Chaotic attractor in two-prey one-predator system originates from interplay of limit cycles

We investigate the appearance of chaos in a microbial 3-species model motivated by a potentially chaotic real world system (as characterized by positive Lyapunov exponents (Becks et al., Nature 435, 2005). This is the first quantitative model that simulates characteristic population dynamics in the system. A striking feature of the experiment was three consecutive regimes of limit cycles, chaotic dynamics and a fixed point. Our model reproduces this pattern. Numerical simulations of the system reveal the presence of a chaotic attractor in the intermediate parameter window between two regimes of periodic coexistence (stable limit cycles). In particular, this intermediate structure can be explained by competition between the two distinct periodic dynamics. It provides the basis for stable coexistence of all three species: environmental perturbations may result in huge fluctuations in species abundances, however, the system at large tolerates those perturbations in the sense that the population abundances quickly fall back onto the chaotic attractor manifold and the system remains. This mechanism explains how chaos helps the system to persist and stabilize against migration. In discrete populations, fluctuations can push the system towards extinction of one or more species. The chaotic attractor protects the system and extinction times scale exponentially with system size in the same way as with limit cycles or in a stable situation.     

Effect of immigration on a chaotic insect population

Recently, the most convincing evidence of complex dynamics and chaos in biological populations has been presented for Tribolium castaneaum, a classic laboratory model insect. In this note, the robustness of this system is investigated and a constant immigration term is added to the adult population equation. It has been found that such perturbation to the model can either have a complicating effect (when the isolated system is periodic) or a simplifying one (when the system is chaotic in isolation).     

Density dynamics of diverse Spiroplasma strains naturally infecting different species of Drosophila

Facultative heritable bacterial endosymbionts can have dramatic effects on their hosts, ranging from mutualistic to parasitic. Within-host bacterial endosymbiont density plays a critical role in maintenance of a symbiotic relationship, as it can affect levels of vertical transmission and expression of phenotypic effects, both of which influence the infection prevalence in host populations. Species of genus Drosophila are infected with Spiroplasma, whose characterized phenotypic effects range from that of a male-killing reproductive parasite to beneficial defensive endosymbiont. For many strains of Spiroplasma infecting at least 17 species of Drosophila, however, the phenotypic effects are obscure. The infection prevalence of these Spiroplasma vary within and among Drosophila species, and little is known about the within-host density dynamics of these diverse strains. To characterize the patterns of Spiroplasma density variation among Drosophila we used quantitative PCR to assess bacterial titer at various life stages of three species of Drosophila naturally-infected with two different types of Spiroplasma. For naturally infected Drosophila species we found that non-male-killing infections had consistently lower densities than the male-killing infection. The patterns of Spiroplasma titer change during aging varied among Drosophila species infected with different Spiroplasma strains. Bacterial density varied within and among populations of Drosophila, with individuals from the population with the highest prevalence of infection having the highest density. This density variation underscores the complex interaction of Spiroplasma strain and host genetic background in determining endosymbiont density.     

Coexistence of two types on a single resource in discrete time

Armstrong and McGehee (1980) have shown that two species modeled in continuous time can coexist on a single resource provided that one species oscillates autonomously. This paper demonstrates the parallel result in discrete time. I consider a deterministic model of two asexual types in a single patch competing for a single resource, and show that such systems generically produce oscillatory coexistence or bistability if one of the types displays periodic or chaotic behavior in isolation. The conditions for coexistence or bistability are derived in terms of the convexity of the functions describing fitness as a function of resource availability. I also analyze whether or not a stable type, a type with a stable equilibrium population size when considered in isolation, can invade a periodic orbit of an unstable type, and show that the same convexity condition distinguishes these two cases. The widely considered exponential or Ricker model for population dynamics lies on the boundary between the two cases and is highly degenerate in this context.     

Density-dependent birth rate, birth pulses and their population dynamic consequences

 In most models of population dynamics, increases in population due to birth are assumed to be time-independent, but many species reproduce only during a single period of the year. We propose a single-species model with stage structure for the dynamics in a wild animal population for which births occur in a single pulse once per time period. Using the discrete dynamical system determined by the stroboscopic map, we obtain an exact periodic solution of systems which are with Ricker functions or Beverton-Holt functions, and obtain the threshold conditions for their stability. Above this threshold, there is a characteristic sequence of bifurcations, leading to chaotic dynamics, which implies that the dynamical behaviors of the single species model with birth pulses are very complex, including small-amplitude annual oscillations, large-amplitude multi-annual cycles, and chaos. This suggests that birth pulse, in effect, provides a natural period or cyclicity that allows for a period-doubling route to chaos. Received: 13 June 2001 / Revised version: 7 September 2001 / Published online: 8 February 2002     

