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4DIFFERENTIAL EQUATIONS AND
CONTROL PROCESSES N 2, 2006 Electronic Journal, reg. N P23275 at 07.03.97
http://www.neva.ru/journal e-mail: diff@osipenko.stu.neva.ru
Computer modeling in dynamical and control systems
ROUT TO CHAOS IN FOOD CHAIN DYNAMICS
George Osipenko Laboratory of Nonlinear Analysis St. Petersburg State Polytechnic University math@math.tu.neva.ru
Abstract
The paper proves that a real discrete system with biological origin possesses a non-oriented invariant manifold — Möbius band. The obtained results demonstrate the existence of multiple attractors in food-chains models. Moreover, the parameter region with three coexisting and closely-spaced attractors was found. It should be noted that such a proximity does not exclude the possibility that a complicated situation may appear, which may lead to more intriguing biological consequences in the system under study or similar systems.
The route to chaos in the food-chain dynamics is investivated. The initial system (as a parameter Mq < 2.9) has a single stable fixed point, when the parameter Mo increases the systems passes through non-trivial cascad of the bifurcations which, when Mq = 3.65, results in the appearance of a minimal chaotic attractor covering a Möbius band.
1 Analytical results
The 3-dimensional dynamical system describes a discrete food chain model. Lind-strom [7] proposed the model that displays a lot of properties commonly known for continuous food-chains [11, 2].
The discrete food chain model is defined by the mapping / of the form
M0Xtexp(-Yt)
X,
t~\~ 1
where
l + Xtmax(exp(-Y,),K(ZT)K(Yt))
Yt+1 = MiXtY, exp(~Zt)K(Yt) ■ K(M3Y,Zt) (1)
Zt+1 = M^YtZti
The detailed description of the model was given in [7]. The variables are related to the different trophic levels of the system, so X is proportional to vegetation abundance whereas Z is proportional to carnivore abundance. Since the relation between herbivores and Y is nonlinear, a more complicated relation describes the situation here. However, such relationships do not change the topological properties of the system under investigation. So, for convenience, we will refer to vegetation, herbivore, and carnivore levels in the sequel.
It should be noted that M2 = M3 in the original equation. The fourth parameter M3 is introduced in order to generate additional cases and obtain the complete analysis of system characteristics. So, the result from [10], p. 207-209 shows the existence of an invariant Mobius band at the parameter position Mq = 4.0, Mi = 1.0, M2 = 3.0, M3 = 4.0. However in this paper we will consider the original model with M2 = M3.
It should be marked that the solutions of (1) remain positive and bounded. Repeating the arguments given in [7] we can show that all solutions starting in the positive cone enter the box 0 < Xt < Mq, 0 < Yt < M0M1, 0 < Zt < M0M1MI/M3 within three iterations.
The system (1) has at most four equilibria [7] which are given by:
Eq — (0,0,0),
Ei = (Mq -1,0,0),
/ Mn log (MiMn\
/MolOg^^J ^
2 y(M0 — i)Mi — 1’ ogvi + mJ’
and E$(X, Y, Z) is given by
M0exp(—— 1 1 /
—m—1--------------7—2------;-----vT ’ XT ’ *°g M1 (M" exP(
,/i(]4)A-(logM1(M„exp(-IL)-l)) V
if M2 = M3 and max(exp(-Y), K(Y)K{Z)) = K(Y)K{Z) at Ez.
Some general features of the system can be described by the Morse spectrum of the determinant det Df [9] which is the rate of change of phase volume.
If £ = {vo? •••, vp — vq} is a p-periodic £-orbit, then the determinant exponent is defined by
1 P~l
A(det Df,g) = - In | det Df (vi) |.
The Morse spectrum of the determinant is defined as the following
E(det Df) = {A £ R : there are 0 and periodic £k — orbits with A(det Df, £k) —> A as k oo}.
It is well known [17] that if Ai, A2, and A3 are Lyapunov exponents of a periodic
orbit £ and A(detD/, £) is its determinant exponent then
M + A2 + A3 = A(det Df,£).
