Chaotic flow: the physics of species coexistence

Fractals and Chaos in the Hemodynamics of Intracranial Aneurysms

Computing the emerging flow in blood vessel sections by means of computational fluid dynamics is an often applied practice in hemodynamics research. One particular area for such investigations is related to the cerebral aneurysms, since their formation, pathogenesis, and the risk of a potential rupture may be flow-related. We present a study on the behavior of small advected particles in cerebral vessel sections in the presence of aneurysmal malformations. These malformations cause strong flow disturbances driving the system toward chaotic behavior. Within these flows, the particle trajectories can form a fractal structure, the properties of which are measurable by quantitative techniques. The measurable quantities are well established chaotic properties, such as the Lyapunov exponent, escape rate, and information dimension. Based on these findings, we propose that chaotic flow within blood vessels in the vicinity of the aneurysm might be relevant for the pathogenesis and development of this malformation.

Coalescent dynamics of planktonic communities

Planktonic communities are extremely diverse and include a vast number of rare species. The dynamics of these rare species is best described by individual-based models. However, individual-based approaches to planktonic diversity face substantial difficulties, due to the large number of individuals required to make realistic predictions. In this paper, we study the diversity of planktonic communities by means of a spatial coalescence model that incorporates transport by oceanic currents. As a main advantage, our approach requires simulating a number of individuals equal to the size of the sample one is interested in, rather than the size of the entire community. By theoretical analysis and simulations, we explore the conditions upon which our coalescence model is equivalent to individual-based dynamics. As an application, we use our model to predict the impact of chaotic advection by oceanic currents on biodiversity. We conclude that the coalescent approach permits one to simulate marine microbial communities much more efficiently than with individual-based models.

Limited Pairwise Synergistic and Antagonistic Interactions Impart Stability to Microbial Communities

One of the central goals of ecology is to explain and predict coexistence of species. In this context, microbial communities provide a model system where community structure can be studied in environmental niches and in laboratory conditions. A community of microbial population is stabilized by interactions between participating species. However, the nature of these stabilizing interactions has remained largely unknown. Theory and experiments have suggested that communities are stabilized by antagonistic interactions between member species, and destabilized by synergistic interactions. However, experiments have also revealed that a large fraction of all the interactions between species in a community are synergistic in nature. To understand the relative significance of the two types of interactions (synergistic vs. antagonistic) between species, we perform simulations of microbial communities with a small number of participating species using two frameworks—a replicator equation and a Lotka-Volterra framework. Our results demonstrate that synergistic interactions between species play a critical role in maintaining diversity in cultures. These interactions are critical for the ability of the communities to survive perturbations and maintain diversity. We follow up the simulations with quantification of the extent to which synergistic and antagonistic interactions are present in a bacterial community present in a soil sample. Overall, our results show that community stability is largely achieved with the help of synergistic interactions between participating species. However, we perform experiments to demonstrate that antagonistic interactions, in specific circumstances, can also contribute toward community stability.

Fitness optimization and evolution of permanent replicator systems

In this paper, we discuss fitness landscape evolution of permanent replicator systems applying the hypothesis that the specific time of evolutionary adaptation of system parameters is much slower than the time of internal evolutionary dynamics. In other words, we suppose that the extremal principle of Darwinian evolution based on Fisher's fundamental theorem of natural selection is valid for the steady-states of permanent replicator systems. Various cases illustrating this concept are considered.

Transport of blood particles: Chaotic advection even in a healthy scenario

We study the advection of blood particles in the carotid bifurcation, a site that is prone to plaque development. Previously, it has been shown that chaotic advection can take place in blood flows with diseases. Here, we show that even in a healthy scenario, chaotic advection can take place. To understand how the particle dynamics is affected by the emergence and growth of a plaque, we study the carotid bifurcation in three cases: a healthy bifurcation, a bifurcation with a mild stenosis, and the another with a severe stenosis. The result is non-intuitive: there is less chaos for the mild stenosis case even when compared to the healthy, non-stenosed, bifurcation. This happens because the partial obstruction of the mild stenosis generates a symmetry in the flow that does not exist for the healthy condition. For the severe stenosis, there is more irregular motion and more particle trapping as expected.

