chaotic dynamics of a nonlinear density dependent...

CHAOTIC DYNAMICS OF A NONLINEAR DENSITY

DEPENDENT POPULATION MODEL

ILIE UGARCOVICI AND HOWARD WEISS

Abstract. We study the dynamics of an overcompensatory Leslie populationmodel where the fertility rates decay exponentially with population size. Wefind a plethora of complicated dynamical behaviors, many of which have notbeen previously observed in population models, and which may give rise tonew paradigms in population biology and demography.

We study the two and three dimensional models and find a large varietyof complicated behaviors: several types of codimension one local bifurcations,period doubling cascades, attracting closed curves which bifurcate into strangeattractors, multiple co-existing strange attractors with large basins (whichcause an intrinsic lack of “ergodicity”), crises – which can cause a discontinuouslarge population swing, merging of attractors, phase locking, and transientchaos. We find (and explain) two different bifurcation cascades transformingan attracting invariant closed curve into a strange attractor. We also find oneparameter families that exhibit most of these phenomena. We show that someof the more exotic phenomena arise from homoclinic tangencies.

1. Introduction

Population biologists and ecologists need accurate population models to predictanimal population sizes and compositions. Models help with wildlife management,establishing harvesting limits, and could be of tremendous help in conservation ef-forts. Governments and businesses need reliable age-structured demographic mod-els to project Social Security outlays, Medicare outlays, education outlays, taxreceipts, set immigration policy, etc.

Virtually all animal population and demographic forecasting models in currentuse are based on variations of the age-structured linear Leslie model (see section2). Perhaps not coincidentally, the 2000 U.S. census “shocked demographers”: itshowed that the best demographic models based on 1990 census data underesti-mated the U.S. population by six million people [35]. Many population biologistsand demographers now look to nonlinear models for more accurate population mod-els and projections.

In this article we study the dynamics of the overcompensatory Leslie populationmodel where the fertility rates decay exponentially with population size [5, 8, 12, 19,34]. Caswell highlights this nonlinear and noninvertible model in the introduction

Date: October 26, 2003.2000 Mathematics Subject Classification. 37D45, 37G35, 37N25, 92D25.Key words and phrases. Leslie population model, chaos, strange attractors, multiple attractors,

crises.Both authors were partially supported by National Science Foundation grant DMS-0100252.

The authors would like to thank Robert Schoen, James Yorke, and Lai-Sang Young for theirinterest in this work.

1

2 ILIE UGARCOVICI AND HOWARD WEISS

of the second edition of his influential treatise [5], as well as in a latter section, andwe think it fair to infer that he is championing this nonlinear model as a prototypefor the next generation of population models.

In the two and three generation versions of this overcompensatory model, throughcareful numerical studies, we find a plethora of complicated dynamical behaviors:several codimension-one local bifurcations, period doubling cascades, attractingclosed curves which bifurcate into strange attractors, three routes to chaos, multipleco-existing strange attractors with large basins (which cause an intrinsic lack of “er-godicity”), crises (interior, boundary – which can cause a discontinuous large pop-ulation swing), merging of attractors, phase locking, transient chaos, and nonuni-form hyperbolicity. We also find one parameter families that exhibit most of thesephenomena. We can not help stress that the variety of complicated dynamicalphenomena in these one-parameter families of smooth maps of R2 and R3 is bewil-dering. We show that some of the more exotic phenomena arise from homoclinictangencies.

Many of these phenomena have not been previously observed in population mod-els and may give rise to new paradigms in population biology and demography. Forinstance, this overcompensatory Leslie model exhibits crises (collisions of (strange)attractors with unstable periodic orbits), which can result in very large and dis-continuous population changes. Grebogi, Ott, and Yorke initiated a comprehensivestudy of these phenomena, see [11]. Crises might help explain some seemingly”bizarre” population explosions or collapses, like the current sardine explosion offthe waters of northern California [6]. We stress that the existence of crises “caus-ing” large and discontinuous population swings result from a special type of chaoticbehavior of the system, and this is a striking consequence to populations of chaos.

We find and explain two different complicated bifurcation cascades transformingan attracting invariant closed curve “into” a strange attractor. The cascade inthe two-dimensional model begins with phase locking and the destruction of aninvariant curve, while the cascade in the three-dimensional case begins with a Hopfbifurcation. Period doubling cascades play an essential role in both cascades. Whileone can find several models in the dynamics literature where an invariant closedcurve seems to bifurcate into a strange attractor [2, 7, 10, 33], these are the firsttwo examples for population models, and we have a complete understanding of thecascades. Crises play an essential role in these cascades.

Even though it is usually impossible to make long term predictions about chaoticsystems, Grebogi, Ott, and Yorke have also pioneered the study of “controllingchaos” [24] and some of their techniques may be useful to control populationsexhibiting chaotic behavior.

Demographers and population biologists call a model (or the population it de-scribes) ergodic if the asymptotic dynamics are independent of the initial conditions.To them, ergodic models are useful because population patterns might reveal some-thing about the underlying process rather than initial conditions. Most populationbiologists seem to strongly believe that what is important is the underlying popula-tion process and the initial conditions are historical accidents, and thus ergodicityis a fundamental element in models. We discovered parameter ranges with coexist-ing strange attractors with large basins, in which this notion of ergodicity fails inan essential way. This may help change population biologists’ views of the roles ofergodicity and initial conditions.

CHAOTIC DYNAMICS OF A NONLINEAR POPULATION MODEL 3

We remark that for this nonlinear model, most of the complicated dynamicsonly occur for large fertility rates (fish, small mammals, insects, etc.). Although weare currently unaware of any animal whose population can be accurately forecastedusing this precise model, this model exhibits many complicated dynamical phenom-ena and paradigms which seem likely to arise in the next generation of populationmodels.

