Chaos theory applied to ecology has shown that chaotic fluctuations in population sizes occur when a population exceeds a certain rate of intrinsic population growth. In nature, populations (metapopulations) are often composed of several to many subpopulations, which are largely (not necessarily completely) segregated from each other. Fuzzy chaos modelling, which considers the dynamics of a metapopulation comprised of many subpopulations, shows that the amplitude of chaotic oscillations is reduced in the metapopulation, even though each subpopulation retains the “normal” degree of chaos. Generally, the greater the number of subpopulations, the smaller the oscillations in the metapopulation. Here we examine whether these findings can be applied to economics. Is a single completely globalized economy perhaps more liable to undergo violent market upheavals than a world economy in which national economies have retained some degree of independence?Chaos theory in biologyMay (e.g. 1975 ) and others after him, using the apparently simple nonlinear equation for population growth xt+1= rxt (1- xt) (where xt= size of population of generation t, xt+1= size of population of generation t+1, r = intrinsic rate of population growth), have shown that chaotic ﬂuctuations in the size of populations arise due to high intrinsic (not affected by environmental influences) rates of population growth r. When population growth is low, populations are at equilibrium, reaching a steady-state with a single value of population size. When reproductive rates exceed 3, there are at first 2 quasi-equilibrium values, followed (with increasing reproductive rates) by 4, 8 and 16 quasi-equilibrium points. At rates exceeding r=3.57, population sizes fluctuate chaotically (Figures 1 and 2). These fluctuations seem to be random, but are in fact strictly deterministic. The fluctuations are highly sensitive to initial conditions, i.e. small differences in the initial population size lead to very large differences in future fluctuations, which makes predictions about the future impossible.
Figure 1: Bifurcation diagram for a single population, showing population sizes x (as the proportion of the crowding level 1) plotted against reproductive rates r. Insets show changes of population sizes of populations with selected reproductive rates plotted against time. Note: the population size is at a stable equilibrium with single values of x until a certain value of r (larger than 3) is reached. Beyond this, populations first alternate between two quasi-equilibrium points, then between 4, 8 and 16. At r=3.57 fluctuations in population size become chaotic. Note: x never exceeds the crowding level 1.
Figure 2. As Figure 1, but shown only for r-values above 3.50.
Fuzzy chaos in ecology
Rohde and Rohde (2001 ), using fuzzy chaos modelling, which sums all values of subpopulation sizes at each particular reproductive rate, have shown that chaotic fluctuations are reduced in metapopulations consisting of several largely independent subpopulations with different reproductive rates, although each single subpopulation still undergoes chaotic fluctuations.
Reduction generally increases with increasing numbers of subpopulations. It decreases when subpopulations are selectively weighted, because a few large subpopulations outweigh a large number of smaller ones. The fact that chaos is reduced in the metapopulation is the effect of how chaos is measured: in the metapopulation, individual values are added and chaos in the resulting net population is necessarily suppressed, in spite of the fact that each subpopulation retains the chaos pattern of a single population. The resulting compression of the chaotic band in the metapopulation is entirely a consequence of the fact that fluctuations in the size of subpopulations are out of phase with each other. An example is given in Figure 3, which shows a bifurcation diagram for a metapopulation consisting of 5000 subpopulations (illustrated only for reproductive rates of r=3.50 and larger). Note that the width (amplitude) of the fluctuations, compared with Figures 1 and 2, is significantly reduced except for a few r-values.
Figure 3. Bifurcation diagram for a metapopulation consisting of 5000 subpopulations shown only for r-values above 3.50. Note: chaotic fluctuations above r=3.57 are still apparent, but much compressed. Strong fluctuations are largely restricted to a few values of r (3.63, 3.74, 3.84).
