Letter to editor

Ecological vs. mathematical chaos

e c o l o g i c a l c o m p l e x i t y 6 ( 2 0 0 9 ) 1 4 7 – 1 4 9

avai lab le at www.sc iencedi rect .com

journal homepage: http://www.elsevier.com/locate/ecocom

The author of this letter suggests that the basic structural

unit of real world ecological systems is a predator-prey

community and other fundamental interactions (mutual-

ism, interference and competition) form connectivity

links between these individual oscillators (Rai, 2004).

Bokelman and Deng indicate that chaos can even be

generated if one of the interacting subsystems rests on a

stable equilibrium point (s. e. p’s) (Bokelman and Deng,

2005). Although this is theoretically true, yet does not have

any relation with how chaos is generated in natural

ecological systems.

In order to understand why this is so, one should note

that there are no stable equilibria in natural ecological

systems as the dynamical systems constantly interact with

their environments (Bauch and Anand, 2005). Even if

parameters of the subsystem have values corresponding

to stable equilibrium behavior, the subsystem driven by

environmental forces displays oscillatory dynamics. Thus,

their criticism does not have any sense. The present author

feels that the reader must not be misled to take that

oscillatory predator-prey interactions are not the basic unit

of natural ecosystems. What is more important is that one

must distinguish mathematical chaos from ecological

chaos. Detection of chaos in a mathematical model has

no meaning in itself unless it corresponds to dynamics of

natural ecological systems. This is a problem with ecological

modeling as suitable data sets are seldom available for

analysis.

The structural complexity of a system in natural

ecologies is built by associating such structural motifs

(shown in Fig. 1). The mutualistic interactions come in to

play at this point. Deng is right in pointing out that chaos

can be generated even when specialist predator-prey

interaction leads to a stable equilibrium. But, such a

dynamics would represent mathematical chaos instead of

ecological chaos which is the object of investigation. The

model studied by Bokelman and Deng (2005) can be

pictorially represented by the following figure.

Predator-prey interactions are well known to generate

oscillatory dynamics (Kendall et al., 1999). In general, two

coupled subsystems of a dynamical system force each other to

generate chaos. Let us consider a case when one of the

subsystems (e.g., X–Y) in Fig. 2 rests on a stable equilibrium

point. In such a case, the other subsystem which is performing

oscillatory motion forces the former and dynamical chaos

results.

A structural complexity of the magnitude equal to the

one shown in Fig. 1 (Polis, 1998) can be created with the

structural motif shown in Fig. 1 using interactions (mutu-

alism and interference) as linking mechanisms. Here, we

must distinguish between interference and interferential

competition. The structural complexity of complex food-

webs (e.g., the one shown by Polis, 1998) can be generated

and its dynamics can be understood in terms of those

shown in Fig. 1 by McCann et al. (1998). The motif shown in

Fig. 2 cannot serve as a basic structural unit of complex

food-webs as the linking mechanisms would not be able to

create a structure of the kind shown by Polis (Polis, 1998).

One may argue that this is treated as an axiom without

caring for actually establishing it. A careful examination

would satisfy any one who can think geometrically. More-

over the fact that there are no s. e. p’s in natural ecologies,

the present author’s contention is reinforced.

The author wishes to impress upon that the target of all

modeling (mathematical or statistical or combined) and

analysis should be ecological chaos. If this suggestion is put

into practice, it will generate interest of ecologists which

study ecological systems in the field or by laboratory

experimentation. However, mathematical chaos in deter-

ministic models (both continuous and discrete) can never be

ecological chaos as ecological systems are weakly dissipa-

tive. There are no chaotic attractors in ecological system,

but unstable invariant sets (Grebogi and Feudel, 1997;

Scheuring and Domokos, 2007) which rest in the basin of

attraction of finite many periodic attractors. These periodic

attractors result due to action of dissipative forces within

the ecological system being described by systems of

difference or ordinary differential equations. Observation

of chaotic behavior in laboratory populations of flour beetle

indicates the presence of chaotic dynamics in non-linear

dynamical systems; it does not establish the existence of

chaotic attractors (Cushing et al., 2002) as chaotic attractors

are not indispensable for the occurrence of chaotic

dynamics. There are many other competing mechanisms

which involve chaotic saddles, noise-induced chaos, etc.

The wonderful stochastic dance in stochastically perturbed

Fig. 2 – Graphical representation of the model ecosystem

described by Eq. (1) in Bokelman and Deng (2005).