In a landmark work, Constantino, Cushing, Dennis, Desharnais, and Hensonmodeled the population growth of laboratory populations of flour beetles [9]. Theyderived a nonlinear model starting from the three generation Leslie model andincorporated the effect of cannibalism caused by overpopulation stress. They showthat for certain parameter ranges, the system has a strange attractor. Remarkably,they observe this chaotic behavior in the laboratory. This work has been hailed asthe first verifiable example of chaos in ecology. We also refer the reader to [9] foran enlightening discussion of the difficulties of biological modeling.

Finally, we outline the structure of the manuscript. In the second section, webriefly describe the (linear) Leslie model. In the third section, we introduce thenonlinear extension of this model and in section four some of the basic properties aredescribed. In section five, we analyze the dynamics for the two generation versionof the model. Almost all of the complicated dynamical phenomena we observefor the three generation model already appear in a one parameter family for thetwo generation model. In section six, we discuss the differences between two andthree generation models. In the Appendix, we briefly review some useful conceptsfrom Dynamical Systems, including the definition of stable and unstable manifolds,Lyapunov exponents, strange attractors, Henon-like attractors, and crises.

2. The linear Leslie model

About 60 years ago, Leslie introduced the following age-structured linear popula-tion model [17, 18]. Consider a population divided into d age-classes or generations,which we call generations 1, 2, . . . , d. Let nk(t) be the number of individuals in thek-th age-class at time t, measured in generations. The individuals in generations2, . . . , d are survivors of the previous generation at time t, and thus one assumesthat for 2, 3, . . . , d

nk(t+ 1) = pk−1nk−1(t),

where pi, i = 1, . . . , pd−1 is the probability that an individual of age class i survivesfor one generation. The new members of age-class 1 can not be survivors of anyother age-class; they must have originated from reproduction. Thus, Leslie assumed

n1(t+ 1) = f1n1(t) + f2n2(t) + · · ·+ fdnd(t),

where fi is the per-capita fertility of generation i. Hence

n1

n2

...nd

(t+ 1) =

f1 f2 · · · fd−1 fdp1 0 · · · 0 0...

... · · ·...

...0 0 · · · pd−1 0

n1

n2

...nd

(t),

or more compactly,

n(t+ 1) = An(t).


This special age-classified matrix A is called the Leslie matrix. The matrix A is non-negative with positive entries only in the first row and the subdiagonal. Populationbiologists and demographers use life table analysis to construct the matrix A.

Since A has non-negative entries and Ad has positive entries (for quite generalconditions on f1, f2, . . . , fd), the Perron-Frobenius theorem gives the stable agedistribution. More precisely, if λ1 is the spectral radius of A, then

limt→∞

(1

λ1A

)t= v ·wT ,

where v and w are normalized left and right eigenvectors for λ1.Most Leslie models used for actual demographic forecasting use five year age

groups instead of three generations, in which case the matrix A becomes a 10× 10matrix. The text “Matrix Population Models” by Caswell [5] contains a compre-hensive treatment of Leslie models.

3. The nonlinear Leslie model

We now consider the Leslie population model where the fertility rates decayexponentially with population size (Ricker-type nonlinearity). In this case, theLeslie matrix A becomes the population-size dependent matrix

A(N) =

f1e−λN f2e

−λN · · · fd−1e−λN fde

−λN

p1 0 · · · 0 0...

... · · ·...

...0 0 · · · pd−1 0

One obtains the following system of d nonlinear equations

n1(t+ 1) = (f1n1(t) + f2n2(t) · · ·+ fdnd(t))e−λ(n1(t)+n2(t)+···+nd(t))

n2(t+ 1) = p1n1(t)

n3(t+ 1) = p2n2(t)

...

nd(t+ 1) = pd−1nd−1(t)

The associated 2d-parameter nonlinear dynamical system T : Rd → Rd is

(3.1) T

x1

x2

...xd

=

(f1x1 + f2x2 + · · ·+ fdxd) · e−λ(x1+x2+···+xd)

p1x1

...pd−1xd−1

.

Notice that when λ = 0 this model reduces to the classical Leslie matrix model.For two and three generations, the corresponding systems are

(3.2) T (x, y) = ((f1x+ f2y) · e−λ(x+y), p1x)

and

(3.3) T (x, y, z) = ((f1x+ f2y + f3z) · e−λ(x+y+z), p1x, p2y)

respectively.


In [12], the authors study a special case of (3.2) when f1 = f2 and p1 = 1; theyanalyze the stability of the nontrivial fixed point, finding four types of codimension-one local bifurcations: transcritical, saddle-node, Hopf, and period doubling. Nu-merically, they also find a “strange attractor” obtained after a period doublingcascade. However, they largely ignore the investigation of multiple stable statesand crises phenomena. Levin and Goodyear effected a local analysis for a relatedfish population model [19]. In [34], the authors rigorously study the stability of thepositive fixed point for two, three, and four generation models. They determinecodimension-one local bifurcations. A somewhat sketchy analysis is performed inthe unstable region of the fixed point.

Some of the complicated dynamical phenomena we observe for the two generationnonlinear Leslie model have also been observed for the famous Henon mapping[14, 31], H : R2 → R2, H(x, y) = (a− by−x2, x) where a and b are real parameters.The mapping H is a quadratic polynomial and is easily seen to be a diffeomorphism.The Jacobian determinant is the constant function b. Benedicks and Carlesonproved that for a positive measure set of parameter values with 1 ≤ a ≤ 2 andb ≈ exp(−20), the mapping H has a nonhyperbolic attractor with a dense orbit andhas a positive Lyapunov exponent. For the parameter values they study, the Henonmapping is a tiny perturbation of a one dimensional quadratic mapping, and this iscrucial for their analysis. Very little has been proven about this mapping for largervalues of b; see [15, 21] for the existence of transversal homoclinic intersections,hence hyperbolic horseshoes.

