period-doubling bifurcations leading to chaos in a model food chain

Ecological Modelling, 69 (1993) 63-77 Elsevier Science Publishers B.V., Amsterdam

63

Period-doubling bifurcations leading to chaos in a model food chain

Vikas Rai a and R. Sreenivasan b

a School o f Physics, Devi Ahilya I, rtshwavidyalaya, Khandwa Road, Indore - 452 001, lndia b School of Environmental Sciences, Jawaharlal Nehru, University, New Delhi - 110 067, India

(Received 24 September 1991; accepted 17 September 1992)

ABSTRACI"

Rai, V. and Sreenivasan, R., 1993. Period-doubling bifurcations leading to chaos in a model food chain. Ecol. Modelling, 69: 63-77.

In this paper, we present a new model for a food chain involving three species and show the presence of a period-doubling scenario leading to chaos. An application of this study to aquaculture research is also discussed.

1. INTRODUCTION

During the past decade much attention has been paid to the study of the transition from periodic to chaotic motion in nonlinear dynamical systems. Understanding the mechanism which leads to such a behaviour constitutes an exercise of paramount importance. Period-doubling bifurcations (Cvitanovi6, 1984; Berg6 et al., 1986; Bai-lin, 1989), Rue l l e -Takens-New- house route (Eckmann and Ruelle, 1985) and intermittency route (Pomeau and Manneville, 1980) are the ones which are reported to be most com- monly observed.

Although chaos has been reported to exist in many ecological systems (Gilpin, 1979; Schaffer and Kot, 1985), there are very few attempts to understand the mechanism which leads to this. Gardini et al. (1989) have studied the transition to chaotic dynamics in a Lotka-Volterra system involving three species. They show that period-doubling cascades are responsible for chaotic dynamics in their system. The purpose of this paper

Correspondence to: V. Rai, School of Physics, Devi Ahilya Vishwavidyalaya, Khandwa Road, Indore - 452 001, India.

0304-3800/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

64 v . RAI AND R. SREENIVASAN

is to present a new food chain model and to report the presence of period-doubling scenario leading to chaos in this model.

2. MODEL SYSTEM

Consider a prey population X and two predator populations Y and Z. The predator Y preys on X and the predator Z preys on Y. Let A be the intrinsic growth rate of the prey X. Then the growth of the prey population X in the absence of the predator Y is given by

d X - A x 1 - (2.1) dt

where K is the maximum prey population allowed by the limited resources provided by the environment. It is evident that self-interaction among the individuals of the prey population is included in Eq. (2.1).

When predators are present, mortality due to predation must be sub- tracted from the right side of Eq. (2.1). In our model, we use the Holling type of functional response for the predators, which is most common (Murdoch and Oaten, 1975). The functional response equation can be written as:

B X

= ( X + D 1 ) ' (2.2)

where q~ is the per capita predation rate. It should be noted that B is the maximum ~ can reach when the predator will not or cannot kill more prey even when they are available. D 1 is a constant which is related to the predator 's handling time of prey (in a general sense). It can be shown that for a given prey density, D 1 is directly proportional to the time required for the predator to search for and find a prey and is, therefore, determined by the protection afforded to the prey by the habitat.

Now, the complete equation for the prey becomes

~ - = A X 1 - ~ ( /_ t_Ol ) (2.3)

Per capita numerical response of the predator is a function of prey population (food availability). It is realistic to assume that the predator 's per capita numerical response is proportional to the prey population. This assumption may not be valid at very high prey densities because predators become inefficient at such densities. However, the possibility of preys attaining such high densities is always excluded due to the presence of the self-interaction term in the rate equation for the prey.

PERIOD-DOUBLING BIFURCATION LEADING TO CHAOS 65

In the Volterra scheme, the growth rate equation for the predator Y can be written as follows:

dY d---t = - C Y + D X Y (2.4)

where D is a proportionality constant and C measures the death rate of the predator Y in the absence of the prey X.

Since the predator Z preys on Y, the complete growth rate equation for the predator Y can be written as:

d Y E Y Z d---[ - - C Y + D X Y ( y + D2 ) (2.5)

where E and D 2 have similar meanings as those of parameters B and DI, respectively.