QUANTUM AND CONTINUOUS EVOLUTION OF DNA BASE COMPOSITION IN THE YEAST GENUS PICHIA

This paper investigates the noncontinuous nature and evolution of the base composition of nuclear DNA (expressed as mol% guanine + cytosine) in species of the yeast genus Pichia (sensu Kurtzman, 1984b). The pattern of change in the G + C contents in species of this genus, which range from about 27 to 52 mol%, was evaluated. When specifically those species of Pichia were analyzed that have evolved in necroses of cactus species and associated Drosophila, a periodic change in the G + C contents of approximately 3.0–3.2 mol% was detected by a “bootstrapping” method, Fourier analysis, and a nonlinear trigonometric model. Pichia species occurring in exudates of broad-leaved deciduous trees or associated Drosophila and substrates such as soil and water (“other”) showed a periodicity of 2.5–2.6 mol%, whereas species associated with conifers and associated bark beetles showed no significant periodicity. Periodicity in the most recent association (cactus and resident Drosophila) as compared to the lack of periodicity in the oldest association (conifer-beetle) may indicate mixed evolutionary processes. Low mol% G + C values appear more frequently in the relatively recent cactus and Drosophila-associated yeast species. In addition, low mol% G + C species do not display the ancestral bud-meiosis mode of sexual reproduction which occurs frequently in medium to high mol% G + C yeasts. It was found that the mol% G + C content of the Drosophila- and cactus-associated Pichia species is positively correlated with the number of compounds fermented or respired by these yeast species. Possible reasons for the periodic changes in mol% G + C content accompanying speciation include aneuploidy, allopolyploidy, the presence of nuclear plasmids, and regular differences in moderately repetitive portions of DNA. Since significant DNA complementarity is virtually limited to species within a relatively narrow G + C group, this suggests that there are at least two processes which alter the G + C content between species, one saltational and one continuous.     

Unmasking Chaotic Attributes in Time Series of Living Cell Populations

Background

Long-range oscillations of the mammalian cell proliferation rate are commonly observed both in vivo and in vitro. Such complicated dynamics are generally the result of a combination of stochastic events and deterministic regulation. Assessing the role, if any, of chaotic regulation is difficult. However, unmasking chaotic dynamics is essential for analysis of cellular processes related to proliferation rate, including metabolic activity, telomere homeostasis, gene expression, and tumor growth.

Methodology/Principal Findings

Using a simple, original, nonlinear method based on return maps, we previously found a geometrical deterministic structure coordinating such fluctuations in populations of various cell types. However, nonlinearity and determinism are only necessary conditions for chaos; they do not by themselves constitute a proof of chaotic dynamics. Therefore, we used the same analytical method to analyze the oscillations of four well-known, low-dimensional, chaotic oscillators, originally designed in diverse settings and all possibly well-adapted to model the fluctuations of cell populations: the Lorenz, Rössler, Verhulst and Duffing oscillators. All four systems also display this geometrical structure, coordinating the oscillations of one or two variables of the oscillator. No such structure could be observed in periodic or stochastic fluctuations.

Conclusion/Significance

Theoretical models predict various cell population dynamics, from stable through periodically oscillating to a chaotic regime. Periodic and stochastic fluctuations were first described long ago in various mammalian cells, but by contrast, chaotic regulation had not previously been evidenced. The findings with our nonlinear geometrical approach are entirely consistent with the notion that fluctuations of cell populations can be chaotically controlled.     

Independent origins of resistance or susceptibility of parasitic wasps to a defensive symbiont

Insect microbe associations are diverse, widespread, and influential. Among the fitness effects of microbes on their hosts, defense against natural enemies is increasingly recognized as ubiquitous, particularly among those associations involving heritable, yet facultative, bacteria. Protective mutualisms generate complex ecological and coevolutionary dynamics that are only beginning to be elucidated. These depend in part on the degree to which symbiont‐mediated protection exhibits specificity to one or more members of the natural enemy community. Recent findings in a well‐studied defensive mutualism system (i.e., aphids, bacteria, parasitoid wasps) reveal repeated instances of evolution of susceptibility or resistance to defensive bacteria by parasitoids. This study searched for similar patterns in an emerging model system for defensive mutualisms: the interaction of Drosophila, bacteria in the genus Spiroplasma, and wasps that parasitize larval stages of Drosophila. Previous work indicated that three divergent species of parasitic wasps are strongly inhibited by the presence of Spiroplasma in three divergent species of Drosophila, including D. melanogaster. The results of this study uncovered two additional wasp species that are susceptible to Spiroplasma and two that are unaffected by Spiroplasma, implying at least two instances of loss or gain of susceptibility to Spiroplasma among larval parasitoids of Drosophila.     