For comparison, the overcompensatory Leslie model is not a polynomial map-ping, it is not a diffeomorphism, and it is not usually dissipative (the modulus ofthe Jacobian matrix does not have determinant less than one). Not surprisingly, wecan not prove very much about the global dynamics. We effect a careful numericalstudy and attempt a theoretical interpretation of our numerical study and numer-ical experiments. A priori, our strange attractors could be periodic orbits havinghuge period (e.g., 107 generations). However, possibilities like this seem totallyirrelevant to the application at hand.

Our goal for the remainder of this paper is to investigate the dynamics of themapping T and the implications for the population.

4. General facts about the nonlinear Leslie model

Before we proceed with the detailed analysis of two and three generation models,we make some general observations about the mapping T . For ease of exposition,we will consider the three generation case.

(1) The nonlinear mapping T is C∞ smooth. It is non-invertible; every pointhas at most two preimages under T . A priori, non-invertibility greatly com-plicates the analysis of a dynamical system. Although the local stable andunstable manifolds exist and have the usual properties (see Appendix), theglobal manifolds are not always well-defined, and even if they are, they couldhave nasty pathologies like cusps and self-intersections. Luckily, in our case,the noninvertibility causes few problems.

(2) If f1 + f2p1 + f3p1p2 < 1, the origin (0, 0, 0) is a global attractor for T (see[8] for details). The quantity f1 + f2p1 + f3p1p2 is the average per capitanumber of offspring that each member produces over their lifetime, assuming


λ = 0, or no overcompensation. The statement implies that if, without theovercompensation, the average number of offspring per individual per lifetimeis less than one, then the population becomes extinct over time.

(3) T is pointwise dissipative [8, 13], i.e., there exists a trapping region R whichabsorbs every orbit after a finite number of iterations of T . Indeed, every orbit,after two iterations, gets mapped into the box

R = {0 ≤ x ≤ r, 0 ≤ y ≤ p1r, 0 ≤ z ≤ p1p2r},where r = (f1 + f2 + f3)/(λe). It follows that the set

Λ =

∞⋂

n=1

Tn(R)

is a global attractor. In general, this attractor is not transitive, but we willtry to identify the transitive components.

(4) If f1 + f2p1 + f3p1p2 > 1, the mapping T has a unique positive fixed point at(a, p1a, p1p2a) where

a =1

λ· log(f1 + f2p1 + f3p1p2)

(1 + p1 + p1p2).

Moreover, the origin (0, 0, 0) is unstable, and the dynamical system is uni-formly persistent with respect to (0, 0, 0), i.e., there exists C > 0 such that if(x0, y0, z0) ∈ R3

+ \ {0, 0, 0}, then lim infn→∞ |x(n) + y(n) + z(n)| > C (see [8]for details).

(5) It is easy to compute the linearization of the map T . The Jacobian JT (x, y, z)of T at (x, y, z) is

e−λ(x+y+z)(f1 − λf) e−λ(x+y+z)(f2 − λf) e−λ(x+y+z)(f3 − λf)p1 0 00 p2 0

,

where f = f(x, y, z) = f1x+ f2y + f3z. The determinant of the Jacobian is

det(JT (x, y, z)) = e−λ(x+y+z)p1p2(f3 − λ(f1x+ f2y + f3z) .

The mapping T is said to be dissipative if | det JT | < 1. In the special casef = f1 = f2 = f3, the map is dissipative if

p1p2f |1− λ(x + y + z)| < eλ(x+y+z) .

Clearly, if p1p2f < 1, then T is dissipative.

(6) The mapping T is a local diffeomorphism at a point (x, y, z) if det JT (x, y, z) 6=0. If det JT (x, y, z) = 0 we call this point degenerate. We define the degeneracylocus D to consist of all the degenerate points. Thus D is the plane f1x+f2y+f3z = f3/λ.

(7) The parameter λ is just a rescaling factor for the mapping T , and thus thedynamics does not depend of the particular choice of λ. In particular

T (λ;x, y, z) =1

λT (1;λx, λy, λz).

In all our numerical simulations, λ = 0.1.


5. The two-generation model

In this section, we study the two generation model with equal fertility ratesf = f1 = f2 and fixed survival probability p1 = 0.7,

T (x, y) = (f · (x+ y)e−0.1(x+y), 0.7x) .

We have chosen this value of p1 to illustrate the complicated dynamics one obtains,and in particular, the existence of multiple strange attractors with large basins ofattraction. This remarkable one parameter family exhibits all of the complicateddynamics we have so far observed for the two generation model, and also containsalmost all of the complicated dynamics we have so far observed for the three-generation model.

We briefly remark that for 0 < p1 < 1/2 the dynamical system undergoes aperiod doubling route to chaos [34].

If 1/2 < p1 < 1, then the fixed point undergoes a supercritical Hopf bifurcationat

fH =1

1 + p1e(1+2p1)/p1 ≈ 18.13 .

This type of bifurcation can not occur for one-dimensional maps. Using the ob-servations made in the previous section, we have, for 0 < f < 1/(1 + p1) = 0.588,the origin (0, 0) is a global attractor. If f > 0.588, there exists a positive fixedpoint (equilibrium solution) which is asymptotically stable for small enough f . For0.588 < f < 14.9, the equilibrium solution is a global attractor.

Figure 1 shows the bifurcation diagram for 10 ≤ f ≤ 35.1 The dotted linesindicate unstable periodic points, which must be computed separately using a rootfinding algorithm.

15 20 25 30 35f

20

40

60

80

100

120

N

Figure 1. Bifurcation Diagram: 10 ≤ f ≤ 35

We now discuss the salient features of the bifurcation diagram. We remark thatsince this is mainly a numerical study, all bifurcation values are approximative.

