chaotic population dynamics and biology of the top-predator

Chaos, Solitons and Fractals 21 (2004) 1195–1204

www.elsevier.com/locate/chaos

Chaotic population dynamics and biology of the top-predator

Vikas Rai a, Ranjit Kumar Upadhyay b,*

a Department of Applied Mathematics, Delhi College of Engineering, Bawana Road, Delhi 110 042, Indiab Department of Applied Mathematics, Indian School of Mines, Dhanbad, Jharkhand 826 004, India

Accepted 8 December 2003

Abstract

We study how the dynamics of a food chain depends on the biology of the top-predator. We consider two model

food chains with specialist and generalist top-predators. Both types of food chains display same type of chaotic

behavior, short-term recurrent chaos; but the generating mechanisms are drastically different. Food chains with spe-

cialist top-predators are dictated by exogenous stochastic factors. On the contrary, the dynamics of those with the

generalist top-predator is governed by deterministic changes in system parameters. The study also suggests that robust

chaos would be a rarity.

� 2004 Elsevier Ltd. All rights reserved.

1. Introduction

Ecology is a parent discipline of chaos. Since the seminal work of Sir Robert May [1] model ecosystems have been

designed and studied [2,3]. These studies conveyed that chaos might be a dominant mode of ecosystem evolution. As

such data requirements to capture chaos in the wild are stringent. One needs a string of precise measurements of species

abundance for a length of time, which cannot be fixed a priori. On contrary to this, the truth of the real world is that

ecological time-series are short and noisy. This severely limits the possibility of chaos getting detected even if it is there.

A couple of years ago, it occurred to us that the origin of the crisis might be intrinsic to organization of ecological

interactions.

Since then, we have made sustained efforts to understand why it is so difficult to observe chaos in nature. Earlier

authors had suggested [4,5] that the narrowness of the boundary between chaos and noise is responsible for the crisis.

All natural systems are noisy. Chaotic dynamics in appearance is noise-like. The noisy character of these systems are

caused either by intrinsic factors or extrinsic forces. When a system is subjected to external influences (deterministic or

stochastic) complex dynamical phenomena (chaos and noisy chaos) emerge. These phenomena are often indistin-

guishable. This leads to failure of attempts to observe chaos. The work [6,7] of the present authors has opened up a new

possibility that the reason for chaos getting poor evidential support might lie elsewhere. Moreover, it also suggested

that the biology of the top-predator has an important role to play as far as the dynamical complexity of a food chain is

concerned. In the present paper, we take up this issue in detail.

The paper is organized as follows. Section 2 presents the model systems that have been chosen for detailed dynamical

investigations. The methodology of the present study is presented in Section 3. We discuss the main findings in Section 4

and record the messages of the study in Section 5.

* Corresponding author. Tel.: +91-326-2200817; fax: +91-326-2210028.

E-mail address: [email protected] (R.K. Upadhyay).

0960-0779/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.chaos.2003.12.065

mail to: [email protected]

1196 V. Rai, R.K. Upadhyay / Chaos, Solitons and Fractals 21 (2004) 1195–1204

2. Model systems

Among four basic interactions (predator–prey, competition, interference and mutualism), predator–prey interaction

is the most common. There are two types of predators: (i) specialists and (ii) generalists. A specialist predator is the one,

which dies out exponentially fast when its favorite food is absent or is in short supply. The latter predators switch over

to alternative food options when its most preferred food is in short supply. The Rosenzweig–MacArthur (RM) model

[8] is the one, which describes the dynamics of a specialist predator and its prey:

dXdt

¼ a1X � b1X 2 � wYXðX þ DÞ ; ð1aÞ

dYdt

¼ �a2Y þ w1YXðX þ D1Þ

; ð1bÞ

where X is a prey for specialist predator Y with Holling type II functional response. a1 is the rate of self-reproductionfor the prey. The parameter a2 measures how fast the predator Y will die when there is no prey to capture, kill and eat.

b1 measures the intensity of competition among individuals of species X for space, food etc. and D measures the

efficiency of the prey in evading a predator’s attack. It depends on the protection afforded by the environment to the

prey. D1 has similar meaning as that of D.A model given by Holling and Tanner (HT) [18] describes the dynamics of a generalist predator and its prey:

dZdt

¼ AZ 1

�� ZK

�� w3UZðZ þ D3Þ

; ð2aÞ

dUdt

¼ cU � w4U 2

Z; ð2bÞ

where Z is the most favorite food for the generalist predator U . In this model, prey and predator both grow logistically.