From self-replication to replicator systems en route to de novo life

The process by which chemistry can give rise to biology remains one of the biggest mysteries in contemporary science. The de novo synthesis and origin of life both require the functional integration of three key characteristics - replication, metabolism and compartmentalization - into a system that is maintained out of equilibrium and is capable of open-ended Darwinian evolution. This Review takes systems of self-replicating molecules as starting points and describes the steps necessary to integrate additional characteristics of life. We analyse how far experimental self-replicators have come in terms of Darwinian evolution. We also cover models of replicator communities that attempt to solve Eigen's paradox, whereby accurate replication needs complex machinery yet obtaining such complex self-replicators through evolution requires accurate replication. Successful models rely on a collective metabolism and a way of (transient) compartmentalization, suggesting that the invention and integration of these two characteristics is driven by evolution. Despite our growing knowledge, there remain numerous key challenges that may be addressed by a combined theoretical and experimental approach.

Ocean currents promote rare species diversity in protists

Oceans host communities of plankton composed of relatively few abundant species and many rare species. The number of rare protist species in these communities, as estimated in metagenomic studies, decays as a steep power law of their abundance. The ecological factors at the origin of this pattern remain elusive. We propose that chaotic advection by oceanic currents affects biodiversity patterns of rare species. To test this hypothesis, we introduce a spatially explicit coalescence model that reconstructs the species diversity of a sample of water. Our model predicts, in the presence of chaotic advection, a steeper power law decay of the species abundance distribution and a steeper increase of the number of observed species with sample size. A comparison of metagenomic studies of planktonic protist communities in oceans and in lakes quantitatively confirms our prediction. Our results support that oceanic currents positively affect the diversity of rare aquatic microbes.

Spontaneous oscillations and negative-conductance transitions in microfluidic networks

The tendency for flows in microfluidic systems to behave linearly poses challenges for designing integrated flow control schemes to carry out complex fluid processing tasks. This hindrance precipitated the use of numerous external control devices to manipulate flows, thereby thwarting the potential scalability and portability of lab-on-a-chip technology. Here, we devise a microfluidic network exhibiting nonlinear flow dynamics that enable new mechanisms for on-chip flow control. This network is shown to exhibit oscillatory output patterns, bistable flow states, hysteresis, signal amplification, and negative-conductance transitions, all without reliance on dedicated external control hardware, movable parts, flexible components, or oscillatory inputs. These dynamics arise from nonlinear fluid inertia effects in laminar flows that we amplify and harness through the design of the network geometry. These results, which are supported by theory and simulations, have the potential to inspire development of new built-in control capabilities, such as on-chip timing and synchronized flow patterns.

Numerical modelling of the effect of intermittent upwelling events on plankton blooms

In the marine environment, biological processes are strongly affected by oceanic currents, particularly by eddies (vortices) formed by the hydrodynamic flow field. Employing a kinematic flow field coupled to a population dynamical model for plankton growth, we study the impact of an intermittent upwelling of nutrients on triggering harmful algal blooms (HABs). Though it is widely believed that additional nutrients boost the formation of HABs or algal blooms in general, we show that the response of the plankton to nutrient plumes depends crucially on the mesoscale hydrodynamic flow structure. In general, nutrients can either be quickly washed out from the observation area, or can be captured by the vortices in the flow. The occurrence of either scenario depends on the relation between the time scales of the vortex formation and nutrient upwelling as well as the time instants at which upwelling pulses occur and how long they last. We show that these two scenarios result in very different responses in plankton dynamics which makes it very difficult to predict whether nutrient upwelling will lead to a HAB or not. This may in part explain why observational data are sometimes inconclusive in establishing a connection between upwelling events and plankton blooms.

Effects of auxin derivatives on phenotypic plasticity and stress tolerance in five species of the green alga Desmodesmus (Chlorophyceae, Chlorophyta)

Green microalgae of the genus Desmodesmus are characterized by a high degree of phenotypic plasticity (i.e. colony morphology), allowing them to be truly cosmopolitan and withstand environmental fluctuations. This flexibility enables Desmodesmus to produce a phenotype–environment match across a range of environments broader compared to algae with more fixed phenotypes. Indoles and their derivatives are a well-known crucial class of heterocyclic compounds and are widespread in different species of plants, animals, and microorganisms. Indole-3-acetic acid (IAA) is the most common, naturally occurring plant hormone of the auxin class. IAA may behave as a signaling molecule in microorganisms, and the physiological cues of IAA may also trigger phenotypic plasticity responses in Desmodesmus. In this study, we demonstrated that the changes in colonial morphs (cells per coenobium) of five species of the green alga Desmodesmus were specific to IAA but not to the chemically more stable synthetic auxins, naphthalene-1-acetic acid and 2,4-dichlorophenoxyacetic acid. Moreover, inhibitors of auxin biosynthesis and polar auxin transport inhibited cell division. Notably, different algal species (even different intraspecific strains) exhibited phenotypic plasticity different to that correlated to IAA. Thus, the plasticity involving individual-level heterogeneity in morphological characteristics may be crucial for microalgae to adapt to changing or novel conditions, and IAA treatment potentially increases the tolerance of Desmodesmus algae to several stress conditions. In summary, our results provide circumstantial evidence for the hypothesized role of IAA as a diffusible signal in the communication between the microalga and microorganisms. This information is crucial for elucidation of the role of plant hormones in plankton ecology.