At f = 14.9, there is a 3-piece saddle-node bifurcation that produces an at-tracting period 3 orbit and its associated period 3 (unstable) saddle orbit. For

1For each f we draw the attractor(s) by plotting the total population size x(t) + y(t) for10000 ≤ t ≤ 10500 starting with 50 random initial conditions


14.9 < f < 18.13, the attracting fixed point coexists with the new period 3 at-tracting orbit. At f = 18.13, the stable fixed point undergoes a supercritical Hopfbifurcation, which creates a smooth invariant attracting closed curve.

The period 3 saddle-node bifurcation and Hopf bifurcation are followed for largeparameter values by various local and global bifurcations which we analyze below.

First, we briefly illustrate the complexity of the bifurcation scenarios with severalexamples for 14.9 < f < 24.195 (figure 2), where there exist two differenttransitive attractors, each with a large basin of attraction.

0 20 40 60 800

10

20

30

40

50

60

++

+

+

(a) f = 19: a period-3 attracting cycle co-

exists with an invariant closed curve

0 20 40 60 800

10

20

30

40

50

60

(b) f = 23.1: a six-piece Henon-like attrac-

tor coexists with an invariant closed curve

0 20 40 60 800

10

20

30

40

50

60

(c) f = 23.6: a three-piece Henon-like at-

tractor coexists with an invariant closedcurve

0 20 40 60 800

10

20

30

40

50

60

(d) f = 24.19: a three-piece Henon-like

attractor coexists with a new strange at-tractor (formerly a closed invariant curve),see fig. 11 for enlargement showing fractalstructure

Figure 2. Multiple attractors and their basins: the basin of theattracting “loop” is the colored region, while the basin of the otherattractor is the white region; the basin boundaries are given by thestable manifolds of the period 3 saddle orbit (marked with “•”)

At f = 19, a period three attracting orbit (marked with “+”) coexists with asmooth invariant attracting closed curve (fig. 2(a)). The boundaries of the basinsof attraction are given by the stable manifolds of the period 3 saddle orbit (theperiod 3 saddle cycle is marked with “•”).


At f = 23.1, a 6-piece Henon-like strange attractor coexists with an attractinginvariant curve (fig. 2(b)). At f = 23.6, a 3 piece Henon–like attractor coexistswith an invariant closed curve (fig. 2(c)). This invariant closed curve bifurcates intoa strange attractor, and the boundaries of the basins of attraction are still givenby the stable manifolds of the period 3 saddle orbit. At f = 24.19, a three-pieceHenon-like attractor coexists with the new strange attractor (fig. 2(d)). See figure11 for an enlargement showing the fractal structure.

The next subsections contain a more thorough discussion of these bifurcationscenarios.

5.1. Evolution of the period 3 attracting orbit. The period 3 attracting orbitundergoes a period doubling cascade at f = 19.9, 22.2, 22.7, . . . , which producesattracting periodic orbits with periods 3 · 2n, 1 ≤ n < ∞. Immediately after that,strange attractors exist. For 23.07 < f < 23.55, the strange attractor consistsof a 6-piece Henon-like attractor (see figure 3(a)). The three pairs of branches ofthe attractor undergo a merging crisis at f = 23.55, which results in a 3–pieceHenon-like attractor (see figures 3(b) and 4). The merging is mediated by a period3 saddle orbit. The 3–piece Henon-like attractor coincides with the closure of theunstable manifold of the period three saddle point.

0 20 40 60 80

0

10

20

30

40

50

60

(a) 6-piece Henon-like attractor (f = 23.07)

0 20 40 60 80

0

10

20

30

40

50

60

(b) Pairwise merging at f = 23.55

Figure 3. Henon-like attractors and merging

0.2 0.3 0.4

59

60

61

62

0.167 0.168 0.16962.291

62.292

62.293

Figure 4. Zoom-in for a Henon-like component at f = 23.55

At f = 31, an explosion of the size of the strange attractor occurs, due to aninterior crisis mediated by another period 3 saddle orbit (see fig. 5(d)). Now, theattractor consists of only one piece. See also [12] for the case p1 = 1.


0 20 40 60 80 1000

10

20

30

40

50

60

70

(a) f = 27

0 20 40 60 80 1000

10

20

30

40

50

60

70

(b) f = 29

0 20 40 60 80 1000

20

40

60

(c) f = 30

0 20 40 60 80 1000

20

40

60

80

(d) f = 31

Figure 5. Strange attractors for f = 27, 29, 30, 31. At f = 31, aninterior merging crisis occurs, mediated by the period three saddleorbit

In the next subsection, we study the evolution of the invariant curve obtainedafter the Hopf bifurcation as the parameter f increases.

5.2. Bifurcation cascade of invariant closed curve to strange attractor.The attracting curve undergoes a complicated transition to chaos (see also [2]for a panorama of possible behaviors for a two parameter family of planar dif-feomorphisms). This transition involves phase locking, homoclinic and heteroclinicintersections, invariant attracting closed curves bifurcating into strange attractors,transient chaos, loss of smoothness, saddle-node bifurcations, appearance of strangeattractors.

At f = 23.7, a period 14 phase locking occurs on the invariant closed curveand persists until f = 23.93. Figure 6 shows the invariant curve in resonanceat f = 23.85. By this, we mean the period 14 attracting orbit connected by theunstable manifolds of the period 14 saddle points (homoclinic connections). Thisclosed curve is not an attractor; most nearby points are attracted to the sinks, andits smoothness is determined by the way the unstable manifolds of two successivesaddle points join at the periodic sink [2].


10 20 30 40

5

10

15

20

25

30

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

Figure 6. Period 14 phase locked invariant curve at f = 23.85

The invariant closed curve in resonance is destroyed at f = 23.9367 due to ahomoclinic tangency: the stable and unstable manifolds of two adjacent period 14saddle points are tangent, see figure 7(b). After the homoclinic tangency, we havetransversal homoclinic intersections (fig. 7(c)), which produce chaotic horseshoes.Although this phenomenon creates chaotic dynamics, the “visible” attractor is stilla period 14 orbit.

