Belyakov Homoclinic Bifurcations in a Tritrophic Food Chain …math.bd.psu.edu/faculty/jprevite/REU07/yu.pdf · dT Dl] , (2.lc) where T is time, R and K are prey intrinsic growth

Belyakov Homoclinic Bifurcations in a Tritrophic Food Chain Model

Yu A Kuznetsov O De Feo S Rinaldi

SIAM Journal on Applied Mathematics Vol 62 No 2 (Oct - Dec 2001) pp 462-487

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SlAhl J APPL MATH 2001 Society for Industrial and Applied Mathematics ol 62 No 2 pp 462-487

BELYAKOV HOMOCLINIC BIFURCATIONS IN A TRITROPHIC FOOD CHAIN MODEL

Abstract Complex dynamics of the most frequently used tritrophic food chain model are investigated in this paper First it is shown that the model admits a sequence of pairs of Belyakov bifurcations (codimension-two homoclinic orbits to a critical node) Then fold and period-doubling cycle bifurcation curves associated to each pair of Belyakov points are computed and analyzed The overall bifurcation scenario explains why stable limit cycles and strange attractors with different geometries can coexist The analysis is conducted by combining numerical continuation techniques with theoretical arguments

Key words homoclinic bifurcations population dynamics continuation

AMS subject classifications 34C37 58F13 92D25

PII SO036139900378542

1 Introduction For several decades after the pioneering work of Lotka [34] and Volterra 1421 one of the topics of major concern in mathematical ecology has been the study of ditrophic food chains This has been accomplished by analyzing a great number of second-order continuous-time dynamical models usually called prey-predator models (see for example [2]) Existence of limit cycles multiplicity of attractors and catastrophic bifurcations are the characteristics of those models which have been used to explain complex behaviors observed in the field It was only in the late seventies that some interest in the mathematics of tritrophic food chain models (composed of prey predator and top-predator) emerged With almost no exception the first contributions dealt with the problem of persistence [18 19 171 and therefore did not provide information on the number and the geometry of the attractors This is a very unfortunate situation because the nature of the attractors is often the most interesting feature of a dynamical system An exception in this respect was an almost unnoticed paper [25] in which it was shown through simulation that a particular food chain model can behave chaotically This property was in practice brought to the attention of the scientific community by a contribution 1241 that appeared much later and showed that food chains behave chaotically on a tea-cup strange attractor provided that the three populations have diversified time responses increasing from bottom to top This condition on the time responses was used in the same years 138 391 to perform a singular perturbation analysis that indeed confirms that the tea- cup geometry is the result of the interactions between high frequency (prey-predator)

Received by the editors September 22 2000 accepted for publication (in revised form) April 16 2001 published electronically October 31 2001 This work was supported by Consiglio Nazionale delle Ricerche Italy Project ST174 hIathematica1 models and methods for the study of biological phenomena

httpwwwsiamorgjournalssiap62-237854html ThIathematical Institute Utrecht University Budapestlaan 6 PO Box 80010 3508 TA Utrecht

The Netherlands and Institute of hlathematical Problems of Biology Russian Academy of Sciences Pushchino hloscow Region 142290 Russia (kuznetQmathuunl)

f ~ a b o r a t o r ~of Nonlinear Systems (DSC-LANOS) Swiss Federal Institute of Technology Lau- sanne (EPFL) 1015 Lausanne Switzerland (oscardefeoQepflch) The research of this author was supported by the Swiss National Science Foundation 2000-05603098

S~ipar t imento di Elettronica e Informazione Politecnico di hlilano Via Ponzio 3415 20133 hlilano Italy (rinaldiQeletpo1imiit)

463 HOhIOCLINIC BIFURCATIONS IN A FOOD CHAIN

oscillations and low frequency (predator-top-predator) oscillations Since then particular effort has been devoted to the study of the complex dynamics of food chain systems and bifurcation analysis has been the major tool of investigation

The most recent studies 128 37 33 11 51 dealing with the so-called Rosenzweig- LIacArthur model show that its bifurcation structure is quite rich In particular it comprises a complex cascade of tangent bifurcations of cycles intersecting with flip bifurcation curves thus delimiting a region of very complex behavior sometimes called the chaotzc regzon [33] Although these analyses were restricted to local bifurcations they clearly indicate the presence of global bifurcations Indeed homoclznzc orbzts ie orbits tending toward the same saddle equilibrium or saddle cycle forward and backward in time have been numerically detected in [37 33 51 and even proved to exist through singular perturbation analysis in the case of trophic levels with time responses increasing from bottom to top [12] Similar analyses have been performed on more complex food chain models 13 23 30 401 and the results are qualitatively the same Homoclinic orbits exist and very complex behavior is possible

Despite the efforts devoted to the analysis of the Rosenzweig-hlacArthur food chain model a systematic study of its chaotic region has not yet been attempted The aim of this paper is to accomplish such study by combining recent numerical techniques for continuing homoclinic bifurcations [6 71 with the analysis of a special codimension-two homoclinic bifurcation first studied in [3] and here referred to as Belyakov bzfurcatzon In particular it will be shown that a family of homoclinic bifurcation curves exists in a two-parameter space and that two Belyakov points are located on each of these curves Since the original analysis in [3] was insufficient for our purposes we have revisited Belyakovs proofs and have shown that three families of subsidiary bifurcation curves (namely tangent flzp and double hornocl~n~c) are rooted at each Belyakov point These points are therefore the organmng centers of the overall bifurcation scenario Another organizing feature of the two-parameter bifurcation diagram is the sharp turn of the primary homoclinic curves