Fig. 1 – Basic structure of a real world ecological system.

e c o l o g i c a l c o m p l e x i t y 6 ( 2 0 0 9 ) 1 4 7 – 1 4 9148

deterministic systems is caused by discrete nature of state

space for biological populations, which is contained in

habitat size and density. The observer sees only the episodic

glimpses of this dance in continuous state space of

deterministic systems. The dance is hidden in the theory

propounded by the present author (Rai, 2004; Rai and

Upadhyay, 2006) and is caused by changes in system

parameters, which themselves are governed by determinis-

tic chaos. What the present author means is that the

values of system parameters can be extracted from the

observed time-series, but the uncertainties and their

magnitude differ as we go along. Sources of noise are

many. One of them is ever changing habitat size and density

(Dennis et al., 2003).

Time-invariance principle (TIP) states that a physical law

has the same mathematical form to every independent

choice of observation time. TIP requires that the model of a

real phenomenon be time-independent. This along with

handling-time principle of Holling implies that no ecological

chaos would result if one increases the reproductive rate.

This is contrary to what is believed (Berryman and Millstein,

1989). Deng concludes that chaos which results with

increasing enrichment and efficiency is mathematical not

ecological (Deng, personal communication). If one leaves

out the one-life rule (Deng, 2008) aside, what results can be

termed as ‘paradox of enrichment’ – efficiency to chaos.

None of the discrete models (logistic, Beverton-Holt maps

(B–H maps and others) display ecological chaos. The chaos

observed in these models is mathematical; it does not

correspond to sensitively dependent dynamics of an

ecological system on initial conditions. The B–H map is a

Poincare map derived from the logistic differential equation.

In the light of Deng’s recent work (Deng, 2008; Deng,

personal communication, 2008) it can be concluded that

chaotic oscillations observed in these discrete maps are

mathematical; meaning that they do not represent an

ecological reality.

The possibility pointed out by Bokelman and Deng (2005)

that chaos can result even without subchain oscillations

appear charming. However, a close scrutiny suggests that

these chaotic oscillations are not real. If a food chain model

violates the one-life rule (by not including the density-

dependent per capita death rates), chaos occurs when the

resource is in abundance or the consumers are reproduc-

tively efficient. In fact, such enrichment and efficiency

only make chaos more predominant. On the contrary, if

all species obey the one-life rule, chaos will be eliminated

with enrichment and efficiency. An alternative scenario

would be that chaos is possible only when predators are

not efficient or the prey is in short supply or both of these

exist.

r e f e r e n c e s

Bauch, C.T., Anand, M., 2005. The role of mathematical modelsin ecological restoration and management. Intern. J. Ecol.Environ. 30, 117–122.

Berryman, A.A., Millstein, J.A., 1989. Are ecological systemschaotic—and if not why not? Trends Ecol. Evol. 4, 26–28.

Bokelman, B., Deng, B., 2005. Food web chaos without sub-chainoscillators. Int. J. Bifur. Chaos 15, 3481–3492.

Kendall, B.E., et al., 1999. Why do populations cycle? Asynthesis of mechanistic and statistical modelingapproaches. Ecology 80, 1789–1805.

Cushing, J.M., Constantino, R., Dennis, B., Desharnais, R.,Henson, S., 2002. Chaos in Ecology: Experimental Non-linearDynamics. Academic Press, NY.

Deng, B., 2008. The time-invariance principle, the absence ofecological chaos, and a fundamental pitfall of discretemodeling. Ecol. Model. 215, 287–292.

Dennis, B., et al., 2003. Can noise induce chaos? Oikos 102, 329–339.

Grebogi, C., Feudel, E., 1997. Multistability and control ofcomplexity. Chaos 7, 597–604.

McCann, K., Hastings, A., Gary, R., Huxel, 1998. Weak trophicinteractions and the balance of nature. Nature 345, 794–798.

Polis, G.A., 1998. Stability is woven by complex webs. Nature395, 744–745.

Rai, V., 2004. Chaos in natural populations: edge or wedge? Ecol.Compl. 1, 127–138.

Rai, V., Upadhyay, R.K., 2006. Evolving to the edge of chaos:chance or necessity? Chaos Solitons Fract. 30, 1074–1087.

e c o l o g i c a l c o m p l e x i t y 6 ( 2 0 0 9 ) 1 4 7 – 1 4 9 149

Scheuring, I., Domokos, G., 2007. Only noise can induce chaos indiscrete populations. Oikos 116, 361–366.

Vikas Rai*

Centre for Bio-medical Engineering,

Indian Institute of Technology at Delhi,

Hauz Khas, New Delhi 110 016, India

*Tel.: +91 11 26704774

E-mail address: rvikas41@hotmail.com

Received 8 April 2008

1476-945X/$ – see front matter

# 2008 Elsevier B.V. All rights reserved.

10.1016/j.ecocom.2008.10.007