Our computing experiments show that in the selected area the considered system has the negative Morse spectrum of det Df. It follows from [9] that the volume tends to zero with negative exponent along each chain recurrent orbit, being its determinant exponent has at least one negative Lyapunov exponent.
2 Numerical results
We limit our discussion to the parameter range Mq G [3.00; 3.65] and fix Mi = 1.0, M2 — M3 = 4.0. We commence an overview about some general features and bifurcations. The selected area is located along the route to chaos.
At Mq ~ 2.93 a Neimark-Sacker bifurcation occurs. The fixed point E3 loses its stability and an invariant circle appears in its vicinity. This curve becomes the minimal attractor of the system. As the parameter Mq increases, the attractor is alternatingly quasi-periodic and periodic, like the dynamics of circle maps [1, 16]. This holds as long as the parameter value stays moderately far from the bifurcation value.
2.1 M0 = 3.000
The Morse spectrum of det(Df) is estimated as [-0.403970,-0.322668]. The system has the fixed point E2 with the coordinates (1.2164, 0.4055, 0) and the Lyapunov
exponents Li = 0.4837; £2,3 = —0.1667. The (xy)-plane is invariant for the differential at £2 and the exponents £2,3 correspond to a focus on this plane. Thus, the unstable manifold WU(E2) is a curve transversal to the (xy)-plane. The system has the fixed point E% with the coordinates (1.7160, 0.2500, 0.2806) and the Lyapunov exponents = 0.0630; L3 = —0.6430. The unstable manifold WU(EZ) is a 2-dimensional surface with an unstable focus. Our computing results show that the closure of WU(EZ) is a global attractor in the positive corner {x >
0, y > 0, z > 0}. In particular, WU(E2) tends to the closure of ^(^3), see Fig.
1. Such a dynamics may be observed when the parameter Mq changes from 3 to
4.
Figure 1: Unstable manifolds WU(E2) and WU(E3), M0 = 3.000.
The closure of the unstable manifold WU(EZ) is diffeomorphic to a standard 2-dimensional closed disc, see Fig. 1. The boundary C of the unstable manifold WU(EZ) is homeomorphic to circle S1. The stable invariant curve C appears at the Neimark-Sacker bifurcation at Mq ph 2.93 and looses its stability at Mq « 3.366. As Mq = 3.0 we observe a quasiperiodic behavior on C. The first approximation of this rotation looks like as 9-periodic. However, Danny Fundinger found the coordinates of a point Xo on C whose 50,000 iterations form a line,being the iterations from 40,000 to 50,000 form a line as well. We conclude from this that the movement on the cycle is not periodic. The coordinates of the iterations are X0 = (1.336740,0.379555,0.253432), X8 = (1.471666,0.417361,0.145821), X9 = (1.372226,0.385229,0.243440), X18 = (1.378605,0.390052,0.234376), X36 = (1.392378,0.397436,0.218947). These results show that if we start from a point Xq, then Xg is shifted a little bit from the position of Xq, Xi$ a little bit further and so on. The chain recurrent sets E2, E3 and C are localized by the symbolic method, the unstable manifolds WU(E2) and WU(EZ) are constructed
by the iterations of broken lines and polytopes, respectively.
When parameter Mq changes from 3 to 3.3, both the topological structure of trajectories and the manifolds WU(E2), WU(EZ) persist.
Figure 2: Unstable manifolds WU(E3), M0 = 3.300.
2.2 M0 = 3.300
The Morse spectrum of det(Df) is estimated as [-0.431418,-0.289534]. The fixed points E2 and E3 have the coordinates (1.27119, 0.50077, 0) and (2.01597, 0.25, 0.39002) respectively. As in previous case they have the same type of stability. Moreover, there is a minimal attractor C with quasiperiodic motion. However, as Mq = 3.3 the manifold WU(E2) tends to the invariant curve C by winding around C, see Fig. 2. Hence C should not be considered as a boundary of the smooth manifold but as its limit set.