The paper is structured as follows In the next section some background information on the Rosenzweig-LlacArthur model is given while in section 3 the simplest local bifurcations relative to equilibria and cycles are discussed Then in section 4 the bifurcation structure of the chaotic region is discussed in detail The basic properties of the Belyakov homoclinic points are presented in the appendix where the asymptotic expressions for the subsidiary bifurcation curves are derived

2 The model and its equilibria The model we analyze in this paper de- scribes a tritrophic food chain composed of a logistic prey (X) a Holling type I1 predator (Y) and a Holling type I1 top-predator (2)It is therefore given by the following system of ordinary differential equations (see [23] for more details)

dT D l ]

(2lc)

where T is time R and K are prey intrinsic growth rate and carrying capacity the Ats are maximum predation rates the Bs are half saturation constants the Dls are death rates and the Es are efficiencies of predator (z = 1) and top-predator

464 YU A KUZNETSOV 0 D E FEO AND S RINALDI

( i = 2) In order to preserve the biological meaning of the model the parameters are assumed to be strictly positive Furthermore to avoid the case where predator and top-predator cannot survive even when their food is infinitely abundant we assume that EA gt D z = 12

By rescaling the variables

one obtains

where

Then the above conditions for predator and top-predator persistence become a gt bd 2 = l 2

The reference parameter values used in this paper are those used in [24] namely

while the two remaining parameters K and r are varied to perform the bifurcation analvsis The reader interested in the biological interpretation of these parameter values can refer to [36]

All coordinate axes and faces of the positive orthant are invariant sets of system (22) There are three trivial equilibria - the origin (OO 0) which is always a saddle - the point ( K 0O) corresponding to prey at carrying capacity and in the absence

of predator and top-predator - the point

which is positive for al gt dl (bl + h) and corresponds to prey-predator coexistence in the absence of top-predator

The point x(O) can be either stable or unstable in the face ( X I 52) When it is unstable it is surrounded by a stable limit cycle [35] which is unique and globally attracting in the plane x3 = 0 (see 191) The transition between the two situations corresponds to a supercritical Hopf bifurcation of the submodel (22a)-(22b) with x3 = 0 and occurs for

465 HOLIOCLINIC BIFURCATIONS IN A FOOD CHAIN

LIoreover a second degeneracy of the point x(O) occurs when the term in the square brackets of ( 2 2 ~ ) vanishes namely when

(25) r (a2 - b2d2) ( K (a1 - bldl) - dl ) = dsK (a1 - bldl)2

It is a transcritical bifurcation giving rise to a strictly positive equilibrium for small perturbations of the parameters

As for nontrivial equilibria it is possible to show that at most two of them can be positive namely

where

Depending upon the parameter values there are three possible cases none of these equilibria is strictly positive only x() is strictly positive or both x() and x ( ~ ) are strictly positive When x ( ~ )is positive it is always a repeller while x() can be either an attractor or a saddle

3 Bifurcations of equilibria and local bifurcations of limit cycles

31 Codimension-two point M If all parameters except K and r are fixed the planar Hopf bifurcation (24) and the transcritical bifurcation of equilibria (25) occur along two curves in the ( K r)-plane labeled by H and TC in Figure 1 These curves intersect at a codimension-two point 31 with coordinates

and the coordinates (see (23)) of the corresponding equilibrium point zg(with one zero eigenvalue and two purely imaginary eigenvalues) are

The analysis of the bifurcations in the vicinity of x(Oj for parameter values close to (KAzl rAzl) can be performed using the normal form technique [I] In particular a parameter-dependent normal form of the system near this point has been derived and used to show [33] that five bifurcation curves emerge from this point None of these curves implies chaos so the codimension-two point A1 can not be considered as the origin of chaos in food chains as first argued in 1281

466 YU A KUZNETSOV 0 DE FEO AND S RINALDI

FIG 1 S o m e local bzfurcation curves of s y s t e m ( 2 2 ) i n t he ( K r ) - p l a n e H-Hopf bzfur-cat ion i n t he plane x3 = 0 TC -transcntzcal bzfurcatzon of equilibrium s ( O ) T C -transcritzcal bzfurcatzon of cycle H and T H o p f and tangen t bzfurcatzons of positive equilzbrza T t a n g e n t

bzfurcatzon of l im i t cycles ~ ( l )and ~ j ~ ) - f i z p bifurcations of l im i t cycles Cod imens ion - two bifurcat ion points Al-zero-Hopf bzfurcation i n t he plane x3 = 0 DH-degenerate Hopf bifurcation C-cusp bifurcation of l im i t cycles D-degenerate transcritical bifurcation of l im i t cycles

32 Bifurcation curves rooted at point M The bifurcation curves emerg- ing from point A l have been continued numerically using the software pacakge LoCBIF

[27] see Figure 1 The curve H is a vertical straight line because r is not present in (24) the curve T is a tangent bifurcation curve for equilibria where x( l ) and x ( ~ ) collide and disappear (annihilation of the radical in (28b)) TC is a transcritical bifurcation curve of equilibria (see (25)) where a strictly positive equilibrium emerges from point x() TC is a transcritical bifurcation curve of cycles where a strictly positive limit cycle emerges from the limit cycle in the plane (xl x2) and finally the curve H = H+ UH is a Hopf bifurcation curve Crossing curve H-the equilibrium x( l ) loses its stability and a stable limit cycle appears around it By contrast crossing curve H+ the equilibrium x() loses its stability while an unstable cycle shrinks on it The first Lyapunov coefficient as~ociat~ed with the Hopf bifurcation H (ie the real part of the cubic coefficient in the normal form [31]) is positive close to 151 and decreases from AI to D H where it vanishes This means that the Hopf bifurcation is subcritical from AI to DH (segment H f ) and supercritical elsewhere (segment H p ) Therefore (see for example [31]) there exists a tangent bifurcation of limit cycles T originating at point DH and corresponding to the collision of two positive limit cycles Numerical continuation shows that curve T has a second codimension-two