42.5 43.5 44.5 45.5

16

16.5

17

17.5

18

+

(a) f = 23.93: before ho-

moclinic tangency

42.5 43.5 44.5 45.5

16

16.5

17

17.5

18

+

(b) f = 23.9367: homo-

clinic tangency

42.5 43.5 44.5 45.5

16

16.5

17

17.5

18

(c) f = 23.95: transversal

homoclinic intersections

Figure 7. Homoclinic intersections

Next, we describe the mechanism that leads to the appearance of the strangeattractor. At f = 23.9866, another saddle-node bifurcation creates a period 31attracting orbit (and its associated period 31 saddle orbit), which coexists untilf = 23.9889, with the period 14 attracting orbit. Figure 8 shows a part of thebifurcation diagram for this parameter interval, with two of the period 14 saddle-node pairs and three saddle-node pairs of period 31.

Remark. The rotation number of the period 14 orbit is 5/14, while the rotationnumber of the period 31 orbit is 11/31. Notice that 5/14 and 11/31 are consecutivenumbers in the Farey sequence (see [2] for details), since 5 · 31 − 14 · 11 = 1. Wethank Henk Bruin for this observation.

At f = 23.9889, the period 14 attracting orbit is annihilated after colliding withits associated period 14 saddle orbit, and at f = 23.989, the period 31 orbit initiatesa periodic doubling cascade. This will lead to a 31−piece Henon-like transitivestrange attractor (see figure 9 for f = 23.992).


23.986 23.988 23.99 23.992 23.994f

42

44

46

48

50

52

N

Figure 8. Part of the bifurcation diagram for 23.984 < f < 23.994

10 20 30 40

5

10

15

20

25

30

(a) 31−piece strange attractor

46 46.2 46.4 46.6

14.5

15

15.5

16

16.5

17

(b) Blow-up of (a)

46 46.1

16.6

16.7

(c) Blow-up of (b)

46.09 46.091 46.09216.65

16.66

16.67

(d) Blow-up of (c)

Figure 9. 31-piece strange attractor and consecutive enlarge-ments (f = 23.992)

For f = 23.99237, an interior crisis occurs (the 31-piece strange attractor collideswith the period 31 saddle orbit created at f = 23.9866). See figure 10. As aconsequence, the 31-piece strange attractor suddenly increases its size (fig. 11),since it will now contain all branches of the unstable manifolds of the period 31


45.8 46 46.2 46.4 46.6

15

15.5

16

16.5

17

46 46.02 46.04 46.06 46.08 46.1

16.6

16.65

16.7

16.75

16.8

Figure 10. Interior crisis at f = 23.99237: the 31-piece strangeattractor collides with a period 31 saddle orbit

10 20 30 40

5

10

15

20

25

30

45.3 45.8 46.3 46.8

15.3

15.8

16.3

16.8

45.45 45.5 45.55 45.6

16.15

16.2

16.25

16.3

45.46 45.4716.22

16.23

16.24

16.25

Figure 11. Strange attractor at f = 23.993, and a sequence ofenlargements showing the fine structure at the tip of the attractor

saddle. Unfortunately we are so far unable to find a homoclinic tangency whichproduces this interior crisis.

Although the former curve in resonance “appears reconstructed”, the collisionresults in a strange attractor (figure 11). Thus the bifurcation route from theinvariant attracting closed curve to strange attractor is subtle, involving the creationand annihilation of period 14 and 31 orbits along the way. It does not occurinstantaneously. This strange attractor will be destroyed by a boundary crisis atf = 24.195 (fig. 12). There is a collision between this attractor and the boundaryof its basin of attraction given by the stable manifold of the period 3 saddle orbitobtained at f = 14.9. At this boundary crisis, the attractor coincides with the


closure of a branch of the unstable manifold of the period 3 saddle orbit. Therefore,the boundary crisis is mediated by a homoclinic tangency (fig. 13(b)).

0 20 40 60 800

10

20

30

40

50

60

Figure 12. Boundary crisis at f = 24.195

2.4 2.5 2.6 2.7

33.1

33.3

33.5

33.7

(a) Before boundary crisis (f = 24.19)

2.4 2.5 2.6

33.3

33.4

33.5

33.6

(b) Boundary crisis at f = 24.195

Figure 13. Homoclinic tangency and boundary crisis

After the homoclinic tangency, the strange attractor disappears, but we havetransient chaos (existence of chaotic horseshoes due to transversal homoclinic in-tersections). Figure 14(b) was obtain by plotting the trajectory of an orbit with aninitial condition taken from the basin of the old attractor. We noticed that the orbitspends a considerable amount of time around the old attractor (for 1 ≤ t ≤ 2200)before moving toward the new attractor. This phenomenon has direct implicationsfor a population model, since, for a significant time period, the behavior is dictatedby transient dynamics.

We can not find any new phenomena appearing in the bifurcation diagram forf ≥ 31. The phase portrait contains a Henon-like strange attractor with occasionalperiodic windows.

We finish this section with two plots representing the maximal Lyapunov expo-nent for 15 ≤ f ≤ 27 (fig. 15) of forward orbits starting from two different initialconditions corresponding to the two coexisting attractors discussed above.


2.4 2.5 2.6

33.3

33.4

33.5

33.6

(a)

0 20 40 60 800

10

20

30

40

50

60

(b)

Figure 14. Transverse homoclinic intersections (a) and transientchaos (b) at f = 24.2

16 18 20 22 24 26f

-1

-0.8

-0.6

-0.4

-0.2

0.2

Max Lyap

16 18 20 22 24 26f

-0.3

-0.2

-0.1

0.1

0.2

Max Lyap

Figure 15. Plots of maximal Lyapunov exponent of orbits withdifferent initial conditions

6. Three-generation model

In this section, we study the three-generation model with equal fertility ratesf = f1 = f2 = f3 and fixed survival probabilities p1 = 0.8, p2 = 0.6. The time-advance mapping is given by

T (x, y, z) = (f · (x+ y + z)e−0.1(x+y+z), 0.8x, 0.6y) .