Shearing in flow environment promotes evolution of social behavior in microbial populations

How producers of public goods persist in microbial communities is a major question in evolutionary biology. Cooperation is evolutionarily unstable, since cheating strains can reproduce quicker and take over. Spatial structure has been shown to be a robust mechanism for the evolution of cooperation. Here we study how spatial assortment might emerge from native dynamics and show that fluid flow shear promotes cooperative behavior. Social structures arise naturally from our advection-diffusion-reaction model as self-reproducing Turing patterns. We computationally study the effects of fluid advection on these patterns as a mechanism to enable or enhance social behavior. Our central finding is that flow shear enables and promotes social behavior in microbes by increasing the group fragmentation rate and thereby limiting the spread of cheating strains. Regions of the flow domain with higher shear admit high cooperativity and large population density, whereas low shear regions are devoid of life due to opportunistic mutations.

According to the principle of the ‘survival of the fittest’, selfish individuals should be better off compared to peers that cooperate with each other. Indeed, even though a population of organisms benefits from working together, selfish members can exploit the cooperative behavior of others without doing their part. These ‘cheaters’ then use their advantage to reproduce faster and take over the population.

Yet, social cooperation is widespread in the natural world, and occurs in creatures as diverse as bacteria and whales. How can it arise and persist then? One idea is that when individuals form distinct groups, the ones with cheaters will perish. Even though a selfish individual will fare better than the rest of its team, overall, cooperating groups will survive more and reproduce faster; ultimately, they will be favored by evolution. This is called group selection.

Here, Uppal and Vural examine how the physical properties of the environment can influence the evolution of social interactions between bacteria. To this end, mathematical models are used to simulate how bacteria grow, evolve and drift in a flowing fluid. These are based on equations worked out from the behavior of real-life populations.

The results show that flow patterns in a fluid habitat govern the social behavior of bacteria. When different regions of the fluid are moving at different speeds, ‘shear forces’ are created that cause bacterial colonies to distort and occasionally break apart to form two groups. As such, cooperative groups will rapidly form new cooperating colonies, whereas groups with cheaters will reproduce slower or perish.

Furthermore, results show that when different areas of the fluid have different shear forces, social cooperation will only prevail in certain places. This makes it possible to use flow patterns to fine tune social evolution so that cooperating bacteria will be confined in a certain region. Outside of this area, these bacteria would be taken over by cheaters and go extinct.

Bacteria are both useful and dangerous to humans: for example, certain species can break down pollutants in the water, when others cause deadly infections. These results show it could be possible to control the activity of these microorganisms to our advantage by changing the flow of the fluids in which they live. More broadly, the simulations developed by Uppal and Vural can be applied to a variety of ecosystems where microscopic organisms inhabit fluids, such as plankton flowing in oceanic currents.

Stirring does not make populations well mixed

In evolutionary dynamics, the notion of a ‘well-mixed’ population is usually associated with all-to-all interactions at all times. This assumption simplifies the mathematics of evolutionary processes, and makes analytical solutions possible. At the same time the term ‘well-mixed’ suggests that this situation can be achieved by physically stirring the population. Using simulations of populations in chaotic flows, we show that in most cases this is not true: conventional well-mixed theories do not predict fixation probabilities correctly, regardless of how fast or thorough the stirring is. We propose a new analytical description in the fast-flow limit. This approach is valid for processes with global and local selection, and accurately predicts the suppression of selection as competition becomes more local. It provides a modelling tool for biological or social systems with individuals in motion.