Similarly, the time evolution of the last predator Z can be described by the equation:

dZ - - = - F Z + G Y Z (2.6) d t

where F and G have similar meanings as those of parameters C and D, respectively.

At this point, we feel that some features of equation (2.6) need further explanation. There also we have assumed that the per capita numerical response of the predator Z is proportional to its prey population (Y). This assumption is valid only when the prey Y does not attain very high densities. The fact that prey Y, which is a predator for X, dies out exponentially when there is no prey (i.e. X = 0) and that its per capita numerical response is proportional to its prey population X, which is limited by the limited resources provided by the environment, ensure that the predator Y will not attain such high densities. It may also be noted that the regulatory mechanism provided by the predator Z also contributes to this effect.

Thus, the complete model system can be given by the following equations:

d X - - A X 1 - (2.7a) dt K (X+D1)

d Y E Y Z d-~ = - C Y + D X Y - ( r + D z ) (2.7b)

d Z - - = - F Z + G Y Z (2.7c) dt

66 v. RAI AND R. SREENIVASAN

where A, B, C, D, E, F, G, D 1, D 2 and K are model parameters, and assume only positive values.

This model system is in many respects different from other, similar systems which are described in the literature. In particular, Schaffer (1985) studied a model which involves two prey and a predator species and employs interaction terms of Lotka-Volterra type. The present system is a food chain which employs a more common functional response term for the predator (May, 1974).

3. LINEAR STABILITY ANALYSIS AND HOPF-BIFURCATION

3.1. Linear stability analysis The elements of the community matrix J are given by the expression:

J/~--~OXk , i , k = 1 , 2 , 3 , (3.1.1)

where the symbol * signifies that the elements of the community matrix are evaluated at the non-degenerate steady states. In our case,

X 1 = X,

X 2 = Y,

X 3 = Z,

and

F I = A X 1 ~ ( X + Ol)

E Y Z F 2 = - CY + D X Y - (3.1.3)

( Y + D 2 )

F 3 = - F Z + GYZ (3:1.4)

Non-degenerate steady states of our system are given by the following expressions:

X * = - ~ ( K - D , ) + ( K - D a ) 2 - e K ~---~7D1 (3.1.5)

F Y* = - - (3.1.6)

G I ( F )

Z * ~ _ _ * E --G + Dz ( D X - C), (3.1.7)

where X * is given by the expression (3.1.5).


The various elements of the community matrix are as follows: 2A BFD a

J~I=A - ~ - X * - G(X* +D1) 2

From Eq. (2.7a), we have A BF

A - - -X* = 0 K G(X+D1)

From Eqs. (3.1.8) and (3.1.9), we get

BF J~ =X* G(X* + D 1 ) 2 - - -

J =0 F

BX * (X*+D1)

A) K

(3.1.8)

(3.1.9)

(3.1.10)

(3.1.11)

(3.1.12)

(3.1.13)

asymptotic stability:

B 1 + D2 G X* 1+ F

< ( 1 + DZG)F (3.1.20) C

J:3=

( - C + DX*) (3.1.14) J2~= (1+ D2G)F

E J2~= (1+ DFG ) (3.1.15)

J3] = 0 (3.1.16)

J~2= GZ (3.1.17)

J3~=0 (3.1.18)

The eigenvalue equation for the community matrix J is

A 3 -4- ( - J l ~ - J 2 ~ ) A2 -4- (J11J22 * * * * * * * - J 1 2 J 2 1 - J 2 3 J 3 2 ) A A- (J1,J23132) =0 ( 3 . 1 . 1 9 )

An application of the Ruth-Hurwitz criteria (May, 1974, appendix II, p. 196) gives the following three conditions for the system to possess local

68 v. RAI AND R. SREENIVASAN

( c) B 1 D - ~ < 0 (3.1.21)

[ c 1 B I ( D X * - C ) F < B1 + i-'--D"-~-~ B2 +--k--)

(3.1.22)

where

BF A Ba = G(X* +D1) 2 K ' and

B 2 - * , • , _ • • - - J12 J21 - J1 a J22 J23 J32

It should be noted that we have used the fact that FX .2 is positive in arriving at condition (3.1.21).