2.3 M0 = 3.3701
The Morse spectrum of det(Df) is estimated as [-0.526124,-0.194662]. The stable invariant curve C looses its stability at Mq ¡=s 3.366. When the parameter Mq becomes 3.3701, this bifurcation results in the appearance of a Mobius band MB(3.3701). Later we prove that the invariant manifold MB(3.3701) is nonoriented. The bifurcation is like period doubling bifurcation which is usually observed in continuous dynamical systems. However, we can not speak of period doubling bifurcations here in the same sense as is usually meant for discrete systems. More precisely, in the discrete system we observe a pattern which is
typical for continuous systems [4, 3]. So we can say about a " Feigenbaum-like bifurcation". As we will see later, several Feigenbaum-like bifurcations of the same kind happen close to the transition to chaos. The manifold MB(3.3701) is the limit set of the unstable manifold WU(E$). Construct the unstable manifold Wu(E2) of the fixed point E2 as above. The boundary L = dMB{3.3701) is a limit set of the unstable manifold WU(E2), see Fig. 3. Center line C of
Figure 3: Unstable manifold WU(E2) and the Möbius band MB(3.3701).
MB{3.3701) is an unstable invariant curve (on MB{3.3701)) with quasiperiodic motion, see Fig. 4. Note that L is a stable invariant curve homeomorphic to circle which is two times longer than the center line C. On L there are the stable 55-periodic orbit {P,... } and the hyperbolic 55-perodic orbit {H,... } which is stable on L. The rotation number [3] of the system on L is 3/55. The points #(1.631969, 0.105806, 0.778837), Q(1.456810, 0.157710, 0.718353) lies on the orbit of P(l.519275, 0.140199, 0.847081), and H has the coordinates (1.582335, 0.118405, 0.815238), see Fig. 4. The eigenvalues of the differential D/55 at P are approximately equal to Ai^ = 0.8584±0.2398i and A3 = 5-10-8. Hence the point P is focus for /55. To prove non-orientability of the band MB(3.3701) we consider the direction of the rotation on the orbit of the point P. Comparing the rotation at points along the center line, i.e. at P, R, etc. (see Fig. 4), we see that the direction of the rotation persists. But the directions of the rotation at the points P and Q are opposites. So, if we go from P to Q along the center line, we save the orientation and if we go transversal to C we get opposite orientation. This is possible if and only if the band is non-oriented. There is only single 2-dimensional non-oriented strip - Möbius band.
Thus, the global dynamics of the system in the positive corner {x > 0, y > 0, z > 0} is the following. The stable 55-periodic orbit of the point P is a single
Figure 4: Detail of the Môbius band MB(3.3701).
attractor A minimal by inclusion. Other trajectories, except the fixed points E3, the center line (7, and the hyperbolic 55-periodic orbit of the point H, tend to A.
2.4 M0 = 3.4001
Figure 5: The unstable manifold WU(E$) and the Möbius band M.5(3.4001).
The Morse spectrum of det(Df) is estimated as [-0.580550,-0.186926]. By using symbolic methods we determined the coordinates of the fixed points E2 (1.288505, 0.5309395, 0.0), E3 (2.11595, 0.24995, 0.42300) and two chain recurrent curves C and L. The eigenvalues at E2 and E3 are Ai(i?2) = 2.124, \2,z(E2) = 0.628869 ± 0.627i, and Xlf2(E3) = 0.806993 ± 0.812688i, A3(E3) = 0.447195. Both C and L are homeomorphic to a circle, being L is two times longer than C. The curve L becomes the minimal attractor of the system. These two curves
belong to a 2-dimensional invariant manifold MB(3.4001) which is homeomorphic to an Möbius band, Fig. 5. On the picture you can see a collection of curves being ustable manifold of an orbit on the line C. Geometrically, the curves resemble the shape of a Möbius strip. The stable invariant curve can be imagined as the edges of the strip, and the unstable curve as its center line. Numerical studies of forward and backward iterations so far indicate that the curve at the center line has saddle type in the space and unstable type on M.B(3.4001). We constructed the unstable manifold WU{E%) and the Möbius band M13(3.4001), see Fig. 5.