Modelling the local dynamics of the zebra mussel (Dreissena polymorpha)

1. The bivalve Dreissena polymorpha has invaded many freshwater ecosystems worldwide in recent decades. Because of their high fecundity and ability to settle on almost any solid substratum, zebra mussels usually outcompete the resident species and cause severe damage to waterworks. Time series of D. polymorpha densities display a variety of dynamical patterns, including very irregular behaviours. Unfortunately, there is a lack of mathematical modelling that could explain these patterns. 2. Here, we propose a very simple discrete‐time population model with age structure and density dependence that can generate realistic dynamics. Most of the model parameters can be derived from existing data on D. polymorpha. Some of them are quite variable: with respect to these we perform a sensitivity analysis of the model behaviour and verify that non‐equilibrial regimes (either periodic or chaotic) are the rule rather than the exception. 3. Even in circumstances where the model dynamics are aperiodic it is possible to predict total density peaks from previous peaks. This turns out to be true also in the presence of environmental stochasticity. 4. Using the stochastic model we explore the effects of age‐selective predation. Quite surprisingly, larger removal rates of adults do not always result in smaller population densities and mussel biomasses. Moreover, non‐selective predation can result in skewed size‐frequency distributions which, therefore, are not necessarily the footprint of predators’ preference for larger or smaller zebra mussels.     

A one‐locus model of androdioecy with two homomorphic self‐incompatibility groups: expected vs. observed male frequencies

Androdioecy, the occurrence of males and hermaphrodites in a single population, is a rare breeding system because the conditions for maintenance of males are restrictive. In the androdioecious shrub Phillyrea angustifolia, high male frequencies are observed in some populations. The species has a sporophytic self‐incompatibility (SI) system with two self‐incompatibility groups, which ensures that two groups of hermaphrodites can each mate only with the other group, whereas males can fertilize hermaphrodites of both groups. Here, we analyse a population genetic model to investigate the dynamics of such an androdioecious species, assuming that self‐incompatibility and sex phenotypes are determined by a single locus. Our model confirms a previous prediction that a slight reproductive advantage of males relative to hermaphrodites allows the maintenance of males at high equilibrium frequencies. The model predicts different equilibria between hermaphrodites of the two SI groups and males, depending on the male advantage, the initial composition of the population and the population size, whose effect is studied through stochastic simulations. Although the model can generate high male frequencies, observed frequencies are considerably higher than the model predicts. We finally discuss how this model may help explain the large male frequency variation observed in other androdioecious species of Oleaceae: some species show only androdioecious populations, as P. angustifolia, whereas others show populations either completely hermaphrodite or androdioecious.     

Complex dynamics of microbial competition in the gradostat

We examine the conditions necessary for the emergence of complex dynamic behavior in systems of microbial competition. In particular, we study the effect of spatial heterogeneity and substrate-inhibition on the dynamics of such a system. This is accomplished through the study of a mathematical model of two microbial populations competing for a single nutrient in a configuration of two interconnected chemostats. Microbial growth is assumed to follow substrate-inhibited kinetics for both species. Such a system with sterile feed has been shown in a previous work to exhibit stable periodic states. In the present work we study the system for the case of non-sterile feed, i.e., when the two species are present in the feed of the chemostats. The analysis is done by numerical bifurcation theory methods. We demonstrate that, in addition to periodic states, the system possesses stable quasi-periodic states resulting from Neimark-Sacker bifurcations of limit cycles. Also, periodic states may undergo successive period doublings leading to periodic states of increasing period and indicating that chaotic states might be possible. Multistability is also observed, consisting in the coexistence of several stable steady states and possibly stable periodic or quasi-periodic states for given operating conditions. It appears that substrate-inhibition, spatial heterogeneity and presence of microorganisms in the inflow are all necessary conditions for complex dynamics to arise in a microbial system of pure and simple competition.     