Sustainability of a Compartmentalized Host-Parasite Replicator System under Periodic Washout-Mixing Cycles

The emergence and dominance of parasitic replicators are among the major hurdles for the proliferation of primitive replicators. Compartmentalization of replicators is proposed to relieve the parasite dominance; however, it remains unclear under what conditions simple compartmentalization uncoupled with internal reaction secures the long-term survival of a population of primitive replicators against incessant parasite emergence. Here, we investigate the sustainability of a compartmentalized host-parasite replicator (CHPR) system undergoing periodic washout-mixing cycles, by constructing a mathematical model and performing extensive simulations. We describe sustainable landscapes of the CHPR system in the parameter space and elucidate the mechanism of phase transitions between sustainable and extinct regions. Our findings revealed that a large population size of compartments, a high mixing intensity, and a modest amount of nutrients are important factors for the robust survival of replicators. We also found two distinctive sustainable phases with different mixing intensities. These results suggest that a population of simple host-parasite replicators assumed before the origin of life can be sustained by a simple compartmentalization with periodic washout-mixing processes.

Ecology and Evolution in the RNA World Dynamics and Stability of Prebiotic Replicator Systems

As of today, the most credible scientific paradigm pertaining to the origin of life on Earth is undoubtedly the RNA World scenario. It is built on the assumption that catalytically active replicators (most probably RNA-like macromolecules) may have been responsible for booting up life almost four billion years ago. The many different incarnations of nucleotide sequence (string) replicator models proposed recently are all attempts to explain on this basis how the genetic information transfer and the functional diversity of prebiotic replicator systems may have emerged, persisted and evolved into the first living cell. We have postulated three necessary conditions for an RNA World model system to be a dynamically feasible representation of prebiotic chemical evolution: (1) it must maintain and transfer a sufficient diversity of information reliably and indefinitely, (2) it must be ecologically stable and (3) it must be evolutionarily stable. In this review, we discuss the best-known prebiotic scenarios and the corresponding models of string-replicator dynamics and assess them against these criteria. We suggest that the most popular of prebiotic replicator systems, the hypercycle, is probably the worst performer in almost all of these respects, whereas a few other model concepts (parabolic replicator, open chaotic flows, stochastic corrector, metabolically coupled replicator system) are promising candidates for development into coherent models that may become experimentally accessible in the future.

Basin entropy: a new tool to analyze uncertainty in dynamical systems

In nonlinear dynamics, basins of attraction link a given set of initial conditions to its corresponding final states. This notion appears in a broad range of applications where several outcomes are possible, which is a common situation in neuroscience, economy, astronomy, ecology and many other disciplines. Depending on the nature of the basins, prediction can be difficult even in systems that evolve under deterministic rules. From this respect, a proper classification of this unpredictability is clearly required. To address this issue, we introduce the basin entropy, a measure to quantify this uncertainty. Its application is illustrated with several paradigmatic examples that allow us to identify the ingredients that hinder the prediction of the final state. The basin entropy provides an efficient method to probe the behavior of a system when different parameters are varied. Additionally, we provide a sufficient condition for the existence of fractal basin boundaries: when the basin entropy of the boundaries is larger than log2, the basin is fractal.

Inhomogeneous distribution of water droplets in cloud turbulence

We consider sedimentation of small particles in the turbulent flow where fluid accelerations are much smaller than acceleration of gravity g. The particles are dragged by the flow by linear friction force. We demonstrate that the pair-correlation function of particles' concentration diverges with decreasing separation as a power law with negative exponent. This manifests fractal distribution of particles in space. We find that the exponent is proportional to ratio of integral of energy spectrum of turbulence times the wave number over g. The proportionality coefficient is a universal number independent of particle size. We derive the spectrum of Lyapunov exponents that describes the evolution of small patches of particles. It is demonstrated that particles separate dominantly in the horizontal plane. This provides a theory for the recently observed vertical columns formed by the particles. We confirm the predictions by direct numerical simulations of Navier-Stokes turbulence. The predictions include conditions that hold for water droplets in warm clouds thus providing a tool for the prediction of rain formation.

Experimental studies of coherent structures in an advection-reaction-diffusion system

We present experimental studies of reaction front propagation in a single vortex flow with an imposed external wind. The fronts are produced by the excitable, ferroin-catalyzed Belousov-Zhabotinsky chemical reaction. The flow is generated using an electromagnetic forcing technique: an almost-radial electrical current interacts with a magnetic field from a magnet below the fluid layer to produce the vortex. The magnet is mounted on crossed translation stages allowing for movement of the vortex through the flow. Reaction fronts triggered in or in front of the moving vortex form persistent structures that are seen experimentally for time-independent (constant motion), time-periodic, and time-aperiodic flows. These results are examined with the use of burning invariant manifolds that act as one-way barriers to front motion in the flows. We also explore the usefulness of finite-time Lyapunov exponent fields as an instrument for analyzing front propagation behavior in a fluid flow.