The Morse spectrum of det(Df) is estimated as [-0.552695,-0.181412]. The Mobius band MB(3A8) persists. It is a global attractor in the positive corner {x > 0, y > 0, z > 0}. The center line C of MB(3A8) is an unstable invariant curve on MB(3A8). The dynamics on C is periodic with the rotation number 7/64. There is a stable 64-perodic orbit of the point P & (1.231941,0.800882,0.137928) on C. The limit set of MB{3A8) is a stable invariant curve L with periodic dynamics of the rotation number 1/18. There is a hyperbolic 18-periodic orbit of the point U « (2.194811,0.356991,0.065345) on L. The differential Df18(U) has the eigenvalues Ai « 1.21, A2 ~ —0.28, A3 « —0.002. The first eigenvalue corresponds to L so the orbit of U is unstable on L. There is a stable 18-periodic orbit of the point S « (1.652079,0.131587,0.502229) on L. The differential Df18(S) has the eigenvalues Ai « 0.81, A2 ~ —0.40, A3 ps —0.003. Since the pair of the eigenvalues at U and S are negative, the Mobius band MB(3A8) tends to L by winding around one. The periodic orbit of S is a single attractor minimal
2.5 M0 = 3.480
Figure 6: The boundary L of MB(3A8) is a limit set of WW(E2).
by inclusion. Thus, almost all trajectories from positive corner tend to this orbit.
Figure 7: Unstable manifold WU(E2) and its limit set LimWu(E2), Mo — 3.532.
The Morse spectrum of det(Df) is estimated as [-0.572940,-0.162688]. The coordinates of the fixed point E2 are (1.305196,0.561747,0). We constructed the unstable manifold WU(E2) and its limit set LimWu(E2). It turns out that LimWu(E2) has nontrivial structure, see Fig. 7. The limit set consists of the 107 circles (C107), the hyperbolic 107-periodic orbit H7 its unstable manifold WU(H)} and the attractor A, see Fig. 8. The unstable 107-periodic orbit U is inside of the set C10T. The limit set of the Möbius band MB(3.532) coincides with LimWu(E2). The attractors A and C107 are minimal by inclusion, so we have a non-ordinary phenomenon —the existence of two minimal attractors in a biology system. It should be noted that the set C107 exists in a short interval for Mq. The circles, the orbits U and H disappear when Mq ph 3.536, whereas the attractor A persists. The paper [6] deals with the appearance of multiple attractors in the chain food dynamics and contains detailed information about the considered system.
2.7 M0 = 3.540
The Morse spectrum of det(Df) is estimated as [-0.578320,-0.173867]. The attractor A is the limit set of the Möbius band MB from Mq « 3.536 to Mq « 3.538, but for all that the invariant curve A looses its stability for the last parameter value. When the parameter Mq becomes 3.54, the bifurcation results in the appearance of the second Möbius band Mi?2(3.54), see Fig. 9. Hence we have
2.6 M0 = 3.532
Figure 9: The second Möbius band MB2(3.54), its unstable center line C2, and the stable 283-periodic orbit S on L2.
" Feigenbaum-like bifurcation".
The second Möbius band MB2(3.54) appears as a limit set of the first Möbius band MI?i(3.54). The center line of the second Möbius band MI?2(3.54) is unstable invariant curve C2 with quasiperiodic motion. The boundary L2 of MB2(3.54) is a stable invariant circle which is two times longer than the center line C2. On the boundary L2 there are two 283-periodic orbits: stable S ~ {(1.354245,0.467325,0.679182),...} and unstable U &
{(1.354245,0.467325,0.679182),... }. The points of these orbits alternate.
Figure 10: The trajectories of the first Möbius band tend to the second Möbius band by winding around it, M0 — 3.570.
2.8 M0 = 3.570
The Morse Spectrum of det(Df) is estimated as [-0.565295,-0.161637]. When the parameter Mq is greater than 3.538, the second Möbius band becomes a limit set of the first Möbius band. More precisely, the trajectories on MB\ tend to MB2 by winding around it. The set Mi?i(3.57), the unstable malifold of an orbit on the center line C\ and its behavior near the limit set MB2 are shown on the Fig.
