Chaotic Populations Research Articles

Delayed female reproduction in equilibrium and chaotic populations

Behavioural and life history polymorphisms are often observed in animal populations. We analyse the timing of maturation and reproduction in risky and resource-limited environments. Field and laboratory evidence suggests that female voles and mice, for example, can adjust their breeding according to the level of risk to their own survival and to survival probabilities and recruitment of young produced under different environmental conditions. Under risky or harsh conditions breeding can be postponed until later in the current breeding season or even to the next breeding season. We develop a population dynamics and life history model for polymorphism in reproduction (co-existence of breeding and non-breeding behaviours) of females in an age-structured population, with two temporally distinct mating events within the breeding season. We assume that, after overwintering, the females can breed in spring and again in summer or they can delay breeding in spring and breed in summer only. Young females born in spring can either mature and breed in summer or stay immature and postpone breeding over the winter to the next breeding season. We show that an evolutionarily stable breeding strategy is either an age-structured combination of pure breeding behaviours (old females breed and young delay maturity) or a mixed breeding behaviour within age-classes (a fraction of females breed and the rest of the age class postpones breeding). Co-occurrence of mixed reproductive behaviour in spring and summer within a single breeding season is observed in fluctuating populations only. The reproductive patterns depend on intraspecific, possibly interspecific, and ecological factors. The density dependence (e.g. social suppression) and predation risk are shown to be possible evolutionary mechanisms in adjusting the relative proportions of the different but co-existing reproductive behaviours.

Lyapunov Exponents and the Mathematics of Invasion in Oscillatory or Chaotic Populations

This paper concisely reviews the mathematical properties of the dominant Lyapunov exponent of a matrix sequence in the context of population biology. The concept of Lyapunov exponent provides a valuable tool for investigating processes of invasion in ecology or genetics, which are crucial in shaping community diversity, determining the spread of epidemics or the fixation of a new mutation. The appeal of the invasibility criterion based on the dominant Lyapunov exponent lies in the opportunity it offers to deal with population structure, complex life cycles, and complex population dynamics resulting from the model nonlinearities (oscillations, chaos), as well as random fluctuations arising from a stochastic environment. We put emphasis on the issues of the existence, numerical approximation, and regularity of the dominant Lyapunov exponent. Our presentation is aimed at showing that, despite our inability to compute the exponent analytically, which adds to its high intrinsic instability, important biological insights can nevertheless be achieved at the cost of fairly mild assumptions on the features of the models considered.

The coupled logistic map: a simple model for the effects of spatial heterogeneity on population dynamics

A simple model consisting of two diffusively coupled logistic maps is used to examine the effects of spatial heterogeneity on population dynamics. Examining the dynamic behaviour of the model using numerical methods, a wide range of behaviours are observed. It is shown that the coupling can stabilize individually chaotic populations, and that under different circumstances the coupling can cause individually stable periodic populations to undergo more complex behaviour. The system exhibits multiple attractors when the qualitative dynamics of the system depend on the initial conditions. Sudden changes in the structure of the attractors are seen for small changes in the parameters; these changes are known as crises. Transient and intermittent behaviours are observed. The implications of all these behaviours for populations are discussed. A linear coupling of the two maps is also considered which leads to counterintuitive behaviour, with chaotic dynamics being obtained by coupling two stable maps. This behaviour occurs because the coupling involves mixing of generations, and is therefore biologically unrealistic.