Eco-genetics of desiccation resistance in Drosophila

Climate change globally perturbs water circulation thereby influencing ecosystems including cultivated land. Both harmful and beneficial species of insects are likely to be vulnerable to such changes in climate. As small animals with a disadvantageous surface area to body mass ratio, they face a risk of desiccation. A number of behavioural, physiological and genetic strategies are deployed to solve these problems during adaptation in various Drosophila species. Over 100 desiccation-related genes have been identified in laboratory and wild populations of the cosmopolitan fruit fly Drosophila melanogaster and its sister species in large-scale and single-gene approaches. These genes are involved in water sensing and homeostasis, and barrier formation and function via the production and composition of surface lipids and via pigmentation. Interestingly, the genetic strategy implemented in a given population appears to be unpredictable. In part, this may be due to different experimental approaches in different studies. The observed variability may also reflect a rich standing genetic variation in Drosophila allowing a quasi-random choice of response strategies through soft-sweep events, although further studies are needed to unravel any underlying principles. These findings underline that D. melanogaster is a robust species well adapted to resist climate change-related desiccation. The rich data obtained in Drosophila research provide a framework to address and understand desiccation resistance in other insects. Through the application of powerful genetic tools in the model organism D. melanogaster, the functions of desiccation-related genes revealed by correlative studies can be tested and the underlying molecular mechanisms of desiccation tolerance understood. The combination of the wealth of available data and its genetic accessibility makes Drosophila an ideal bioindicator. Accumulation of data on desiccation resistance in Drosophila may allow us to create a world map of genetic evolution in response to climate change in an insect genome. Ultimately these efforts may provide guidelines for dealing with the effects of climate-related perturbations on insect population dynamics in the future.     

Dynamics of a physiologically structured population in a time-varying environment

Physiologically structured population models have become a valuable tool to model the dynamics of populations. In a stationary environment such models can exhibit equilibrium solutions as well as periodic solutions. However, for many organisms the environment is not stationary, but varies more or less regularly. In order to understand the interaction between an external environmental forcing and the internal dynamics in a population, we examine the response of a physiologically structured population model to a periodic variation in the food resource. We explore the addition of forcing in two cases: (A) where the population dynamics is in equilibrium in a stationary environment, and (B) where the population dynamics exhibits a periodic solution in a stationary environment. When forcing is applied in case A, the solutions are mainly periodic. In case B the forcing signal interacts with the oscillations of the unforced system, and both periodic and irregular (quasi-periodic or chaotic) solutions occur. In both cases the periodic solutions include one and multiple period cycles, and each cycle can have several reproduction pulses.     

The effect of seasonal harvesting on stage-structured population models

In most models of population dynamics, increases in population due to birth are assumed to be time-independent, but many species reproduce only during a single period of the year. We propose an exploited single-species model with stage structure for the dynamics in a fish population for which births occur in a single pulse once per time period. Since birth pulse populations are often characterized with a discrete time dynamical system determined by its Poincaré map, we explore the consequences of harvest timing to equilibrium population sizes under seasonal dependence and obtain threshold conditions for their stability, and show that the timing of harvesting has a strong impact on the persistence of the fish population, on the volume of mature fish stock and on the maximum annual-sustainable yield. Moreover, our results imply that the population can sustain much higher harvest rates if the mature fish is removed as early in the season (after the birth pulse) as possible. Further, the effects of harvesting effort and harvest timing on the dynamical complexity are also investigated. Bifurcation diagrams are constructed with the birth rate (or harvesting effort or harvest timing) as the bifurcation parameter, and these are observed to display rich structure, including chaotic bands with periodic windows, pitch-fork and tangent bifurcations, non-unique dynamics (meaning that several attractors coexist) and attractor crisis. This suggests that birth pulse, in effect, provides a natural period or cyclicity that makes the dynamical behavior more complex.This work is supported by National Natural Science Foundation of China (10171106)     

The dynamics of two diffusively coupled predator-prey populations

I analyze the dynamics of predator and prey populations living in two patches. Within a patch the prey grow logistically and the predators have a Holling type II functional response. The two patches are coupled through predator migration. The system can be interpreted as a simple predator-prey metapopulation or as a spatially explicit predator-prey system. Asynchronous local dynamics are presumed by metapopulation theory. The main question I address is when synchronous and when asynchronous dynamics arise. Contrary to biological intuition, for very small migration rates the oscillations always synchronize. For intermediate migration rates the synchronous oscillations are unstable and I found periodic, quasi-periodic, and intermittently chaotic attractors with asynchronous dynamics. For large predator migration rates, attractors in the form of equilibria or limit cycles exist in which one of the patches contains no prey. The dynamical behavior of the system is described using bifurcation diagrams. The model shows that spatial predator-prey populations can be regulated through the interplay of local dynamics and migration.     