A and K are respectively the rate of self-reproduction and carrying capacity for the prey Z. c is the growth rate of the

generalist predator due to sexual reproduction. The last term in Eq. (2b) describes how loss in species U depends on per

capita availability of its prey (Z). The other parameters have their usual meaning. Eqs. (2a) and (2b) together represent

this model. In what follows, we attempt to arrive at a graphical representation, which gives us an idea about the

parameter regimes displaying distinct dynamical possibilities.

A predator–prey system qualifies as a Kolmogorov (K) system if it can be cast into the following form:

dXdt

¼ XF ðX ; Y Þ;

dYdt

¼ YGðX ; Y Þ;

where F and G are continuous and analytic functions in the domain X P 0, Y P 0.

It can be easily seen that both the models qualify as a K-system. Kolmogorov analysis of RM model yields following

conditions:

w1 > a2; ð3aÞa1b1

>D1a2

ðw1 � a2Þ: ð3bÞ

The local stability analysis puts the additional constraint

2b1D1a2

w1 � a2

� �þ b1D� a1 > 0: ð4Þ

One obtains a stable limit cycle in the phase space, if inequality (4) is violated and those from Kolmogorov analysis

[9,10] are honored while making the choices of the parameter values. The two conditions can be combined to give a

straight line with slope ð2D1=ðw1 � a2ÞÞ. This line is located at a distance D from the origin of the coordinate system.

a1=b1 and a2 are the ordinate and abscissa. The straight line divides the space between two regions: the region above

presents parameter values, which correspond to stable limit cycle solutions. The region below the dividing line and the

straight line (a1=b1 ¼ D) is populated by stable equilibrium solutions (cf. Fig. 1a).

Fig. 1. (a) Two stability regions are separated by a straight line with slope 2D1=ðw1 � a2Þ and meets the abscissa at D [15]. (b)

Graphical representation of criterion (5). The figure is drawn for A=c ¼ w3=w4. Similar stability boundaries exist for other relation-

ships. This figure has been adapted from May [9].

V. Rai, R.K. Upadhyay / Chaos, Solitons and Fractals 21 (2004) 1195–1204 1197

Local stability analysis of HT model [9] provides a criterion

cA>

2ða� RÞ1þ aþ bþ R

; ð5Þ

where

a ¼ w3cw4A

; b ¼ D3

Kand R ¼ ½ð1� a� bÞ2 þ 4b�1=2: ð6Þ

The criterion when graphically represented, provides an idea about what parameter values will lead to a limit cycle.

A graphical representation of the criterion for the case Ac ¼

w3

w4is given in Fig. 1b.

The unstable region signifies stable limit cycle solutions, which emanate from stable ones through super critical Hopf

bifurcations. The assumed relationship is biologically realistic and was chosen for the shake of simplicity. Similar

stability boundaries exist for other relationships. It should be noted that a Kolmogorov analysis of this system does not

put any constraint on the parameter values of the system (Eqs. (2a) and (2b)).

When parameters are chosen in such a way that both RM and HT systems display regular persistent periodic

behavior, these systems act as oscillators. Let us call RM oscillator as LCO (1) and HT oscillator as LCO (2). When

suitably coupled, these systems will force each to generate chaos. An ecologically sound coupling of these systems gives

us the following model:

dXdt


dYdt


� w2YUðY þ D2Þ

; ð7bÞ

dZdt

¼ AZ 1

�� ZK

�� w3UZðZ þ D3Þ

; ð7cÞ

dUdt

¼ cU � w4U 2

ðY þ ZÞ ; ð7dÞ

where parameter a1 is the rate of self-reproduction of species X , b1 measures the intensity of competition among

individuals of species X for space, food, etc. w=ðX þ DÞ is the per capita rate of removal of species X by Y . Y is a

specialist predator, i.e., X is the only food for it. Therefore, Y dies out exponentially in the absence of X . w is the

maximum value that function wX=ðX þ DÞ can attain. a2 is the rate at which Y dies out exponentially in the absence of

its prey X . w1X=ðX þ D1Þ denotes gain in the specialist predator population due to proportionate loss in its prey. A and

K are respectively the rate of self-reproduction and carrying capacity for prey Z. The last term in Eq. (7b) represents the


functional response of the predator U , which is a generalist predator. It switches its prey whenever its favorite food

option Z is in short supply. The first term in the evolution equation for U describes the growth of U due to self-

reproduction. The last term in Eq. (7d) describes how loss in species U depends on per capita availability of its preys (Zand Y ). This gives us a model which was studied by us in Upadhyay et al. [11] with the functional response of the

vertebrate predator. w2 serves as the coupling parameter.