Constructing minimal models for complex system dynamics

One of the strengths of statistical physics is the ability to reduce macroscopic observations into microscopic models, offering a mechanistic description of a system's dynamics. This paradigm, rooted in Boltzmann's gas theory, has found applications from magnetic phenomena to subcellular processes and epidemic spreading. Yet, each of these advances were the result of decades of meticulous model building and validation, which are impossible to replicate in most complex biological, social or technological systems that lack accurate microscopic models. Here we develop a method to infer the microscopic dynamics of a complex system from observations of its response to external perturbations, allowing us to construct the most general class of nonlinear pairwise dynamics that are guaranteed to recover the observed behaviour. The result, which we test against both numerical and empirical data, is an effective dynamic model that can predict the system's behaviour and provide crucial insights into its inner workings.

The dynamics of the RNA world: insights and challenges

The RNA world hypothesis of the origin of life, in which RNA emerged as both enzyme and information carrier, is receiving solid experimental support. The prebiotic synthesis of biomolecules, the catalytic aid offered by mineral surfaces, and the vast enzymatic repertoire of ribozymes are only pieces of the origin of life puzzle; the full picture can only emerge if the pieces fit together by either following from one another or coexisting with each other. Here, we review the theory of the origin, maintenance, and enhancement of the RNA world as an evolving population of dynamical systems. The dynamical view of the origin of life allows us to pinpoint the missing and the not fitting pieces: (1) How can the first self-replicating ribozyme emerge in the absence of template-directed information replication? (2) How can nucleotide replicators avoid competitive exclusion despite utilizing the very same resources (nucleobases)? (3) How can the information catastrophe be avoided? (4) How can enough genes integrate into a cohesive system in order to transition to a cellular stage? (5) How can the way information is stored and metabolic complexity coevolve to pave to road leading out of the RNA world to the present protein-DNA world?

Many ways to stay in the game: individual variability maintains high biodiversity in planktonic microorganisms

In apparent contradiction to competition theory, the number of known, coexisting plankton species far exceeds their explicable biodiversity-a discrepancy termed the Paradox of the Plankton. We introduce a new game-theoretic model for competing microorganisms in which one player consists of all organisms of one species. The stable points for the population dynamics in our model, known as strategic behaviour distributions (SBDs), are probability distributions of behaviours across all organisms which imply a stable population of the species as a whole. We find that intra-specific variability is the key characteristic that ultimately allows coexistence because the outcomes of competitions between individuals with variable competitive abilities are unpredictable. Our simulations based on the theoretical model show that up to 100 species can coexist for at least 10,000 generations, and that even small population sizes or species with inferior competitive ability can survive when there is intra-specific variability. In nature, this variability can be observed as niche differentiation, variability in environmental and ecological factors, and variability of individual behaviours or physiology. Therefore, previous specific explanations of the paradox are consistent with and provide specific examples of our suggestion that individual variability is the mechanism which solves the paradox.

Is chaotic advection inherent to porous media flow?

We show that chaotic advection is inherent to flow through all types of porous media, from granular and packed media to fractured and open networks. The basic topological complexity inherent to all porous media gives rise to chaotic flow dynamics under steady flow conditions, where fluid deformation local to stagnation points imparts a 3D fluid mechanical analog of the baker's map. The ubiquitous nature of chaotic advection has significant implications for the description of transport, mixing, chemical reaction and biological activity in porous media.

Stirring effects in models of oceanic plankton populations

We present an overview and extend previous results on the effects of large scale oceanic transport processes on plankton population dynamics, considering different types of ecosystem models. We find that increasing stirring rate in an environment where the carrying capacity is non-uniformly distributed leads to an overall decrease of the effective carrying capacity of the system. This may lead to sharp regime shifts induced by stirring in systems with multiple steady states. In prey-predator type systems, stirring leads to resonant response of the population dynamics to fluctuations enhancing the spatial variability-patchiness-in a certain range of stirring rates. Oscillatory population models produce strongly heterogeneous patchy distribution of plankton blooms when the stirring is weak, while strong stirring may either synchronise the oscillatory dynamics, when the inhomogeneity is relatively weak, or suppress oscillations completely (oscillator death) by reducing the effective carrying capacity below the bifurcation point.