467 HOMOCLINIC BIFURCATIONS IN A FOOD CHAIN

singularity namely a cusp C where three limit cycles collide simultaneously The curve T terminates at a point D on the transcritical bifurcation curve TC where a cycle passes through the invariant plane x3 = 0 when approaching point D along T the two colliding cycles hit the invariant face Thus the curve T connects the codimension-two bifurcation points D H and D

In a parameter region delimited by the bifurcation curves connecting the points 121D C and D H there is a saddle cycle This cycle disappears via a transcritical bifurcation (on a segment of TC between lll and D ) or a tangent bifurcation (on T) or a subcritical Hopf bifurcation (on a piece of H+ between M and D H )

33 Cascades of flip bifurcations The bifurcation curves described so far form a bifurcation set connected with point A l However the actual bifurcation diagram is much more complex and involves many other bifurcation curves that are disconnected from the previous ones Figure 1 shows four such curves F)F(~)Fj3)

and F ( ~ ) [Id] These curves computed using the programs LOCBIF [27] and amp ~ ~ 0 9 7 are part of a bifurcation scenario composed of Feigenbaum-like (period-doubling) cascades alternating with chaotic windows The continuation for decreasing values of K of the stable limit cycle existing in the right-upper corner of Figure 1 reveals a flip bifurcation curve F() followed by a Feigenbaum cascade of flips F(~) Fj2)F~ ~ ) ending with a curve ~ 2 )after which the attractor is a strange attractor Notice that only the first flip F(~)of this Feigenbaums cascade is shown in Figure 1 The chaotic

region delimited on the right by ~ 2 )ends on the left with an attractor crisis namely with the sudden disappearance of the strange attractor which is substituted by a period-3 cycle namely by a cycle characterized by three prey-predator oscillations per cycle ie by three minima of the prey x1 per cycle (see Figure 2) Decreasing

K further the period-3 periodic window ends with the flip bifurcation F~)shown in Figures 1 and 2 Such a bifurcation is the first period-doubling of a new Feigenbaum cascade F(~) F ~ ~ ) ending at FE) where a new strange attractor appears Fi3) And the story repeats The second chaotic region is followed by a period-4 periodic window which is then interrupted by the flip curve F(~)which is the first period-

doubling of a Feigenbaum cascade F(~)Fi4)Fj4) ~ 2 ) Figure 2 shows that the attractors (cycles and strange attractors) of the system are obtained from generating cycles through a series of bifurcations and that each generating cycle is characterized by a different number i of prey-predator oscillations namely by a different number i of minima of the prey (x l ) per cycle

It will be shown later that the generating cycles organize the overall bifurcation structure This is why a superscript (i) will characterize all bifurcation curves For example the kth flip bifurcation of the period-i generating cycle is called FLi)There is however a hidden drawback in this notation since the number i can change in the continuation (see below)

Coming back to Figure 1 we can notice that the left side of the chaotic region is quite complex because on that side the flip curves intersect with each other (and with other bifurcation curves not shown in the figure) This problem will be studied in the next section by focusing on the rectangular subregion indicated in Figure 1

4 Homoclinic orbits and associated bifurcations We show in this section that limit cycle bifurcations characterizing the chaotic region are organized by an infinite family of U-shaped bifurcation curves h() z = 12 corresponding to the presence of orbits homoclinic to the saddle (or saddle-focus) x() For simplicity the

YU A KUZNETSOV 0 DE FEO AND S RINALDI

FIG 2 One-parameter bifurcation scenario with respect to K for the cycle existing i n the right-upper corner of Figure 1 (T = 12)

first one of these bifurcation curves is called primary and all the others secondary We can anticipate that each homoclinic bifurcation corresponds to homoclinic orbits that differ in the number of minima of the prey These bifurcation curves are computed using the numerical toolbox for homoclinic bifurcation analysis HOMCONT 16 71 incorporated into AUTO^^ 1141 It turns out that when the equilibrium x() is a saddle-focus its complex-conjugate eigenvalues have positive real part and are closer to the imaginary axis than the real eigenvalue so that Shilnikovs theorem 1311 implies the existence of an infinite number of saddle limit cycles for parameter values near the homoclinic bifurcation curves As shown in [21 15 22 201 under the same conditions a t least three countable families of subsidiary bifurcations (flip tangent and homoclinic) accumulate on each homoclinic curve Moreover two Belyakov points ie two codimension-two homoclinic bifurcation points where the transition from saddle- focus to saddle of the equilibrium occurs lie on each homoclinic bifurcation curve and are the roots of the subsidiary bifurcations Finally the geometry of the subsidiary bifurcation curves is determined by the sharp U-turn of the homoclinic curves h(i)

All these facts imply that the chaotic region has a very complex structure and is actually fractalized in regions where chaotic attractors coexist with cycles with different numbers of prey-predator oscillations per cycle

41 Primary homoclinic and subsidiary bifurcations Through the numerical continuation in (K r ) of the flip curve ~ ( l ) (see Figure I) one can easily discover that the period of the cycle becomes very large on the left branch of the curve when r becomes slightly bigger than 4 This is a clear indication that the cycle