Let us imagine a situation that arises when one of the prey species (Z) leaves out the patch for a better habitat. The

resulting system is described by the following equations:

dXdt


dYdt


� w2YZðY þ D2Þ

; ð8bÞ

dZdt

¼ cZ � w3Z2

ðY Þ : ð8cÞ

It should be noted that generalist predator is denoted by Z, instead of U . This is for the shake of the continuity of the

notations for different species in the model.

The generalist predator Z (in Eq. (8c)) is a sexually reproducing species. For most sexual species over most popu-

lation densities, reproduction is determined primarily by female numbers. Because of the female-biased demography of

the sexual species the population growth rate is linear in population size, which is well below the carrying capacity.

However, the growth of a sexually reproducing population is proportional to the square of the number of individuals

present, in situations when the population is at very low densities. This is known as the Allee effect [12]. In such species,

the decline of the population to low densities occurs due to a variety of causes [13]. Most important of these is

inbreeding depression, which results from physical or chemical modification of the environment of organisms by social

interaction or by density-dependent mating success. For example, some aquatic organisms condition their medium by

releasing substances that stimulate growth of species, which have similar genetic make-up. Sparse populations rarely

provide sufficient opportunities for social interaction necessary for reproduction. The Allee effect variant of this model

can be written by replacing Z by Z2 in the first term of Eq. (8c) and adding an additional constant (D3) in the

denominator of the last term of the same equation. In this case, the last equation of the model system is modified to:

dZdt

¼ cZ2 � w3Z2

ðY þ D3Þ: ð9Þ

Some insect top-predators very often switch to alternative prey in situations when their favorite food is in short

supply. Eqs. (8a)–(8c) define the linear phase of model 1. The non-linear phase is described by Eqs. (8a), (8b) and (9).

These equations describe model 1.

We chose to study the non-linear phase as the linear phase does not support the chaotic behavior at all. The sexually

reproducing populations are covered by this phase when they are under Allee effect. A dynamical representation of

these model systems are given in Fig. 2.

Limit cycle oscillator of the first kind (LCO (1)) is obtained when parameters of Rosenzweig–McArthur model (RM;

first subsystem) [8] are set at limit cycle values derived from an application of pseudo–prey method [6]. Likewise, the

prey

Specialistpredator

Generalistpredator

pseudo-prey prey

Specialistpredator

L.C.O.(2)L. C. O.(1)

pseudo-prey

Generalistpredator

(a)

(b)

Fig. 2. (a) A representation of the biological system. (b) Two coupled limit cycle oscillators.

L.C.O.(1)L. C. O.(1)

Fig. 3. Dynamical representation of model 2.


other subsystem (obtained by cutting the link between the specialist and generalist predators in the second diagram of

Fig. 2a when set at appropriate parameter values serves as a limit cycle oscillator of the second kind. The predator–prey

interaction in RM model is formulated in Volterra scheme, which assumes exponential decay of a predator population

in the absence of its sole prey. In contrast to this, the second subsystem is a valid representation of predator–prey

interaction when the predator has other food options in addition to its most favorite prey.

When we couple two LCO (1)’s, we get a model proposed by Rosenzweig and MacArthur [8] and studied by

Hastings and Powell [19] and by Rinaldi et al. [3]. The model is described by the following equations:

dXdt


dYdt


� w2YZðY þ D2Þ

; ð10bÞ

dZdt

¼ �cZ þ w3YZðY þ D3Þ

: ð10cÞ

We call this model 2. The same can be dynamically represented as done in Fig. 3.

3. The methodology

The two models dynamically differ in one essential way: model 2 represents two coupled oscillators of type 1 whereas

model 1 results from a coupling of LCO (1) and LCO (2). Model parameters are selected in accordance with a method

given in Upadhyay et al. [6]. For 2D scans, two parameters are varied on both sides of their limit cycle values. An

important part of the methodology of selection of parameter values for simulation experiments is the following theorem:

Two coupled K-systems in the oscillatory mode would yield either cyclic (stable limit cycles and quasi-periodicity)

or chaotic solutions depending on the strength of coupling between the two.

If key parameters of a model system are known, one can study the dynamics by varying these system parameters on

both sides of their basic value for which both subsystems yield a stable limit cycle in the phase space. Rest of the

parameters are kept constant. For example, a1 and c are two key parameters for model 1. The ranges for variation of

these parameters in simulation experiments are decided on the available biological information [16].

4. Results and discussion

As far as observability of chaos in nature is concerned, it can be divided into two distinct categories: (i) robust chaos,

and (ii) short-term recurrent chaos. We give precise definitions for both of them and present results, which suggest that

robust chaos should be rarely found in nature.