We observe many of the same global bifurcations and complicated dynamics as wedid for the two-generation model. The main differences involve an interesting newbifurcation scenario of attracting closed curve into strange attractor (see subsection(6.1)) and the ”filling-in” phenomenon resulting in a sudden increase of the fractaldimension for the strange attractor.

The bifurcation diagrams for 20 ≤ f ≤ 45 and 45 ≤ f ≤ 75 are shown on figure16. In what follows, we list the important dynamical properties which appear inthe parameter range 0 < f ≤ 75.

(i) The trivial fixed point (0, 0, 0) is globally asymptotically stable for 0 < f <1/(1 + p1 + p1p2) = 0.438. For f > 0.438 there exists a non-zero fixed pointwhich is asymptotically stable for small enough f .


25 30 35 40 45f

20

40

60

80

100

120

140

N

(a) 20 ≤ f ≤ 45

50 55 60 65 70 75f0

50

100

150

200

N

(b) 45 ≤ f ≤ 75

Figure 16. Bifurcation diagrams for p1 = 0.8, p2 = 0.6

(ii) A saddle-node bifurcation occurs at f = 26.7, creating a period 4 attractingorbit which coexists with the stable fixed point. The period 4 orbit under-goes a period doubling cascade (f = 31.6, 34.1, 34.7, 34.75, . . . ). Strangeattractors appear after the period doubling limit.

(iii) Multiple attractors coexist for 26.7 ≤ f ≤ 38.3. Figure 17(a) shows a four-piece Henon-like attractor and a period 2 attracting cycle (marked by “+”).The period 4 saddle orbit is marked by “•”.

(iv) The fixed point undergoes a period doubling bifurcation at f = 34.

(v) At f = 38.3, the strange attractor is destroyed due to a collision with theperiod 4 saddle. This is a boundary crisis and results in a large discontinuousdecrease in population (figure 17(b)). After the boundary crisis, the onlyattractor left is the period 2 orbit.

0

50

100

025

5075

100

0

20

40

60

++++

0

50

100

025

5075

100

(a) A four-piece Henon-like attractor

and a period 2 cycle coexist at f = 37.5

0

50

100

025

5075

100

0

20

40

60

++++

0

50

100

025

5075

100

(b) Boundary crisis at f = 38.3: the

four-piece Henon-like attractor is de-stroyed

Figure 17. Coexistence of two attractors (a) and boundary crisis (b)

(vi) The period 2 sink bifurcates into a period 4 sink (period doubling bifurcationat f = 48.5).

(vii) The period 4 sink undergoes a Hopf bifurcation at f = 55.5; four attractingsmooth closed curves are born (see fig. 18 for f = 58).


10

20

30

40

1020

30

5

10

15

20

10

20

30

40

5

10

15

20

Figure 18. f = 58: the transitive attractor consists of foursmooth closed curves visited in a cyclic manner

(viii) A period 5 orbit is born at f = 59.4, after a saddle-node bifurcation.

(ix) Multiple transitive attractors coexist for 59.4 ≤ f ≤ 63.1 (see fig. 19 and 20).

0 50 100 150 200

050100

150

0

25

50

75

100 +

++

+

+

+

0 50 100 150 200

050100

150

(a) A period 5 orbit coexists with

two closed curves

1020

3040

10 20 30

5

10

15

20

5

10

15

20

(b) Zoom-in of the two closedcurves

Figure 19. Coexisting attractors at f = 60

050

100150

200

050100

150

0

25

50

75

100

050

100150

200

050100

150

(a) A 5-piece Henon-like strange

attractor coexists with twoclosed curves

210220

230240

0

5

1015

0

0.5

1

1.5

2

210220

230240

0

5

1015

(b) Zoom-in of one Henon-likecomponent

Figure 20. Coexisting attractors at f = 63

(x) The period 5 sink undergoes a period doubling cascade, which will lead toHenon-like strange attractors.


(xi) There is a merging crisis of two pairs of attracting closed curves at f = 59.

(xii) There is a boundary crisis at f = 63.1. The Henon-like strange attractor isdestroyed.

(xiii) At f = 66 there are two attracting T 2-invariant closed curves. As f increasesthese two closed curves bifurcate into a pair of strange attractors. As ffurther increases, the two strange attractors fill-in and merge. Figure 21shows the corresponding sequence of bifurcations. In the following subsectionwe explain the bifurcation route leading from two smooth invariant curves toa strange attractor.

1020304050

10 20 30 40

5

10

15

20

5

10

15

20

(a) f = 66

1020304050

10 20 30 40

5

10

15

20

5

10

15

20

(b) f = 66.7

2040

10 20 30 40

5

10

15

20

25

5

10

15

20

25

(c) f = 67

2040

10 20 30 40

5

10

15

20

25

5

10

15

20

25

(d) f = 67.084

2040

10 20 30 40

10

20

10

20

(e) f = 68.2

2040

10 20 30 40

10

20

10

20

(f) f = 74

Figure 21. Bifurcation of closed curves to strange attractor


6.1. Bifurcation cascade of attracting closed curves to strange attractor.At f = 66.5 a period 52 saddle-node bifurcations occurs. Figure 22(a) shows thecoexistence of two attracting T 2-invariant closed curves and a period 52 attract-ing orbit. At f = 66.935 a new period 48 saddle-node bifurcation occurs. For66.935 ≤ f ≤ 67.11 at least three transitive attractors coexists: two T 2-invariantclosed curves, a period 52 orbit, and a period 48 orbit. At f = 67.11 there is aboundary crisis between the closed curves and the period 48 saddle, which leads tothe destruction of the closed curves (figure 23).