F I G 3 Ske tch of bzfurcatzon curves associated w i t h the first Belyakov pazr BA1) and B)

h( ) -p r zmary homoclznic bifurcatzon t i t i - t angen t bzfurcation of l im i t cycles F ( ) and f1(21)-$~p

bzfurcatzons of lzmzt cycles T h e upper i n d e s (1 indicates t he n u m b e r of prey-predator osczllations per cycle

is very close to a homoclinic orbit Further simulations combined with suitable perturbations of the parameters allow one to detect a homoclinic bifurcation point with an associated homoclinic orbit characterized by a single minimum of the prey Then through the two-parameter continuation an entire homoclinic bifurcation curve h( l ) can be produced Such a curve is U-shaped as is qualitatively sketched in Figure 3

For sufficiently high values of r the right branch of h() corresponds to homoclinic orbits to a saddle with a single minimum of the prey Going down along the right branch we pass the first Belyakov point B$)(K = 12202954903 r =

40263103008) and below that point we have homoclinic orbits to a saddle-focus

Proceeding further after a turning point we encounter the second Belyakov point B) after which we have again homoclinic orbits to a saddle While making the U-turn the geometry of the homoclinic orbit changes significantly because a second minimum of the prey appears the homoclinic orbit then makes two global turns involving two oscillations of the prey-predator subsystem Figure 4 shows how the homoclinic orbits vary along the bifurcation curve h() The homoclinic orbits associated to the right branch of h() have a single prey-predator oscillation while those associated to the left branch have two oscillations

It has been proved in [3] that each Belyakov point is the origin of two infinite families of subsidiary bifurcation curves One is a family of tangent bifurcations of cycles and the other is a family of homoclinic bifurcations associated to homoclinic


FIG 4 Deformation of the homoclinic orbit along the curve h( ) the homoclinic orbits associated to the right (left) branch of h( ) have one global minimum (two deep minima) of zl ( t ) The second minimum is added in passing the U-turn The outermost loop of the local spiral close t o the equilibrium grows and becomes a global turn

orbits (called double) characterized by a number of global turns which is twice that of the primary homoclinic orbit We prove in the appendix that an infinite family of flip bifurcation curves is also rooted there All these curves accumulate exponentially


on the primary homoclinic curve h() and have infinite-order tangency to it at the Belyakov point These accumulation properties are so strong that it is very difficult to numerically produce more than a few of these subsidiary curves In the present case we were able to compute (through continuation) only the first tangent and the first flip bifurcation curve of the corresponding families as sketched in Figure 3 The tangent bifurcation titi starts from point and has two cusps while the flip bifurcation fl(2j starts and returns to the same Belyakov point B) Note that the cycles associated to these bifurcation curves have one and two minima of the prey per cycle and this is why the curves are identified with the superscripts (1 and ( 2 ) respectively In reality the U-turn is very sharp (as noticed in [30] for a similar model) and the two Belyakov points almost coincide in the (Kr)-plane so that we were unable to resolve them However it is possible to distinguish these points by zooming in on the corresponding homoclinic orbits in the vicinity of the saddle equilibrium x() as shown in Figure 5 Moreover the four bifurcation curves F) h() t i t i and fi(2jshown in Figure 3

practically coincide in the vicinity of the Belyakov points while the flip curve F) is well separated from h()

F I G 5 Resolution of the Belyakov points by zooming i n o n the equilibrium z() ( a ) B) (b )

B y )

In conclusion the bifurcation diagram associated to the primary homoclinic curve (1) (1) (1)h() includes h( ) itself the subsidiary bifurcation curves fz30 tz O and h30 z =

1 2 associated with Bill and the subsidiary bifurcation curves f) t y and

hj2j i = 1 2 associated with Bill These results are in agreement with the two- parameter analysis performed in [20] where nevertheless the sharp geometry of the homoclinic curve was not fully understood since homoclinic orbits with two global turns were not even taken into account It should be noted that there are many other bifurcation curves in a neighborhood of the Belyakov points corresponding for example to trzple homoclznzc loops Figure 6 shows the partial bifurcation diagram we were able to obtain At that scale the two Belyakov points appear as a single point and the two branches of the primary homoclinic are not distinguishable

42 Secondary homoclinics and subsidiary bifurcations Numerical con- tinuations show that the bifurcation diagrams associated to the secondary homoclinics


F I G 6 Computed bifurcatzon curves associated with the first Belyakov pazr Labelling as i n Figure 3 The two Belyakov points B) and B) are indistinguishable at this scale

h(2) h(3) have the same structure as the diagram associated to the primary homoclinic h() The homoclinic orbits associated to the homoclinic bifurcation curves h(i) involve i or (i+1)minima of the prey per cycle instead of one or two Figure 7 shows a qualitative sketch of the diagram associated to h() The homoclinic bifurcation curve h(2) is U-shaped and has two Belyakov points B r ) and Bi2) The homoclinic orbits associated to the right branch of h(2) make two global turns while those associated to the left branch make three global turns as clearly detectable in Figure 8 where the homoclinic orbits a t the Belyakov points are shown Notice that these two orbits are more easily distinguishable than in the case of the primary homoclinic h()

The main difference between the bifurcation scenario associated with the primary homoclinic (Figure 3) and the scenario associated with the secondary homoclinics (Figure 7) is that in the latter a tangent bifurcation curve t g l rooted at the left

Belyakov point Bi2)is also present As in the primary case the two Belyakov points are so close as to appear to be a single point as shown in Figure 9 which reports actual results of our computations At the scale of the figure the two branches of h(2) cannot be distinguished and the bifurcation curves h() f(3 t f j and ti2appear as a

single curve in the vicinity of the Belyakov points The flip F(~)tends asymptotically

to tj2i as r increases

The same results can be obtained for a few other secondary homoclinic curves h() Indeed we have been able to perform the computations up to the fifth homoclinic bifurcation h(5) Superimposing the five corresponding diagrams we have obtained