4.1. Robust chaos

If chaotic dynamics exists in a region in 2D parameter space spanned by two crucial parameters of a model system

and also if the basin of attraction for the associated attractor is large; one understands that the chaotic dynamics

displayed by the system is robust. If this is the case, chaos would be a dominant dynamical mode of the corresponding

real system.

4.2. Short-term recurrent chaos

There exist two mechanisms which generate this kind of chaotic behavior. If chaotic dynamics is displayed in a

region in 2D parameter space of a model system, it can still support short-term recurrent chaos (strc) provided the

1.92 1.93 1.94 1.95 1.96 1.97 1.980.028

0.030

0.032

0.034

0.036

0.038

0.040

0.042

0.044

0.046

c

a1

Fig. 4. Distribution of points, where chaos was observed in model 1 with invertebrate top-predator. a1 was varied in the range

½0:5; 2:5�. The parameter c was varied in the range ½0:005; 0:05�. a2 was set at 1.


basins of attraction of different attractors are intermixed. Alternatively, it can be generated when chaos exists at discrete

and isolated points in 2D parameter space. As chaos (exponential sensitivity of system’s trajectories on ‘‘initial con-

ditions’’) takes time (inversely proportional to the max. Lyapunov exponent of the system) to develop, system gets

needed amount of time to lock itself on to a non-chaotic attractor before it settles to a fully developed chaotic state. In

the former case the agency which generates strc is exogenous stochastic influences. It is purely deterministic changes in

the crucial parameters of the system, which cause this behavior in the latter.

In the present paper, we explore chaotic dynamics of two model systems, which differ from each other in one

essential way; i.e., the top-predator in model 1 is a generalist predator whereas that of model 2 is a specialist one. In

these model systems, the top-predator is assumed to be an invertebrate.

Fig. 4 present results of simulation experiments on model 1. It evident that chaotic dynamics exist at discrete and

isolated points in these systems. The secular changes in model parameters cause frequent interruptions in chaotic

dynamics, at which time, the system evolves on non-chaotic attractors (periodic and fixed point).

Chaotic dynamics in model 1 was observed only when a2 was fixed at 1. For other values of this parameter, no chaos

was found in the model. This suggests that there does not exist a window in parameter space for chaotic dynamics.

Chaotic dynamics does not exist for other values of a2. Fig. 4 suggests that there does not exist a window in parameter

space for chaotic dynamics.

We present basin boundary calculations for chaotic attractors supported by model 1 at bottom-left corner

ða1 ¼ 1:92; c ¼ 0:03Þ, top-right corner ða1 ¼ 1:975; c ¼ 0:045Þ and middle ða1 ¼ 1:95; c ¼ 0:035Þ of chaotic region in

Figs. 5–7, respectively. All the three figures are the XY view of the basin boundary structure (�1006X 6 100,

�1006 Y 6 100) of chaotic attractor (shown in yellow colour 1). The basin boundary calculations are performed using

the basins and attractors structure (BAS) routine developed by Maryland chaos group. We have used the Dynamics

software package of Nusse and Yorke [20] for all the basin boundary calculations. The basin boundary calculations are

presented as colour diagrams (for detail description see [11,21]). It is clear from Figs. 5–7 that the chaotic attractor is the

dominant one. No other attractor with appreciable basin size exist. The encroachment into the basin of chaotic

attractor by basin of attractor at infinity (shown in green colour here) can be observed. The basin boundary diagram

also shows that there is no intermixing of basins of different attractors. This establishes that strc in this model is

generated by deterministic changes in the crucial parameters of the system.

Fig. 8 present results of 2D parameter scan on model 2. It can be seen from this figure that chaos exists in a region

(non-zero area) in a1 � w3 parameter space. But the chaotic region is intruded by limit cycle attractors. We show results

of basin boundary calculations at the top-left corner ða1 ¼ 1:75;w3 ¼ 3:75Þ, top-right corner ða1 ¼ 3:0;w3 ¼ 3:75Þ andin the middle ða1 ¼ 2:5;w3 ¼ 3:0Þ of the chaotic region in Figs. 9–11, respectively. All these figures reveal that the basins

of different attractors are intermixed. In such a case, the dynamics of the system is decided by external stochastic

1 For interpretation of color in Figs. 5–11, the reader is referred to the web version of this article.

Fig. 5. Model 1. Basin boundary calculation for bottom-left corner point (a1 ¼ 1:92; c ¼ 0:03) in Fig. 4 for the chaotic attractor. The

values of other parameters are a2 ¼ 1:0, w ¼ 1, D ¼ 10, b1 ¼ 0:05, w1 ¼ 2, D1 ¼ 10, w2 ¼ 1:55, D2 ¼ 10, w3 ¼ 1, D3 ¼ 20.