Application of fuzzy chaos modelling to economics
Can these findings be applied to economics? They certainly suggest that chaotic fluctuations, because they are all in phase (i.e., all troughs occur together, as do the peaks), are much stronger in single large economies or in one completely globalized economy, than in the world economy consisting of national economies that are, to a degree, separated. Because “peaks” and “troughs” of the fluctuations of individual economies are out of phase in the latter case, resources – on a world-wide basis – during a recession (bust) would not be devastated to the same degree as they would be in a single global economy, at least not at the same particular point in time. In a nutshell: imagine a world economy which is completely globalized, without any economic boundaries, and imagine a world-wide economic collapse caused by excessive debts (or some other factors) at the centre of that economy, for example the United States. The social consequences may be disastrous, in particular for poorer countries, much more so than in a situation where national economies would have retained some degree of independence and are therefore less affected by a simultaneous collapse. Naturally, booms in a fragmented world economy also would be reduced relative to those in a single globalized economy.Of course, one can only speculate how such segregation could be maintained. Would the existence of different financial systems in the world (such as the traditional Western capitalist system and the Islamic finance system) perhaps be useful? After all, during the Asian financial crisis in the 1990’s Malaysia escaped the worst consequences of the collapse because it sought refuge in Islamic financing, thus isolating itself from the Western banking system and particular the IMF. Reasons inherent in the philosophy behind Islamic financing may be partly responsible, but in our context it is important that two different systems existed which were responsible for a “decoupling” of the economies.- On a smaller scale: would the breaking up of the few oversized financial institutions (banks) into smaller units, as recently suggested by Paul Volcker, the former Chairman of the Federal Reserve, perhaps help in avoiding future large-scale collapses ? Serious errors committed by managers of one bank would not affect other banks, at least if the break-up would go hand in hand with the introduction of regulations that ensures a certain degree of segregation in the banks’ practices. After all, larger is not always better. Perhaps it is time to have another look at Schumacher’s (1973) classic: Small is Beautiful: Economics As If People Mattered . Possible objectionsIt is important to note that in the fuzzy chaos model, individual subpopulations (or national economies) still exhibit the usual degree of chaos, even though the metapopulation (or world economy) shows reduced chaos. This raises the question as to whether the chaos exhibited by the net world economy is a valid metric to use. One could argue that what matters to the individual in the world economy is not how chaotic the sum of all economies is, but how much chaos they are subjected to. – An objection to this objection is: it all depends on the mechanisms available for transferring funds (“help”) from one economy to another. In a single globalized economy all individual economies would be in a bust (and boom) at the same point of time, and transfer between economies in a bust would be severely limited. In economies at different stages of the cycle, transfer should at least in principle be possible, if appropriate mechanisms were developed. – This corresponds to what we find in nature. In species of animals and plants threatened by extinction, those consisting of metapopulations with many subpopulations have a greater chance of survival than species with a single population. In times of severe crisis, some subpopulations of the former will disappear, but their habitats can be resettled by a few individuals from other habitats, because subpopulations are never completely isolated from each other. In the latter, once the population is gone, it cannot recover. Another possible objection is that all this is an exercise in theory, but that it will be impossible to change the world’s economic system in such a way that effective interactions between economies will be, to a degree, prevented. However, there may be cultural and environmental reasons in addition to economic ones, which may make it desirable to maintain a certain degree of separation between national identities, enhancing our motivation for introducing appropriate measures. At this stage we do not know how a total globalization would affect cultural diversity and the environment, both essential for humanity’s survival. South Korea’s plan to lease 1 1/2 million hectares of land in Madagascar for growing corn to be exported to South Korea, may be an indication of things to come. Madagaskar has one of the most diverse ecosystems on Earth and the consequences of such huge monocultures would be disastrous. Other examples, already existent now, are the vast palmoil plantations in Borneo and the huge cattle ranches in the Amazon. Borneo and the Amazon have the Earth’s richest and second richest tropical rainforests, respectively.
I realize that this approach is highly unconventional and may not be applicable to economics. Nevertheless, it may induce people to think the problem over, perhaps initiating a discussion on this site.
1. May, R.M. 1975. Deterministic models with chaotic dynamics. Nature 261, 903-910.
2. Rohde, K. and Rohde, P.P. 2001. Fuzzy chaos: reduced chaos in the combined dynamics of several independently chaotic populations. American Naturalist 158, 553-556.
3. Schumacher, E. F. 1973. Small Is Beautiful: Economics As If People Mattered. Harper & Row. N.Y.
I thank Peter Rohde for many very helpful comments. Peter was involved in developing the fuzzy chaos model and wrote all the programs for it.
© Klaus Rohde