HOMOCLINIC BIFURCATIONS IN A FOOD CHAIN 473

FIG 7 Sketch of bifurcation curves associated with the second Belyakov pair ~ r ) and ~ 1 ~ )

h ( 2 ) -secondary homoclinic bifurcation t(lA and t t l -tangent bifurcations of limit cycles F ~ ) and

fl( -flip bifurcations of limit cycles T h e upper index (i) indicates the number of prey oscillations per cycle

FIG 8 Homoclinic orbits corresponding t o the Belyakov points ( a ) B) (b) B i 2 )

the bifurcation subset shown in Figure 10 In such a diagram the ten Belyakov points appear as a single point and the

five homoclinic curves h(i) i = 1 5 can hardly be distinguished By contrast the subsidiary bifurcation curves tjIo t) f) Fii) can be fairly well identified Nevertheless we like to stress that these curves represent only a very small fraction of


F I G 9 Computed bzfurcatzon curues associated with the second Belyakov pair Labelling is as zn Figure 3

the complete bifurcation set Indeed each curve is only the first member of an infinite family of similar bifurcation curves The subsidiary homoclinic curves are missing since we were unable to produce them numerically hforeover we must also mention that there are other global bifurcations involved such as the recently discovered [5] heteroclinic bifurcations associated with orbits connecting the saddle point x() to a saddle limit cycle

5 Discussion We have shown in the previous sections (see in particular Fig- ure 10) that a family of homoclinic bifurcations organize the structure of the so-called chaotic region This region is fractalized in subregions of chaotic andor periodic behavior and the coexisting attractors (cycles and strange attractors) are characterized by different geometries namely by a different number of prey-predator oscillations The coexistence of different attractors is due to the overlapping of the basic bifurcation structures sketched in Figures 3 and 7 The series of Feigenbaum-like cascades that exists on the right side of the chaotic region is also organized by the same bifurcation structure Indeed the curves t) and F() on the right of Figure 10 form the skeleton of the series of Feigenbaums cascades described in section 33 and in Figure 2 In fact the curve tgt ) is the tangent bifurcation that opens the periodic

window of period-(i + 1)and the curve F() is the first flip of the period-(i + 1) cycle

In order to show how the attractors depend upon K and r we have plotted in Figure 11 the period T of the cycle born on the Hopf bifurcation curve H- of Figure 1 The period T has been computed through continuation with respect to r

475 HOhfOCLINIC BIFURCATIONS IN A FOOD CHAIN

FIG 10 Detailed bifurcation structure of the chaotic region One-parameter bzfurcation diagrams correspondzng to vertical segments ( a ) ( b ) ( c ) and ( d ) are shown in Figure 12

for different values of K The points marked with a triangle are flip points while those marked with a circle are tangent points and the number of prey-predator oscillations present in each cycle is indicated within parentheses Moreover Figure 12 reports for four different values of K the bifurcation scenarios of the minima of x l on the attractors Each scenario is accompanied by the two-parameter bifurcation diagram in the neighborhood of the K value characterizing the scenario

For K lt 087 ie when the bifurcations of Figure 10 are not involved there exists only one stable cycle Its period Tas well as the number of prey-predator oscillations increases with r as indicated in Figure 11 Consistently Figure 12(a) obtained for K = 085 shows that there is only one cycle and that the number of minima of x1 per cycle increases from 1to 5 in the interval 09 5 r 5 16 The values of r a t which the number of minima of xl changes are values for which the periodic function X I ( t )has an inflection point with i1= 0 The locus where these inflections occur is reported in the two-parameter bifurcation diagram with a dotted line

For 087 lt K lt 105 ie from the first overlapping of flip and tangent bifurcation curves to the (primary and secondary) homoclinic bifurcation curves h() h() (see Figure lo ) the period T of the cycle and the number of global turns still increase with r (see Figure 11) but coexistence of different attractors with different number of global turns per cycle is possible The bifurcation scenario of Figure 12(b) obtained for K = 096 clearly points out this possibility

For 105 lt K lt 117 ie from the homoclinic bifurcations h() to the end of the

YU A KUZNETSOV 0DE FEO AND S RINALDI

F I G 11 T h e perzod T of the cycles i n the chaotzc regzon czrcles and triangles represent tangent and P i p bifurcations respectively

flip and tangent overlapping (again see Figure 10) the number of global turns of xl ( t ) per cycle still increases with r while the period T of the cycle increases and decreases alternately (see Figure 11) The scenario in Figure 12(c) shows that the previous well-organized structure is no longer present and that the minima of xl in the strange attractor do not belong to separated segments This means that the geometry of the strange attractor is no longer simple

Finally for K gt 117 ie when there is no flip and tangent overlapping (see Figure 10) a series of Feigenbaum cascades alternating with chaotic windows can be observed (see Figure 12(d)) The fact that there is also a series of reversed Feigenbaum cascades is due to the curvature of the flip and tangent bifurcations

All the results that we have found through continuation are in agreement with simulation experiments which are summarized in Figure 13 This figure is obtained by numerical integration of the system starting near the equilibrium x() In the figure darker gray levels are associated with more complex attractors characterized by higher numbers of prey-predator oscillations The figure clearly shows that the right side of the chaotic region is regularly organized in bands of simple and complex attractors By contrast the left side of the chaotic region is fractalized in subregions with simple and complex behaviors The figure also points out the existence of an island of simple behavior inside the chaotic region This island first discovered in [40] has been recently shown in [5 ] to be related to the homoclinic orbits to the saddle cycle mentioned at the end of section 32