Now we observe the next "Feigenbaum-like bifurcation" near the boundary of the second Möbius band MB2. Here the bifurcation is the following. The invariant curve L2 = dMB2 loses its stability and a strange invariant set MB% appears, see Fig. 11. The set MB^(3.570) consists of 71 pieces,
10
p
Figure 11: The second Möbius band and the detail of its limit set, M0 = 3.570
each piece has both 2- and 1-dimensional parts. This decomposition is invariant, i.e. an image of a piece part is a piece part. The 2-dimensional part is the closure of the unstable manifold WU(U) of the hyperbolic 71-periodic orbit U ft {(1.323374,0.186463,0.646940),...}. The eigenvalues of the differential Df7\U) are Ai ft 1.374292, A2 ft -1.501502, A3 ft -0.794848. The eigenvalues Ai and A2 correspond to the unstable manifold WU(U). The system behavior on WU(U) is similar to its dynamics on a Möbius band. As A1A2 < 0, the 71-th iteration /71 inverts the orientation on WU(U). The 1-dimensional part is formed by the unstable manifold WU(H) of the hyperbolic 71-periodic orbit H ft {(1.576668,0.44846,0.72410),... }. The manifold WU(H) ends at the stable 71-periodic cycles Q ft {(1.913398,0.677286,0.093128),...} and P ft {(2.489430,0.086803,0.027202),...}, see Fig. 11. The limit set of the 2-dimensional manifold WU(U) is a stable invariant curve which forms by the 1-dimensional unstable invariant manifold WU(F) of the hyperbolic 142-periodic orbit F. The structure of the both Möbius bands persists near M0 ft 3.5708. Moreover, the structure of the invariant set MB$ persists as well. It should be noted that in this case we detect a new phenomenon. When Mq = 3.5708 the set MB$ contains at least three stable periodic orbits: the known 71-periodic orbits Q ft {(1.560941,0.447309,0.731145),...} and P « {(1.601819,0.436388,0.74072),... } and a new stable 142-periodic orbit S ft {(1.547730,0.450719,0.727846),...}. Thus, we obtain three minimal attractors in the biology system.
2.9 M0 = 3.571
Figure 12: The invariant set of Af£>3(3.571) and the unstable manifold Wu(Hq) on it.
In this case the Morse spectrum of the det(Df) is estimated as [-0.596287,-0.163302]. The structure of MP3(3.571) is the following. There are two attractors minimal by inclusion. One of them — P — is a stable 71-periodic cycle generated by the point Po & (1.6014,0.4362,0.7415), see Fig. 12. The second attractor is a stable cycle S formed by 142-periodic orbit of the point So ~ (1.93364,0.60068,0.0002). The eigenvalues of these cycles are following:
Ai,2(P) « 0.6477 ± 0.3656z, |Ai,2(P)| = 0.7438 < 1, A3(571) « 0
and
Ai)2(5) « 0.5079 ± 0.3626i, |Ai,2(S)| = 0.6241 < 1, A3(S) « 0.
Thus, the both cycles are of focus type and obviously, they are attractors minimal by inclusion. Additionally, there are two unstable periodic cycles. One of them H is 71-periodic orbit of the point Hq « (1.5889,0.4438,0.7305) and the other Q is 142-periodic orbit of the point Qo « (1.578620,0,410556,0.798312), f71(Qo) = Q71 ph (1.649636,0.279537,1.066632). Again, we analyze the eigenvalues of these cycles:
Ai(H) « 1.4817, A2{H) « 0.5603, A3(H) « 0;
and
Ai(Q) « 1.8779, A2(Q) « -0.0669, A3(Q) « -0.