Reacting particles in open chaotic flows

We study the collision probability p of particles advected by open flows with chaotic advection. We show that p scales with the particle size (or, alternatively, reaction distance) δ as a power law whose coefficient is determined by the fractal dimensions of the invariant sets defined by the advection dynamics. We also argue that this same scaling also holds for the reaction rate of active particles in the low-density regime. These analytical results are compared to numerical simulations, and they are found to agree very well.

The evolution of enzyme specificity in the metabolic replicator model of prebiotic evolution

The chemical machinery of life must have been catalytic from the outset. Models of the chemical origins have attempted to explain the ecological mechanisms maintaining a minimum necessary diversity of prebiotic replicator enzymes, but little attention has been paid so far to the evolutionary initiation of that diversity. We propose a possible first step in this direction: based on our previous model of a surface-bound metabolic replicator system we try to explain how the adaptive specialization of enzymatic replicator populations might have led to more diverse and more efficient communities of cooperating replicators with two different enzyme activities. The key assumptions of the model are that mutations in the replicator population can lead towards a) both of the two different enzyme specificities in separate replicators: efficient “specialists” or b) a “generalist” replicator type with both enzyme specificities working at less efficiency, or c) a fast-replicating, non-enzymatic “parasite”. We show that under realistic trade-off constraints on the phenotypic effects of these mutations the evolved replicator community will be usually composed of both types of specialists and of a limited abundance of parasites, provided that the replicators can slowly migrate on the mineral surface. It is only at very weak trade-offs that generalists take over in a phase-transition-like manner. The parasites do not seriously harm the system but can freely mutate, therefore they can be considered as pre-adaptations to later, useful functions that the metabolic system can adopt to increase its own fitness.

Fractal structures in stenoses and aneurysms in blood vessels

Recent advances in the field of chaotic advection provide the impetus to revisit the dynamics of particles transported by blood flow in the presence of vessel wall irregularities. The irregularity, being either a narrowing or expansion of the vessel, mimicking stenoses or aneurysms, generates abnormal flow patterns that lead to a peculiar filamentary distribution of advected particles, which, in the blood, would include platelets. Using a simple model, we show how the filamentary distribution depends on the size of the vessel wall irregularity, and how it varies under resting or exercise conditions. The particles transported by blood flow that spend a long time around a disturbance either stick to the vessel wall or reside on fractal filaments. We show that the faster flow associated with exercise creates widespread filaments where particles can get trapped for a longer time, thus allowing for the possible activation of such particles. We argue, based on previous results in the field of active processes in flows, that the non-trivial long-time distribution of transported particles has the potential to have major effects on biochemical processes occurring in blood flow, including the activation and deposition of platelets. One aspect of the generality of our approach is that it also applies to other relevant biological processes, an example being the coexistence of plankton species investigated previously.

Chaotic advection in blood flow

In this paper we argue that the effects of irregular chaotic motion of particles transported by blood can play a major role in the development of serious circulatory diseases. Vessel wall irregularities modify the flow field, changing in a nontrivial way the transport and activation of biochemically active particles. We argue that blood particle transport is often chaotic in realistic physiological conditions. We also argue that this chaotic behavior of the flow has crucial consequences for the dynamics of important processes in the blood, such as the activation of platelets which are involved in the thrombus formation.

Phototactic clustering of swimming microorganisms in a turbulent velocity field

We study the distribution of swimming microorganisms advected by a two-dimensional smooth turbulent flow and attracted towards a light source through phototaxis. It is shown that particles aggregate along a dynamical attractor with fractal measure whose dimension depends on the strength of the phototaxis. Using an effective diffusion approximation for the flow, we derive an analytic expression for the increase in light exposure over the aggregate and by extension an accurate prediction for the fractal dimension based on the properties of the advection and the statistics of the attracting field.

Onset of chaotic advection in open flows

In this paper we investigate the transition to chaos in the motion of particles advected by open flows with obstacles. By means of a topological argument, we show that the separation points on the surface of the obstacle imply the existence of a saddle point downstream from the obstacle, with an associated heteroclinic orbit. We argue that as soon as the flow becomes time periodic, these orbits give rise to heteroclinic tangles, causing passively advected particles to experience transient chaos. The transition to chaos thus coincides with the onset of time dependence in open flows with stagnant points, in contrast with flows with no stagnant points. We also show that the nonhyperbolic nature of the dynamics near the walls causes anomalous scalings in the vicinity of the transition. These results are confirmed by numerical simulations of the two-dimensional flow around a cylinder.