Multiple attractors in stage-structured population models with birth pulses

In most models of population dynamics, increases in population due to birth are assumed to be time-independent, but many species reproduce only during a single period of the year. A single species stage-structured model with density-dependent maturation rate and birth pulse is formulated. Using the discrete dynamical system determined by its Poincaré map, we report a detailed study of the various dynamics, including (a) existence and stability of nonnegative equilibria, (b) nonunique dynamics, meaning that several attractors coexist, (c) basins of attraction (defined as the set of the initial conditions leading to a certain type of attractor), (d) supertransients, and (e) chaotic attractors. The occurrence of these complex dynamic behaviour is related to the fact that minor changes in parameter or initial values can strikingly change the dynamic behaviours of system. Further, it is shown that periodic birth pulse, in effect, provides a natural period or cyclicity that allows multiple oscillatory solutions in the continuous dynamical systems.     

Population dynamics of three songbird species in a nestbox population in Central Europe show effects of density,climate and competitive interactions

Unravelling the contributions of density‐dependent and density‐independent factors in determining species population dynamics is a challenge, especially if the two factors interact. One approach is to apply stochastic population models to long‐term data, yet few studies have included interactions between density‐dependent and density‐independent factors, or explored more than one type of stochastic population model. However, both are important because model choice critically affects inference on population dynamics and stability. Here, we used a multiple models approach and applied log‐linear and non‐linear stochastic population models to time series (spanning 29 years) on the population growth rates of Blue Tits Cyanistes caeruleus, Great Tits Parus major and Pied Flycatchers Ficedula hypoleuca breeding in two nestbox populations in southern Germany. We focused on the roles of climate conditions and intra‐ and interspecific competition in determining population growth rates. Density dependence was evident in all populations. For Blue Tits in one population and for Great Tits in both populations, addition of a density‐independent factor improved model fit. At one location, Blue Tit population growth rate increased following warmer winters, whereas Great Tit population growth rates decreased following warmer springs. Importantly, Great Tit population growth rate also decreased following years of high Blue Tit abundance, but not vice versa. This finding is consistent with asymmetric interspecific competition and implies that competition could carry over to influence population dynamics. At the other location, Great Tit population growth rate decreased following years of high Pied Flycatcher abundance but only when Great Tit population numbers were low, illustrating that the roles of density‐dependent and density‐independent factors are not necessarily mutually exclusive. The dynamics of this Great Tit population, in contrast to the other populations, were unstable and chaotic, raising the question of whether interactions between density‐dependent and density‐independent factors play a role in determining the (in) stability of the dynamics of species populations.     

Spatio-temporal pattern formation in Rosenzweig–MacArthur model: Effect of nonlocal interactions

Spatio-temporal pattern formation in reaction–diffusion models of interacting populations is an active area of research due to various ecological aspects. Instability of homogeneous steady-states can lead to various types of patterns, which can be classified as stationary, periodic, quasi-periodic, chaotic, etc. The reaction–diffusion model with Rosenzweig–MacArthur type reaction kinetics for prey–predator type interaction is unable to produce Turing patterns but some non-Turing patterns can be observed for it. This scenario changes if we incorporate non-local interactions in the model. The main objective of the present work is to reveal possible patterns generated by the reaction–diffusion model with Rosenzweig–MacArthur type prey–predator interaction and non-local consumption of resources by the prey species. We are interested in the existence of Turing patterns in this model and in the effect of the non-local interaction on the periodic travelling wave and spatio-temporal chaotic patterns. Global bifurcation diagrams are constructed to describe the transition from one pattern to another one.     

Evolutionary Stable Strategies and Trade-Offs in Generalized Beverton and Holt Growth Models

A generalized Beverton–Holt model is considered in which a parameterγcharacterizes the onset of density dependence. An evolutionary stable strategy analysis of this parameter, reported in Getz (1996), is developed further here, using invasion exponents and the strategy dynamics of Vincentet al.(1993). The parameterγis also allowed to be density dependent, and it is shown that the most successful strategies of this type are those for whichγis large for low densities and close to its minimum for high densities. A biological interpretation is given in the context of mobile females depositing their relatively sessile young on patches of resource, namely, females should overdisperse their young on resources when adult densities are high and underdisperse them when these densities are low. Finally the per capita growth rate parameter is also allowed to depend onγ. It is shown that this dependence provides a mechanism by which periodic or chaotic attractor dynamics could evolve towards equilibrium attractor dynamics.