Local and Ensemble Dynamics of Linked Populations

In a recent issue of the Journal of Animal Ecology, Bascompte & Sole (1994) investigated the effect of introducing immigration between a number of populations each represented by a simple one-step map. They conclude that this linkage generally serves to destabilize individual populations, increasing their propensity to exhibit chaotic fluctuations, but tending to reduce the variance of the ensemble. Although not mentioned by Bascompte & Sole, there have been several other recent papers on linkage between potentially chaotic populations (e.g. McCallum 1992; Gonzalez-Andujar & Perry 1993a,b; Stone 1993). These papers suggest that linkage between simple maps tends to decrease the likelihood of chaotic fluctuations, rather than increase it, as Bascompte & Sole suggest. These studies use a smaller number of populations than considered by Bascompte & Sole or simply consider immigration to be a time-independent constant. Even so, they suggest that Bascompte & Soles conclusion about local dynamics should be tempered. While linkage does have an effect on local dynamics, it is not clear that there is a general rule governing the nature of this effect. For the simplest case, where immigration is a constant, Ruxton (1993) provides an example where introducing low levels of immigration can induce chaos in a model population which would show a regular cycle if isolated. However, higher levels of immigration recover a simple cycle again. Bascompte & Sole also conclude that ensemble dynamics are 'stabilized' by the addition of linkage, in that the variance decreases and the resultant total population resembles a steady state with added noise. This appears at odds with my own observation of 10 linked maps where the global dynamics commonly appear as a simple cycle (Ruxton 1994). However, a reconciliation of these two, apparently contradictory, observations may be found in the work of de Roos, McCauley & Wilson (1991) who considered a predator-prey interaction on a discrete lattice. If movement takes place only between adjacent sites then they observed a noisy signal around a roughly constant population. However, when all lattice sites were equally available for movement, then large amplitude cycles were obtained. A similar situation appears to apply to our linked populations. I allow dispersal to all populations equally and obtain regular cycles whereas Bascompte and Sole allow movement only between neighbouring populations and obtain a 'noisy equilibrium'. Hence, although Bascompte and Sole are correct in concluding that the number of subpopulations has an important effect on the behaviour of the ensemble, the degree of linkage between individual populations should also be emphasized as an equally important ecological parameter. One of the most developed areas for exploration of the way linkage between populations affects dynamics lies in the field of human childhood diseases. Here, in contrast to most other areas, firmly grounded theoretical models can be combined with a considerable amount of data (e.g. Sugihara, Grenfell & May 1990; Grenfell 1992).

Low levels of immigration between chaotic populations can reduce system extinctions by inducing asynchronous regular cycles

Populations whose behaviour is described as chaotic commonly fall to very low values. Hence such populations have a high risk of extinction. However, recent research has shown that the probability of extinction for a system of linked populations is reduced when parameters are chosen such that the populations would behave chaotically in isolation. This effect is suggested to be the result of desynchronization between populations due to the amplification of external noise. Here, I investigate the effect on extinction risk of introducing weak coupling between chaotic populations in the absence of such external noise. Populations in the weakly linked system often do not behave chaotically even when parameter values are such that they would do so if isolated. Instead, they can exhibit simple cycles or remain at a stable level, sometimes with a reduced probability of falling to low population levels. More importantly, although subsets of the group of populations often follow the same dynamic pattern, all the populations never become synchronized, hence recolonization following extinction of a subset of the populations is a common occurrence.

The Effect of Dispersal between Chaotic and Non-Chaotic Populations within a Metapopulation

The Effect of Dispersal between Chaotic and Non-Chaotic Populations within a Metapopulation

The Dynamics of Host-Parasitoid-Pathogen Interactions

The growing importance of pathogens in biological control warrants an investigation into the likely effects of disease on the widely successful class of biological-control agents, the parasitoid. Our models consider a parasitoid that can exhibit a range of attack behaviors ranging from clumped to random. The pathogen is assumed to be transmitted by external stages that can span periods when the host is rare or absent. When these two natural enemies occur simultaneously in a given host, the result can be the elimination of one enemy by the competitor in all coinfected hosts or it can be some intermediate outcome. We show through an analysis of invasion that both natural enemies may coexist with constant, cyclic, or chaotic populations. Other outcomes are also possible. One enemy may exclude its competitor either unilaterally or on the basis of the initial densities of the host and the competing natural enemies. The particular outcome depends on four main factors: (1) the finite rate of increase of the host, (2) the clumping of parasitoid (and pathogen) attacks, (3) the relative extrinsic potentials of the competitors, and (4) the relative intrinsic potentials of the competitors. We conclude that although situations may arise in which the host is best controlled by only one natural enemy, multiple introductions are likely to be a sound strategy for a wide variety of systems.

Managing unpredictable resources: Traditional policies applied to chaotic populations

Conventional theory for the management of living ocean resources assumes a predictable link between current management actions and the future state of managed populations. As a practical matter, however, it is very hard to establish this kind of predictable relationship. It is possible that the dynamics of these populations exhibit chaotic variation. This paper addresses the question of appropriate management policies in a regime characterized by chaotic population dynamics. The problem is approached through a bioeconomic simulator that has chaotic properties. With light fishing, policies that alter the conditions of fishing perform better than policies dependent upon population predictions.

On the resolution of chaos in population models

Some simple equations representing the dynamics of a single population over time are known to be capable of generating extremely complicated behavior. This complicated behavior has been termed chaotic (Li and Yorke, 1975; May, 1974, 1975; May and Oster, 1976; Guckenheimer et al., 1976). The practical consequence of chaotic behavior is a certain inability to describe particular population trajectories with any degree of accuracy. The purpose of this communication is to demonstrate that for certain models, when properly viewed, extremely chaotic populations can behave in very regular and predictable ways, similar to that noted by May (1975). Furthermore, it appears that, in a sense, the more chaotic a population, the more predictable its behavior, at least for a certain class of models.