If the choice of our parametric value is such that all the above con- straints (3.1.20-3.1.22) are simultaneously respected, then the system will be asymptotically stable in a local sense. On the other hand, violation of these conditions would result in the system jumping into the non-linear regime where phenomena like bifurcations and chaos are observed.

3.2. Hopf-bifurcation analysis For a Hopf-bifurcation to occur, a pair of eigenvalues must cross the

imaginary axis with the third eigenvalue being real and negative (Marsden and Mccracken, 1976). Hence, we assume that at critical value of the parameters, the set of three zeros of Eq. (3.2.1) consists of a pair of pure imaginary conjugates (+ i t / ) and one real negative root ( - r ) :

e ( , ~ ) __.,~3 _1._ ( _ J 2 2 _ J l l ) A 2 -I- (J11J22-J23J32-J12J21)A + (Ja lJ23J32)=0

(3.2.1)

This equation is nothing but the eigenvalue equation for the Jacobian matrix. Jn , J = , . . . are the elements of the Jacobian matrix at the non-degenerate steady states.

Applying the above mentioned criterion for Hopf-bifurcation to occur we get

P(A) = (A + r)(A - i-q)(A + it/) = A 3 +rA 2 +r/2A + rr/2 = 0

Equations (3.2.1) and (3.2.2) are identical only when

r = - (Jla +J12)

~7 2 = Jll J22 - J23 J32 - J12 J21 > 0

rrl 2 = J11J23J32

(3.2.2)

(3.2.3) (3.2.4) (3.2.5)


From Eqs. (3.2.3) and (3.2.5), it is clear that

( Jla J23 J32 ) r/2 = > 0 (3.2.6)

(Jl + J22)

Equat ions (3.2.4) and (3.2.6) give us the Hopf-bifurcat ion condi t ion as

( J l l "[- J22)( - J l a J 2 2 -{- J23J32 "q- J12J21) = JllJ23J32 (3.2.7)

If the choice of our parametr ic values is such that the condit ion (3.2.7) is satisfied, then a stable fixed point will bifurcate to give a stable limit cycle.

4. P E R I O D - D O U B L I N G C A S C A D E S L E A D I N G T O C H A O S IN T H E P R E S E N T S Y S T E M

A Hopf-bifurcat ion occurs (stable equl ibr ium point ~ stable limit cycle) when the parameters of our mode l system take on the values:

A = 1 . 1 1 8 , B = I , C = 1 , D = 0 . 0 5 , E = I , F = I , G = 0 . 0 5 ,

D 1 = 10 , D 2 = 10 and K = 50.

The stable limit cycle at tractor at A = 1.2 is shown in Fig. 1. As pa ramete r A is increased (while the rest of the parameters are kept

5 ~

Z F

fi~ )<p

Fig. 1. X - Z projection of the limit cycle attractor at A = 1.2.

Gt~

70

1 2 0

Z F

0

(a)

V. RAI AND R. SREENIVASAN

0

- 0 . 0 5 9

rr

0

0 _1

- i . 9

"(b)

/

XP 9 0

't•••••h••'h•'•h•••h•'••••••h'••h•••h•''••'"h'••h••'h'•'h•••h•••••'''h•'•••h•••••

FREGUENCY 0 0 . 0 6

Fig. 2. (a) X - Z projection of the limit cycle at tractor at A = 2.3. (b) The corresponding power spectrum.

constant), a hierarchy of limit cycle attractors of various periods appear. One such attractor is given in Fig. 2a. Fig. 2b gives the corresponding power spectrum. All the power spectra were designed by taking the fast Fourier transform (FFT) of the auto-correlated data in X-variable gener-

P E R I O D - D O U B L I N G B I F U R C A T I O N L E A D I N G T O C H A O S 71

ated by integrating the model equations (Press et al., 1986). If we call the frequency of the limit cycle attractor at A = 1.2 as the fundamental frequency ( f ) , then it is clear from Fig. 2b that the first subharmonic generation has taken place. Subsequent limit cycle attractors of various periods and their corresponding power spectra are shown in Figs. 3a and

150

ZF =

0

(a)

0

- 0 . 1 t0

XP 120

or tU

0 ~L

0 _J

i ! i ! - i .8

FREOUENCY 0 0 . 0 6

Fig. 3. (a) X - Z projection of the limit cycle attractor at A = 2.85. (b) The corresponding power spectrum.