FIG 12 Bifurcation diagrams in subregions of the chaotic region and bifurcation scenarios of q with respect to r for four values of K (a) K = 085 (b) K = 096 (c) K = 1135 (d) K = 119

6 Concluding remarks In this paper we have studied the most common model of tritrophic food chains by focusing on its local and global bifurcations We have discovered that the model has an infinite number of homoclinic bifurcation curves


FIG 13 Experimental two-parameter bifurcation diagram showing the complexity of the attractors Darkness levels correspond to attractors with a high number of prey-predator oscillations

and that on each one of them there are two special points namely codimension-two Belyakov homoclinic bifurcation points We have proved that three infinite families of subsidiary (flip tangent and homoclinic) bifurcation curves emerge from each Belyakov point The numerical computation of these subsidiary bifurcations and the analysis of their intertwining has allowed us to understand the structure of the so- called chaotic region In particular we have discovered that the number of oscillations per cycle of one of the three state variables can be a convenient complexity index for encoding the attractors and that one side of the chaotic region is nicely organized in bands of alternate high and low complexity while the other side is completely fractalized in terms of complexity

From a theoretical point of view our analysis is interesting because it contains new results concerning flip bifurcation curves near Belyakov points (cf [3]) Moreover the basic bifurcation scenario near the U-turn of each homoclinic curve (see Figures 3 and 7) adds some details to the results described in 1201 particularly about homoclinic orbits with several global turns However our study is also interesting from the computational point of view because it shows how powerful the combination of thee retical analysis and continuation techniques can be for understanding the behavior of nonlinear dynamical systems

The results pointed out in this paper can be interpreted biologically by noticing that one of the two parameters of our discussion namely the prey carrying capacity K can be controlled through enrichment or impoverishment of the habitat of the prey population In particular our analysis shows that the dynamic complexity of


the ecosystem first increases and then decreases with enrichment This result is particularly interesting because extensive simulations of the same model have pointed out [ll]that in the case when the top-predator is harvested at constant effort the mean yield first increases and then decreases with enrichment and reaches its maximum roughly on the right border of the chaotic region We can therefore hope that our results can be used for proving this important and intriguing property of exploited renewable resources

Finally we would like to mention that the study we have performed in this paper could be repeated for a variety of different but similar models that have been used to describe cyclical phenomena such as the alternation between despotism and anarchy in ancient China 1161 electrical activity of pancreatic cells 18 411 microbial dynamics in the chemostat [29] autocatalytic enzymatic reactions [13 261 and the use of electronic oscillators as chaos generators for communication and artificial intelligence purposes [lo]

Appendix A Belyakov bifurcation revisited In this appendix we apply the scaling techniques used in [22] to analyze subsidiary bifurcations near the Belyakov point The resulting analysis is simpler and more complete than the one in the original paper [3] The following theorem will be proved

THEOREMA 1 Consider a generic two-parameter family of smooth autonomous ordinary differential equations i n R3 having at some parameter values a n orbit r l homoclinic to a n equilibrium 0 with eigenvalues

T h e n the corresponding point i n the parameter plane is the origin of three countable sets of bifurcation curves namely

(1) t ( 1 )) t a n g e n t bifurcation curves of periodic orbits making one global passage near r l

(2) f ~ l ) ) - - j l i p bifurcation curves of periodic orbits making one global passage near r1

(3) h))--bifurcatzon curves corresponding t o the existence of saddle-focus homoclinic orbits making two global passages near r l

The theorem is illustrated in Figure 14 The curves t) and f i l ) accumulate exponentially fast at both sides on the saddle-focus part of a bifurcation curve h ( ) corresponding to the existence of a homoclinic orbit to the equilibrium making one global passage near T I while the curves hi2) do so at one side only (cf (A12) and (A15) below) The critical orbits corresponding to the curves with bigger integer n E N make more local turns near the equilibrium before the global passage The existence of the flip bifurcation curves was not reported in [3] although probably known to experts There are many other bifurcation curves in a neighborhood of the Belyakov point corresponding for example to triple homoclinic loops Alloreover such homoclinic curves may be not rooted at the Belyakov point [22]

A1 New coordinates and parameters Any generic system satisfying the conditions listed in the theorem can be transformed near the critical parameter values ( p = 0) in a neighborhood of the equilibrium 0 into the form


FIG14 Bifurcat ion curves rooted a t t he Belyakov point t i 1 ) - p n m a r y tangen t bzjurcatzon curves f i l l -pr imary flip bifurcation curves h() and h i 2 ) -pr imary and secondary (doub le ) homoclinzc bijurcatzon curves

where p = are small parameters the smooth functions ~ ( p ) (PI L L ~ ) ~ and X(p) satisfy ~ ( 0 )= lt 0 X(0) = vo gt 0 while f l 2 and g12 are smooth functions of their arguments The transformation into the form (Al) is achieved by smooth coordinate and parameter changes and by a time reparametrization [3] At p = 0 the system (Al) has a critical node 0 at x = y = z = 0with eigenvalues

For p1 lt 0 the eigenvalues of the equilibrium O are real and simple while for L L ~ gt 0 there is a pair of complex-conjugate eigenvalues and a positive eigenvalue

For all sufficiently small 1 1 p 1 the equilibrium O has a one-dimensional unstable manifold W u ( 0 ) composed of two outgoing orbits rl and r2and a two-dimensional stable manifold W s ( 0 ) composed of all incoming orbits (see Figure 15) In the coordinates (x y z) the manifold W u ( 0 ) locally coincides with the z-axis while the manifold W 3 ( 0 ) is locally represented by z = 0 Let Fl depart from the equilibrium O along the positive half of the z-axis