1020

3040

50

1020

3040

5

10

15

20

+

+

+

++

+

+

++

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+ +

+

+

+

++

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

++

5

10

15

20

(a) A period 52 attracting orbit appears

off the invariant curves at f = 66.5, aftera saddle-node bifurcation

2040

1020

3040

5

10

15

20

**

*

*

*

*

*

*

**

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

**

*

*

**

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

5

10

15

20

(b) A period 48 attracting orbit appears off

the invariant curves at f = 66.935, after asaddle-node bifurcation

Figure 22. Periodic orbits coexist with T 2-invariant curves

20

40

1020

3040

5

10

15

20

25

5

10

15

20

25

Figure 23. Boundary crisis at f = 67.11: the two closed curvescollide with a period 48 saddle orbit

Between 67.11 < f < 67.7 the period 48 attracting orbit coexists with the period52 attracting orbit (fig. 24).

The period 48 cycle exhibits a Hopf bifurcation at f = 67.97 (fig. 25(a)). Figure26 shows a transitive attractor consisting of 48 T 48-invariant closed curves. The 48invariant loops are destroyed in a boundary crisis at f = 67.99 (they collide with aperiod 48 saddle orbit). The period 52 stable orbit starts a period doubling cascadeat f = 67.7 (fig. 25(b)). After the period doubling limit, a 2×52−piece Henon-liketransitive strange attractor is born (fig. 27(a)).


20

40

1020

3040

5

10

15

20 +

+

+

+

+

+

+

+

+ +

+

+

+

++

+

+

++

+

+

+

+

+

+

+

+

+

+

+

++

+

+

++

+

+

++

+

+

+

+

+

+

+

+

+

+

+

+

+

*

*

*

*

*

*

**

*

*

*

*

*

*

**

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

**

*

*

* **

*

*

*

*

*

*

*

*

*

*

5

10

15

20

Figure 24. Coexistence of period 48 and 52 attracting orbits afterdestruction of attracting closed curves (f = 67.5)

67.95 68 68.05 68.1f

41.5

42

42.5

N

(a) Bifurcation diagram for T (48) shows a

Hopf bifurcation at f = 67.97: the dottedline represents a period 48 saddle

67.7 67.8 67.9 68 68.1f

42.5

43

43.5

N

(b) Bifurcation diagram for T (52): a period

doubling cascade starts at f = 67.7

Figure 25. Evolution of period 48 and period 52 attracting orbits

2040

1020

3040

10

20

10

20

7

8

3031

32

3

3.5

3

3.5

Figure 26. Transitive attractor consisting of 48 closed curves atf = 67.98 (left), and enlargement of one closed curve (right)

We are currently investigating the bifurcation route from the Henon-like attrac-tors to the attractor shown in figure 27(b). In the two dimensional model we show


that this explosion in the size of the strange attractor is due to an interior crisis.This case is more subtle, because the fractal dimension of the strange attractorseems to jump (and visually becomes “thicker”), which does not occur in the two-dimensioal model. This phenomenon might be a manifestation of the noninvertibil-ity of the mapping T ; it might result from the interaction of the degenerate locuswith the basin boundary (see [20] for possible scenarios for two dimensional nonin-vertible maps). Figure 28 shows the existence of “fat” strange attractors obtainedfor larger values of the parameter f .

2040

1020

3040

10

20

10

20

(a) 2 × 52−piece Henon-like transitive at-

tractor at f = 68.06

2040

1020

3040

10

20

10

20

(b) Strange attractor at f = 68.08

Figure 27. Strange attractors

020

4060

020

40

0

10

20

30

0

10

20

30

020

4060

020

4060

0

10

20

30

0

10

20

30

Figure 28. Strange “fat” attractors at f = 77 (left) and f = 100 (right)

We observe some key differences between the three-dimensional model and thetwo-dimensional model:

• In the two dimensional system the Henon-like attractor obtained after theperiod doubling cascade is the one that will persist for large values of theparameter f , and the “invariant loop” strange attractor is destroyed in aboundary crisis. In the three dimensional case, the Henon-like attractorsare destroyed in a boundary crisis, and for large values of parameter f , the“invariant loop” attractor persists.• The invariant curve in the two dimensional model is born after a Hopf bi-

furcation of the positive equilibrium point, while in the three dimensionalsituation, the positive fixed point first undergoes two period doubling bifur-cations and then the stable 4 cycle undergoes a Hopf bifurcation creatingfour attracting closed curves.


7. Appendix

7.1. Stable and unstable manifolds. Let us recall that if F : U ⊂ Rd → Rd is aC1 mapping and if p is a hyperbolic fixed point with linear splitting TpU = Es⊕Eu,then there exists a local stable manifold W s

loc(p) and a local unstable manifoldW u

loc(p) tangent at p to Es and Eu, and satisfying the following properties:

(1) W sloc(p) and W u

loc(p) are F−invariant, i.e.,

F (W sloc(p)) ⊂W s

loc(p) and F (W uloc(p)) ⊃W u

loc(p);

(2) There are constants C > 0 and 0 < λ < 1 such that for any n ∈ N onehas that d(F nx, Fny) ≤ Cλnd(x, y) for x, y ∈ W s

loc(p) and d(F nx, Fny) ≥Cλ−nd(x, y) if F kx, F ky ∈ W u

loc(p) for k = 1, · · · , n;(3) There exists δ > 0 such that

W sloc(p) = {x ∈ U : d(F nx, p) ≤ δ for all n ∈ N}

and

W uloc(p) = {x ∈ U : there exists xn ∈ F−n(x) such that

d(F kxn, p) ≤ δ for all n ∈ N and k = 1, · · · , n}The global stable and unstable manifolds (see e.g., [20], [25]) are defined by

W s(p) =⋃

n∈NF−n(W s

loc(p)) = {x ∈ U : d(F n(x), p)→ 0}

W u(p) =⋃

n∈NFn(W u

loc(p)) = {x ∈ U : ∃xn ∈ F−n(x) such that d(xn, p)→ 0} .

