

-Y3 -PY3 Y2IY3 -Y3

~ 2 / ~ 3 - Y 1 -Y1 -BY1 JF(plx*)= -By3 Y ~ / Y Z - Y ~ -Y3

-Y2 -BY2 1 -Y2

and

y s = l / ( l - P ) , ~1,2=(&+2*' ) ' ) ~ 3 , ' ) ' = d m , & = [ ( R - 2 ) - p ( R f 1)]/2.

After straightforward algebraic computations, we find

Appendix C. Bifurcations in the two-dimensional L.V. map. The restriction of the L.V. map (1.3D) to the coordinate plane ( x i , xi) reads

with (A,B ) = ( a ,p ) or ( A , B ) = ( p ,a ) . The spectrum of the Jacobian matrix of the right side of (C.l) at the equilibria Ei = ( l ,O), Ej = (0, 1 ) and Eij=

( 1 -A , 1-B ) / ( l - A B ) are given by

A(Ei)= (1-R, 1+R ( l - B ) } ,

A(E, )={l -R , l + R ( l - A ) } ,

A (E i j )={ l+R[(a+p-2)* (a+p-2ap)] /2 (1 -ap)} .

For any R E (O,2) the eq. p. Ei(Ej) is stable for 1 <B < ( 2 + R ) / R ( l < A < ( 2 + R ) / R ) and bifurcates at B = l ( A = 1 ) as p2=+l, and at B = (2+ R ) / R as p2= -1 (see Fig. 19(a)).The eq. p. Eij is stable in the shaded region of

BIFURCATIONS IN THE LOTKA-VOLTERRA MAP

Frc. 19. Stability domains Sand bifurcation curves of the eq. ps. E,, E,, E , ofthe map (C.l). S(E,) and S(E,) in (a). S(E,) in (b).

Fig. 19(b), bounded by the straight lines A = 1and B = 1, loci of transcritical bifurcation of Eij with E, and Ei respectively, except for the point A = B = 1, and by the curve of equation qZ(A, B, R ) =0 (given in Appendix A), on which one eigenvalue is equal to -1. We have also reported in Fig. 19(b) the curve of equation cp,(A, B, R ) =0 to show that it belongs to the stability domain of E,.

Acknowledgments. Substantial improvements to the first version of the paper have been stimulated by the referee's comments and suggestions, for which we want to thank them.

REFERENCES

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Bifurcations and Transitions to Chaos in the Three-Dimensional Lotka- Volterra MapL. Gardini; R. Lupini; C. Mammana; M. G. MessiaSIAM Journal on Applied Mathematics, Vol. 47, No. 3. (Jun., 1987), pp. 455-482.Stable URL:

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References

8 Nonlinear Aspects of Competition Between Three SpeciesRobert M. May; Warren J. LeonardSIAM Journal on Applied Mathematics, Vol. 29, No. 2. (Sep., 1975), pp. 243-253.Stable URL:

http://links.jstor.org/sici?sici=0036-1399%28197509%2929%3A2%3C243%3ANAOCBT%3E2.0.CO%3B2-E

9 On #-Limits for Competition Between Three SpeciesP. Schuster; K. Sigmund; R. WolffSIAM Journal on Applied Mathematics, Vol. 37, No. 1. (Aug., 1979), pp. 49-54.Stable URL:

http://links.jstor.org/sici?sici=0036-1399%28197908%2937%3A1%3C49%3AOFCBTS%3E2.0.CO%3B2-B

10 Limit Cycles in Competition CommunitiesMichael E. GilpinThe American Naturalist, Vol. 109, No. 965. (Jan. - Feb., 1975), pp. 51-60.Stable URL:

http://links.jstor.org/sici?sici=0003-0147%28197501%2F02%29109%3A965%3C51%3ALCICC%3E2.0.CO%3B2-F

11 Asymptotic Behaviors in the Dynamics of Competing SpeciesJ. Coste; J. Peyraud; P. CoulletSIAM Journal on Applied Mathematics, Vol. 36, No. 3. (Jun., 1979), pp. 516-543.Stable URL:

http://links.jstor.org/sici?sici=0036-1399%28197906%2936%3A3%3C516%3AABITDO%3E2.0.CO%3B2-S

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