By the theorems conditions at p = 0 the system (A 1) has a homoclinic orbit Fl returns to the equilibrium 0 Generically upon return it does not coincide with the x-axis Also generically Fl misses the stable manifold W s ( 0 ) near O by the p2-shift in the z-direction when 12 0 This means in particular that for all Ipl I small and p2 = 0 the system (Al) has a homoclinic orbit to 0 It is homoclinic to the saddle for p1 lt 0 and to the saddle-focus for p1 gt 0

Our aim is to analyze the bifurcation diagram of (A l ) for small pll in the half- plane p1 gt 0 in the Shilnikov case

In the opposite case the bifurcation behavior of (A l ) is rather simple A unique stable limit cycle bifurcates from the homoclinic orbit for p2 gt 0 In both cases crossing the line L L ~ = 0when p l lt 0 results in the appearance of a single limit cycle

A2 Poincar6 map The technique for analyzing the behavior of (A l ) near the bifurcation is rather standard and consists of reducing its analysis to that of a Poincark map near the homoclinic orbit

HOMOCLINIC BIFURCATIONS IN A FOOD CHAIN

FIG 15 Homoclinic orbit t o a critical node at the Belyalcov bifurcation Local planar sections II are transversal to the homoclinic orbit at points M

Let M - = (00 z-) and M+ = (0 y+ 0) be points on the departing and incoming parts of the homoclinic orbit rl at p = 0 We introduce two local cross-sections to rl at these points nP(x1 yl z-) and II+(O yo zo) (see Figure 15) Here the pairs of coordinates (yo zo) and (xi yl) are used to parameterize the cross-sections As in [3] the Poincar6 map P II+ + II+ along orbits of the system can be defined for all small llpll as a composition of the singular map A II+ -+ n- (near the equilibrium 0 ) and the regular map Q II- + II+ (near the global part of the homoclinic orbit)

The singular map A is mainly determined by the linear part of (Al) and has the form

where

The rescaling

brings A to the form

The regular map Q which is a diffeomorphism can be written in the rescaled coordinates in the form


where a b c and d are smooth functions of p such that a(O)d(O) - b(O)c(O) 0 Taking the product of the maps A and Q defined respectively by (A4) and (A5)

we get

where Q and are functions of I namely

b d cos 9 = -fi+ o ( f i ) a sin = -amp+ o ( f i ) C

Finally the substitution

l o o exp (-m) p2 ++ 12 exp (g)fi brings the Poincari map P to the form

with O = 9 - being some function depending on p and

A3 Tangent and flip curves A fixed point (y a) of (A6) satisfies the system

Applying the implicit function theorem to the first equation in a neighborhood of y = 1 a = O weget

Thus the a-component of the fixed point satisfies the scalar equation

or

(A7)

where

(A8)

483 HO~IOCLINICBIFURCATIONS IN A FOOD CHAIN

Tangent bifurcations of the fixed points in (A6) correspond to double roots of (A7)gt iegt

whose leading term coincides with the tangent bifurcation condition for the map (A8) Similarly one can see that flip bifurcations of the fixed points in (A6) happen when

Also in this case the leading term coincides with that of the flip bifurcation condition for the map (A8) For this reason we call (A8) the normal form of the Belyakov bifurcation

Due to assumption (A3) equation (A7) has countably many roots near the origin ( z = 0) which can be isolated by writing

where n E N is sufficiently big and Q E [- ) Taking (A l l ) into account both tangent condition (A9) and flip condition (AlO) can be rewritten in terms of ( 8 n) as

Therefore in view of (A3)

By substituting this expression into the fixed point equation (A7) and by taking (A l l ) into account we obtain the following asymptotic representation for the tangent bifurcation curve

where Pl(p1) -+ 0 as p1 + 0 Thus there is a countable set of tangent bifurcatzon curves t p ) of (A8) accumulating on the primary homoclinic curve h(l) (p2 = 0) for small p1 gt 0 All these curves have infinite-order tangency with that curve at p1 = 0

Since the leading terms of the tangent and flip bifurcation conditions coincide (A12) also gives (with another amp(p l ) ) a representation of the flip bifurcation curves fil)near the origin ( p = 0) This has been noticed in a different context in [20] Therefore the Belyakov point is also the origin of a countable set of jlzp bzfurcatzon

(1)curves fA1) which have the same properties as t

A4 Secondary homoclinic curves The point if- of the intersection of r1with the plane IT has the coordinates (xlyl) = (0O) and is mapped by the global map Q (see (A5)) into a point = amp(if-) E nf with the coordinates

YU A KUZNETSOV 0 DE FEO AND S WNALDI

FIG 16 Double homoclinic loop t o a saddle-focus near the Belyakov bifurcation M2 E W S ( 0 )

(yo zo) = (1p2) If the image point M2 = P(Ml) returns to the stable manifold of 0 ie its z-coordinate happens to be zero we have a double homoclinic orbit (see Figure 16) Therefore the condition for (Al) to have a double homoclinic orbit to 0 can be expressed using (A6) as

Provided (A3) holds this equation defines countably many functions representing the double homoclinic bifurcation curves hi2) for small p1 gt 0 Indeed writing

(A 14) -- f i X

lnp2 = ~n + 0

where n E N is sufficiently large and 0 E [- ) from (A14) and (A13) it follows that

which gives

Substituting this expression back into (A13) and using (A14) we get the following asymptotic expression for the double homoclinic bifurcation curves

A5 Bifurcations in a transversal one-parameter family To get more in- sights into possible bifurcations near the Belyakov point consider bifurcations of fixed points and cycles in the normal form (A8) under variation of p2 with fixed p1 gt 0