ZF =

0

-0.200

n - LU

0 0.

0 d

b)

, , . i , , , , h , , , I . . . .

- 1 .B

(a) /x -~

i i /

\ \ !I "\\\

72

1 5 0

X P

V. RAI AND R. SREENIVASAN

1 2 0

FREQUENCY 0 0 . 0 5

Fig. 4. (a) A - Z projection of the stable limit cycle attractor at A =3.05. (b) The corresponding power spectrum.

4a, and 3b and 4b, respectively. It should be noted that power spectrum corresponding to the attractor at A = 2.85 (Fig. 3b) has frequency f/4 and its harmonics which clearly shows that second subharmonic generation has occurred. Similarly, frequencies arising in the power spectrum (Fig. 4b) could be explained. It is evident from the power spectrum (Fig. 4b) that the

PERIOD-DOUBLING BIFURCATION LEADING TO CHAOS 73 (a)

3 ~

Z P

- - 1

- - I

- 0 . 3 5 0

I11

0 I1

0 J

!I Y':>,\

q / ",,X, \'~\%

X P 2 5 B

- -~ . 5 iI'hi'`h`'*h'~`h'''h'~d'~*'h'''h''1~''~''~`h~t'~I~'`h`'d''''~'`jhI''h`'i~°'iih'`~

FREQUENCY 0 0 .06

Fig. 5. (a) X - Z projection of the strange chaotic attractor at A = 4.0. (b) The corresponding power spectrum.

level of background noise has risen considerably. It will grow as one approaches the limit cycles of higher periods. This is why it becomes very difficult to observe higher period-doublings in the present system. The strange chaotic attractor is shown in Fig. 5a and the corresponding power spectra in Fig. 5b. For the sake of clarity of presentation, various attractors

74

TABLE 1

V. R A I A N D R. S R E E N I V A S A N

Fig. Attractor Spectral line Remarks

1 Stable limit cycle of period 1

2a,b Stable limit cycle of period 2



fundamental frequency (f)

f /2 and its harmonics



5a,b Strange chaotic Continuous power attractor spectrum

Corresponding power spectrum not shown

Occurrence of first period doubling

Occurrence of second period doubling

Occurrence of third period doubling amidst rising level of background noise

Exponential decay of power with frequency

and important features of their power spectra are listed in Table 1. Our model system exhibits chaotic behaviour in the parameter regime (A = 4 to 4.5), and beyond this, the chaotic behaviour is interrupted by the "windows of periodicity". To show that chaotic behaviour exists in our model system in the aforementioned parameter regime we show, in Fig. 6, the variation of maximum Lyapunov exponents with the different values of the model parameter A. An algorithm designed by Wolf et al. (1985) was employed to extract the maximum Lyapunov exponents from the time series.

0 .B~:

O . ' I C . . . . i

3 . g A 4 . 6

Fig. 6. Max Lyapunov exponents (Amax) in the units of bits of information per time-step are plotted against model parameter A.


5C

X N + I

5

5

f

,,,~ ,, ~ , , I ~ ~ ~ f 5 0 X N

Fig. 7. Poincar6 first-return map calculated by slicing the strange chaotic attractor at A = 4.5 with the plane Z = 11.

It is evident from the power spectra that the underlying mechanism which leads to chaotic behaviour in our system is characteristic of period- doubling scenario• We show the Poincar6 first-return map calculated for the chaotic attractor at A = 4.5 in Fig. 7. It has a geometrical dimension be tween 1 and 2. Keeping in view the fact that the nature of the steady states at the parameter values A = 4 and A = 4.5 are the same (two saddle points with two stable and one unstable manifold each and an unstable focus) and so will be the organization of the flow around them, one would not hope to get Feigenbaum's universal number which describes the doubling of the period for a large class of unimodal one-dimensional maps.