Fig. 6. Model 1. Basin boundary calculation for top-right corner point (a1 ¼ 1:975; c ¼ 0:045) in Fig. 4 for the chaotic attractor. The

values of other parameters are a2 ¼ 1:0, w ¼ 1, D ¼ 10, b1 ¼ 0:05, w1 ¼ 2, D1 ¼ 10, w2 ¼ 1:55, D2 ¼ 10, w3 ¼ 1, D3 ¼ 20.


influences and the result is strc; a form of chaotic behavior, wherein it is interrupted by other kind of dynamics at

irregular and unpredictable intervals. Therefore, we see that short-term recurrent chaos can be caused either by

deterministic changes in the system parameters (model 1) or by exogenous stochastic influences (model 2). In the

former, chaotic dynamics is interrupted by smooth changes in system’s parameters whereas stochastic influences affect

such interruptions in the latter one.

5. Conclusions

Thus, it is clear that robust chaos is less likely to be found in nature. Instead, what nature seems to abound in has

recently been called strc [14,15]. The strc can occur through two mechanisms. The first mechanism involves deter-

ministic changes in the system parameters. The other invokes exclusive role of abrupt changes in initial conditions. The

only systems, wherein robust chaos has been observed so far, are diffusively coupled predator–prey systems [17].

Diffusion couples stable limit cycle oscillators on a spatial gradient. These coupled oscillators force each other at

‘‘incommensurate’’ frequencies to generate chaos. The chaotic dynamics exists in a region of non-zero measure in 2D

Fig. 7. Model 1. Basin boundary calculation for middle point (a1 ¼ 1:95; c ¼ 0:035) in Fig. 4 for the chaotic attractor. The values of

other parameters are a2 ¼ 1:0, w ¼ 1, D ¼ 10, b1 ¼ 0:05, w1 ¼ 2, D1 ¼ 10, w2 ¼ 1:55, D2 ¼ 10, w3 ¼ 1, D3 ¼ 20.

Fig. 9. Model 2. Basin boundary calculation for top-left corner point (a1 ¼ 1:75;w3 ¼ 3:75) in Fig. 8 for the chaotic attractor. The

values of other parameters are a2 ¼ 1:0, w ¼ 1, D ¼ 10, b1 ¼ 0:05, w1 ¼ 2, D1 ¼ 10, w2 ¼ 1:5, D2 ¼ 10, c ¼ 0:7, D3 ¼ 20.

0

0.75

1.5

2.25

3

3.75

4.5

0 0.5 1 1.5 2 2.5 3 3.5

a1

w3

SFLCChaos

Fig. 8. Model 2. 2D scan diagram in (a1;w3) parameter space. The parametric values for limit cycle solution are (using pseudo-prey

method) chosen as a1 ¼ 2, b1 ¼ 0:05, w ¼ 1, D ¼ 10 ¼ D1 ¼ D2, a2 ¼ 1, w1 ¼ 2, w2 ¼ 1:5c ¼ 0:7, w3 ¼ 1, D3 ¼ 20. The parameter space

for the simulation experiment is chosen as 0:56 a1 6 3, 0:16w3 6 4.


Fig. 11. Model 2. Basin boundary calculation for middle point of chaotic region (a1 ¼ 2:25;w3 ¼ 3:0) in Fig. 8 for the chaotic

attractor. The values of other parameters are as in Fig. 9.

Fig. 10. Model 2. Basin boundary calculation for top-right corner point (a1 ¼ 3:0;w3 ¼ 3:75) in Fig. 8 for the chaotic attractor. The

values of other parameters are as in Fig. 9.


parameter space and there is no other competing attractor in the initial condition space. Thus, there is no intermixing of

the attractor basins.

In our earlier work [6], we had suggested that the biology of the top-predator would be a crucial factor for the

determination of food chain dynamics. The results of the present study indicate that chains ending at specialist pre-

dators are dictated by exogenous stochastic influences. On the contrary, those with generalist top-predators are gov-

erned by deterministic changes in system parameters. We opine that the natural systems with first kind of food chains

would present difficult challenges as far as program of quantification of their dynamical complexity is concerned. The

other kind of systems seems to allow such a program to be implemented smoothly. This conjecture is to be tested in the

laboratory and in the field.

Acknowledgements

The research described in this paper is supported by Department of Science and Technology, New Delhi, grant

under SERC Fast Track scheme for young scientists 2001–2002, to the second author (RKU).


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chaotic population dynamics and biology of the top-predator

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