485 HOhlOCLINIC BIFURCATIONS IN A FOOD CHAIN

F I G 17 One-parameter continuation of the jxed point and period-2 cycles of the normal form (A8)f o r p l = 1 B = l p = A= a

This corresponds to bifurcations in a one-parameter family of systems (Al) transversal to the saddle-focus homoclinic branch of h() Figure 17 obtained numerically using the software package CONTENT [32] shows such bifurcations

In the figure a wiggly curve originating in the upper right corner and approaching the line p2 = 0 is the branch of fixed points of (A8) Each turning point of this curve gives a tangent bifurcation (ie collision of two fixed points) The critical parameter values corresponding to the tangent bifurcations t) clearly accumulate on p2 = 0 from both sides As predicted by the theory very close to each turning point there exists a period-doubling bifurcation fA1) These points are marked with dots in the figure The corresponding parameter values also accumulate on p2 = 0 from both sides The numerical continuation of period-2 cycles bifurcating from the flip points shows that these cycles approach some values of p2 gt 0 as z -+ 0 (In z -+ -m)

These values correspond to the double homoclinic bifurcations hk2)and accumulate on p2 = 0 from the right ( p z gt 0) Notice that only a point with the minimal z-value on the period-2 cycle is plotted The period-2 branches do not intersect and there is one double homoclinic bifurcation hi2)between each two tangent bifurcations t c ) and t Z 2 There are also sequences of tangent and flip bifurcations of period-2 cycles accumulating on each double homoclinic bifurcation curve etc

Acknowledgments The authors would like to thank Dr A J Homburg (Uni- versity of Amsterdam) for useful discussions on Belyakov points


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[ll] 0 DE FEO A N D S RINALDI Yzeld and dynamics of trztrophic food chains Am Nat 150 (1997) pp 328-345

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[13] 0 DECROLY A N D A GOLDBETER FTOm szmple to complex osczllatory behaviour Analyszs of burstzng i n a multiply regulated biochemical system J Theoret Biol 124 (1987) pp 219- 250

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[18] H FREEDMAN Mathematical analysis of some three-species food-chain mod- A N D P ILT~IAN els hlath Biosci 33 (1977) pp 257-276

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SlAhl J APPL MATH 2001 Society for Industrial and Applied Mathematics ol 62 No 2 pp 462-487

BELYAKOV HOMOCLINIC BIFURCATIONS IN A TRITROPHIC FOOD CHAIN MODEL

Abstract Complex dynamics of the most frequently used tritrophic food chain model are investigated in this paper First it is shown that the model admits a sequence of pairs of Belyakov bifurcations (codimension-two homoclinic orbits to a critical node) Then fold and period-doubling cycle bifurcation curves associated to each pair of Belyakov points are computed and analyzed The overall bifurcation scenario explains why stable limit cycles and strange attractors with different geometries can coexist The analysis is conducted by combining numerical continuation techniques with theoretical arguments

Key words homoclinic bifurcations population dynamics continuation

AMS subject classifications 34C37 58F13 92D25

PII SO036139900378542

1 Introduction For several decades after the pioneering work of Lotka [34] and Volterra 1421 one of the topics of major concern in mathematical ecology has been the study of ditrophic food chains This has been accomplished by analyzing a great number of second-order continuous-time dynamical models usually called prey-predator models (see for example [2]) Existence of limit cycles multiplicity of attractors and catastrophic bifurcations are the characteristics of those models which have been used to explain complex behaviors observed in the field It was only in the late seventies that some interest in the mathematics of tritrophic food chain models (composed of prey predator and top-predator) emerged With almost no exception the first contributions dealt with the problem of persistence [18 19 171 and therefore did not provide information on the number and the geometry of the attractors This is a very unfortunate situation because the nature of the attractors is often the most interesting feature of a dynamical system An exception in this respect was an almost unnoticed paper [25] in which it was shown through simulation that a particular food chain model can behave chaotically This property was in practice brought to the attention of the scientific community by a contribution 1241 that appeared much later and showed that food chains behave chaotically on a tea-cup strange attractor provided that the three populations have diversified time responses increasing from bottom to top This condition on the time responses was used in the same years 138 391 to perform a singular perturbation analysis that indeed confirms that the tea- cup geometry is the result of the interactions between high frequency (prey-predator)

Received by the editors September 22 2000 accepted for publication (in revised form) April 16 2001 published electronically October 31 2001 This work was supported by Consiglio Nazionale delle Ricerche Italy Project ST174 hIathematica1 models and methods for the study of biological phenomena

httpwwwsiamorgjournalssiap62-237854html ThIathematical Institute Utrecht University Budapestlaan 6 PO Box 80010 3508 TA Utrecht

The Netherlands and Institute of hlathematical Problems of Biology Russian Academy of Sciences Pushchino hloscow Region 142290 Russia (kuznetQmathuunl)

f ~ a b o r a t o r ~of Nonlinear Systems (DSC-LANOS) Swiss Federal Institute of Technology Lau- sanne (EPFL) 1015 Lausanne Switzerland (oscardefeoQepflch) The research of this author was supported by the Swiss National Science Foundation 2000-05603098

S~ipar t imento di Elettronica e Informazione Politecnico di hlilano Via Ponzio 3415 20133 hlilano Italy (rinaldiQeletpo1imiit)







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Documents

Belyakov Homoclinic Bifurcations in a Tritrophic Food Chain …math.bd.psu.edu/faculty/jprevite/REU07/yu.pdf · dT Dl] , (2.lc) where T is time, R and K are prey intrinsic growth