Moreover, we estimate the corresponding eigenvectors and eigenspaces. So the periodic cycles H and Q are of hyperbolic type with 1-dimensional unstable manifolds. The unstable manifold WU(H) are constructed and we see (Fig. 12) that the right part of Wu(Ho) has a simple behavior and ends at the point Po of the orbit P. The left part of Wu(Hq) has more complex oscillated behavior. The limit set of Wu(Hq) contains the points So and 671 = f71(So) of the stable orbit
S. The limit set of Wu(Hq) contains the points Qq and Q71 of the hyperbolic orbit Q, see Fig. 13. We estimate the angle between the stable subspace of the orbit Q at Qo and Wu(Hq) and verify that their intersection is transversal near Qo. Thus, we have the heteroclinic transversal connection H —>■ Q.
The next step is the construction of the unstable manifold Wu(Qo) at the point Qo- It turn out that here we have a similar behavior: one part of Wu(Qo) ends at So of the stable orbit S. The other part of Wu(Qo) has more complex oscillated behavior such that its limit set contains the point of the hyperbolic orbit H and the point P36 of the stable orbit P. Here we also obtain the heteroclinic transversal connection Q —)• H. From this it follows that there is the homoclinic connections Q —>• Q and H —»• H. To check this conclusion we construct the global unstable manifold WU(H) of the hyperbolic orbit H. The result of our
Figure 13: The closure of the unstable manifold WU(H), the stable orbits S and F.
computing displayed on Fig. 13 shows that the limit set of Wu{Hq) contains = /36(i^o)- Thus, there exists the homoclinic orbit H —> H, which usually leads to chaotic dynamics near this orbit [12, 13, 14, 15, 8, 3]. Our numerical investigation shows that a chaotic dynamics is located inside of the closure of WU(H). This closure is not minimal by inclusion because it contains the stable orbits S and P, see Fig. 12.
2.10 Chaos
Figure 14: The chaotic attractors for M0 = 3.573 and M0 = 3.58.
Consider the system behaviour for Mq = 3.573, 3.580 and 3.650. The Morse spectrum of the differential was obtained by the symbolic analysis methods. For Mq — 3.573 the Morse spectrum of det(Df) is estimated as [-0.601288,-0.167536]. In the case Mo = 3.58 the spectrum is estimated as [-0.582492,-0.156852] and for Mq = 3.65 the estimate is [-0.601972,-0.136850]. Thus, we have negative spectrum in all cases. It results in zero volume of the chain recurrent set and, as a consequence, in zero volume of the chaotic attractor. It was mentioned above that the chaotic dynamics appears when Mo ~ 3.571 and is located in very small domain. When Mq e [3.573,3.650] several local and global bifurcations lead to a number of subsequent changes in the dynamics. This involves the appearance of chaotic attractor which grows as Mq increases. Our construction of a chaotic attractor is trivial: we pick up a point from its domain of attraction and consider its iterations. If the iteration number is huge, we obtain the desired attractor. In our cases the chaotic attractor is minimal by inclusion, so almost all points from the positive corner are in the basin of the desired attractor. When Mo = 3.573 we can observe (see Fig. 14) that the chaotic attractor coincides with the invariant set Mi?3(3.573). If the parameter Mq increases and reaches value Mq = 3.58, the chaos occupies the second Möbius band MZ?2(3.58), see Fig. 14. After all, as Mq = 3.650 the chaos covers the first Möbius band MI?i(3.65), see Fig. 15.
Figure 15: The chaotic attractor for M0 = 3.65.
3 Conclusion
In this paper we have proved that a real discrete system with biological origin possesses a non-oriented invariant manifold — Môbius band. The obtained results
demonstrate the existence of multiple attractors in food-chains models. We have found several parameter regions where attractors of this kind exist. Moreover, the parameter region with three coexisting and closely-spaced attractors was found. It should be noted that such a proximity does not exclude the possibility that a complicated situation may appear, which may lead to more intriguing biological consequences in the system under study or similar systems.
We have shown the route to chaos in the food-chain dynamics. The initial system (Mq < 2.9) has a single stable fixed point, when the parameter Mq increases the systems passes through non-trivial cascad of the bifurcations which, when Mq = 3.65, results in the appearance of a minimal chaotic attractor covering a Mobius band.
Acknowledgement.
This research was supported by the Royal Swedish Academy of Science.
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