5. DISCUSSION AND CONCLUSION

It can be easily seen that our system as described by equations (2.7a), (2.7b) and (2.7c), is asymmetric under the symmetry operations: (X, Y, Z)

( - X , -Y , Z) and (X,, Y, Z ) ~ ( - X , -Y , - Z ) . Therefore, we hope that the present food chain can provide a more realistic model for a class of ecological systems, as any realistic model of natural ecosystems is believed to be asymmetric (Cox, 1990). The study of the mechanism leading to chaotic dynamics in our system gains more importance when one keeps in

76 v . R A I A N D R. S R E E N I V A S A N

mind that this hypothetical food chain can model a real-world ecological system which involves a blue-green algae (Chlorella) as a prey, rotifers (zooplankton) as predator, which, in turn, are preyed by fish larvae. This constitutes one of the current problems of paramount importance in aquaculture research.

Our study suggests that the present system will go chaotic when the rate of self-reproduction of the prey is large. Thus, one can hope that the chaotic dynamics will occur for those ecological systems in which the rate of self-reproduction of the top prey assumes a sufficiently high value. Moreover, our control parameter A is the one which can be very easily controlled in laboratory experiments. Therefore, it is desirable to set up a laboratory experiment involving aforementioned species. This will be helpful in verifying one of the predictions of our study, e.g., presence of period-doubling scenario leading to chaos. Thus, it can be said that the present study has potential to provide guidelines to set up laboratory experiments to simulate and study the dynamical behaviour of natural ecosystems.

The real importance of this study lies in the fact that it may be helpful in studying the dynamics of various interacting populations of a food chain in the laboratory and extract some information about the parameter regime where the chaotic dynamics sets in the same. This will be helpful in avoiding chaos in a real-world situation (e.g., fish pond) by manipulating parameteric values of the system. The basis of this hope is that chaos is something which is not desirable when one is interested in managing a system because chaos allows only short-term predictions.

Recently, Sugihara and May (1990) have shown that there underlies a three-dimensional chaotic attractor in the dynamics of marine planktonic diatoms. Although the corresponding time series is very noisy, yet they have been able to extract the information that some of the dynamics could be described by deterministic chaos. This is the first concrete example of occurrence of chaos in nature. Keeping in view the fact that the population biology of our model species is of the similar nature as that of the marine planktonic diatoms and the inherent assumptions of our model could be realistic for the population dynamics of planktonic diatoms, we hope that the deterministic part of the planktonic dynamics can be modelled by our system.

ACKNOWLEDGEMENTS

We wish to thank Profs. R. Ramaswamy and J. Subbarao for helpful discussions. The authors are grateful to Prof. Alan Wolf for sending the FET code, which extracts maximum Lyapunov exponents from a given time


series. Dr . A.J . Ur f i is t h a n k e d for m a k i n g us aware o f a p o n d l a b o r a t o r y

e c o s y s t e m which exempl i f i e s ou r m o d e l food chain .

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Berg6, P., Pomeau, Y. and Vidal, C., 1986. Order within Chaos: Towards a Deterministic Approach to Turbulence. Hermann-Wiley, Paris.

Cox, S.M., 1990. The transition to chaos in an asymmetric perturbation of the Lorentz system. Phys. Lett. A, 144: 325-328.

Cvitanovi6, P., 1984. Universality in Chaos. Adam Hilger Ltd., Bristol. Eckmann, J.-P. and Ruelle, D., 1985. Routes to turbulence in dissipative dynamical systems.

Rev. Mod. Phys., 57: 617-656. Gardini, L., Lupini, R. and Messia, M.G., 1989. Hopf bifurcation and transition to chaos in

Lotka-Volterra equation. J. Math. Biol., 27: 259-272. Gilpin, M.E., 1979. Spiral chaos in a predator-prey model. Am. Nat., 113: 306-308. Marsden, J.E. and Mccracken, M., 1976. The Hopf Bifurcation and its Applications.

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period-doubling bifurcations leading to chaos in a model food chain

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