7.2. Lyapunov Exponents. The dominant Lyapunov exponent associated to anorbit {Tn(x)} is defined by

λ(x) = lim supn→∞

1

nlog ||DTn(x)|| .

A positive Lyapunov exponent indicates an infinitesimal chaotic orbit.

7.3. Strange attractors. There exist several definitions, in the literature, forstrange (chaotic) attractors. In this paper, by strange attractor we mean a compactinvariant set Λ having a dense orbit with positive Lyapunov exponent (sensitive de-pendence on initial conditions), and whose basin of attraction has a non-emptyinterior (or even more restrictive, Λ is included in the interior of its basin).

Following [25], a strange attractor Λ is Henon-like if besides the properties men-tioned above, Λ coincides with the closure of the unstable manifold of a hyperbolic(saddle) periodic point, and Λ is not a hyperbolic set.

7.4. Crises. In a series of highly cited papers, Grebogi, Ott, and Yorke have under-taken a systematic study of crises. Following [11], a crisis is defined to be a collisionbetween a (strange) attractor and a coexisting unstable fixed point or periodic or-bit. Crises appear to be the cause of most sudden changes in chaotic dynamics.There are three types of crises:

• boundary crisis: leads to sudden destruction of the chaotic attractor andits basin. The unstable orbit which collides with the strange attractor ison the boundary of the basin.


• interior crisis: leads to sudden increases in the size of the attractor. Theunstable orbit which collides with the strange attractor is situated withinthe basin of attraction.• merging crisis: two or more chaotic attractors merge to form one chaotic

attractor.

In almost all known two dimensional examples, an interior or boundary crisis occursat a parameter value where there is a homoclinic (heteroclinic) tangency of thestable and unstable manifolds.

References

[1] K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos: An introduction to Dynamical Systems,Springer-Verlag, 1997

[2] D. G. Aronson, M. A. Chory, G. R. Hall, and R. P. McGehee, Bifurcations from an invariantcircle for two-parameters families of maps of the plane: a computer assisted study, Comm.Math. Phys., 83 (1982), 303–354.

[3] L. Barreira, Ya. Pesin, Lyapunov exponents and smooth ergodic theory, Amer. Math. Soc.,2001.

[4] M. Benedicks and L. Carleson, The dynamics of the Henon map, Annals of Math., 133 (1991),73–169.

[5] H. Caswell, Matrix Population Models, second ed., Sinauer, (2002).[6] R. J. Conser, K. T. Hill, P. R. Crone, N. C. H. Lo, and D. Bergen, Stock assessment of Pacific

sardine with management recommendations for 2002, Pacific Fishery Management Council,Portland, Oregon.

[7] J. H. Curry and J. A. Yorke, A transition from Hopf bifurcation to chaos: computer experi-ments with maps on R2 (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977), 48–66,Lecture Notes in Math., 668, Springer, 1978.

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ecology: Experimental Nonlinear Dynamics, Academic Press, 2002[10] C. E. Frouzakis, I. G. Kevrekidis and B. B. Peckham, A route to computational chaos re-

visited: noninvertibility and the breakup of an invariant circle, Phys. D 177 (2003), no. 1-4,101–121.

[11] C. Grebogi, E. Ott, and J. Yorke, Crisis, sudden changes in chaotic attractors and transientchaos, Physica D (1983), 181–200.

[12] J. Guckenheimer, G. Osher, and A. Ipaktchi, The dynamics of density dependent models, J.Math. Biology 4 (1977), 101–147.

[13] J. K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Mono-graphs 25, AMS, 1985.

[14] M. Henon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys., 50(1976), 69–77.

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[17] P. Leslie, On the use of matrices in population mathematics, Biometrika 33 (1945), 183–212.[18] P. Leslie, Some further notes on the use of matrices in population mathematics, Biometrika

35 (1948), 213–245.[19] S. A. Levin, C. P. Goodyear, Analysis of an age-structured fishery model, J. Math. Biol. 9

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invertible maps, World Scientific, 1996.

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[22] S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology 13 (1974), 9–18.[23] E. Ott, Chaos in Dynamical Systems, Cambridge, 1993.


[24] E. Ott, T. Sauer, and J. Yorke (editors), Coping with chaos, Wiley Series on NonlinearScience, 1994.

[25] J. Palis and F. Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurca-tions, Cambridge, 1993.

[26] T. S. Parker and L. O. Chua, Practical numerical algorithms for chaotic systems, Springer-Verlag, 1989.

[27] M. Pollicott, Lectures on ergodic theory and Pesin theory on compact manifolds, CambridgeUniversity Press, 1993.

[28] Y. Pykh and S. Efremova, Equilibrium, stability and chaotic behavior in Leslie matrix mod-els with different density-dependent birth and survival rates, Math. Comput. Simulation 52(2000), 87–112.

[29] C. Robinson, Bifurcation to infinitely many sinks, Comm. Math. Phys. 90 (1983), no. 3,433–459.

[30] M. Sandri, Numerical calculations of Lyapunov exponents, The MATHEMATICA Journal(1996), 78–84.

[31] C. Simo, On the Henon-Pomeau attractor, J. Stat. Physics 21 (1979), 465–494.[32] S. Tuljapurkar and H. Caswell, Structured population models in marine, terrestrial, and

freshwater systems, Chapman and Hall, 1997.[33] Q. Wang and L.-S. Young, From invariant curves to strange attractors, Commun. Math.

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Department of Mathematics, The Pennsylvania State University, University Park,PA 16802

E-mail address: [email protected] and [email protected]

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