Does local competition increase the coexistence of species in intransitive networks?

Competitive intransitivity, a situation in which species' competitive ranks cannot be listed in a strict hierarchy, promotes species coexistence through "enemy's enemy indirect facilitation." Theory suggests that intransitivity-mediated coexistence is enhanced when competitive interactions occur at local spatial scales, although this hypothesis has not been thoroughly tested. Here, we use a lattice model to investigate the effect of local vs. global competition on intransitivity-mediated coexistence across a range of species richness values and levels of intransitivity. Our simulations show that local competition can enhance intransitivity-mediated coexistence in the short-term, yet hinder it in the long-term, when compared to global competition. This occurs because local competition slows species disaggregation, allowing weaker competitors to persist longer in the shifting spatial refuges of intransitive networks, enhancing short-term coexistence. Conversely, our simulations show that, in the long-term, local competition traps disaggregated species in unfavorable areas of the competitive arena, where they are excluded by superior competitors. As a result, in the long-term, global intransitive competition allows a greater number of species to coexist than local intransitive competition.

Rate of mixing controls rate and outcome of autocatalytic processes: theory and microfluidic experiments with chemical reactions and blood coagulation

This article demonstrates that the rate of mixing can regulate the rate and outcome of both biological and nonbiological autocatalytic reaction systems that display a threshold response to the concentration of an activator. Plug-based microfluidics was used to control the timing of reactions, the rate of mixing, and surface chemistry in blood clotting and its chemical model. Initiation of clotting of human blood plasma required addition of a critical concentration of thrombin. Clotting could be prevented by rapid mixing when thrombin was added near the critical concentration, and mixing also affected the rate of clotting when thrombin was added at concentrations far above the critical concentration in two clinical clotting assays for human plasma. This phenomenon was modeled by a simple mechanism--local and global competition between the clotting reaction, which autocatalytically produces an activator, and mixing, which removes the activator. Numerical simulations showed that the Damköhler number, which describes this competition, predicts the effects of mixing. Many biological systems are controlled by thresholds, and these results shed light on the dynamics of these systems in the presence of spatial heterogeneities and provide simple guidelines for designing and interpreting experiments with such systems.

Mixing of supersaturated assemblages and the precipitous loss of species

Mechanisms influencing species richness are many. Recent theoretical research revealed additional mechanisms that involved neutral and lumpy coexistence and alternating assemblage states. These mechanisms can lead to conditions where the number of coexisting species is greater than the number of limiting resources, that is, species supersaturation. Our research focused on the role of disturbances (migration and pulsed through-flows) in supersaturated plankton systems. Our simulations employed 30 different supersaturated assemblages generated by using various ecological principals. Our findings indicated that immigration rates as low as 0.1% of total biomass per day generally led to regional homogenization of species and dramatic extinction events, with assemblages characteristic of lumpy coexistence being more resilient than those characteristic of neutral coexistence or alternating states. Generally, pulsed through-flows tended to offset, to some extent, the negative effects of migration. The precipitous loss of species with the onset of migration is observed in other systems as well, for example, cichlid fish communities of East Africa rift lakes and songbird assemblages from Indian Ocean islands. While many explanations have been offered to explain postimmigration extinctions in species-rich systems, another explanation might be that the assemblages in these systems are in a fragile state of supersaturated coexistence.

Coexistence of inertial competitors in chaotic flows

We investigate the dynamics of inertial particles immersed in open chaotic flows. We consider the generic problem of competition between different species, e.g., phytoplankton populations in oceans. The strong influence from inertial effects is shown to result in the persistence of different species even in cases when the passively advected species cannot coexist. Multispecies coexistence in the ocean can be explained by the fact that the unstable manifold is different for each advected competitor of different size.

Synchronization of oscillating reactions in an extended fluid system

We present experiments on the synchronization of a dynamical, chemical process in an extended, flowing, fluid system. The oscillatory Belousov-Zhabotinsky chemical reaction is the process studied, and the flow is an annular chain of counterrotating vortices. Azimuthal motion of the vortices is controlled externally, enabling us to vary the type of transport. We find that oscillations of the Belousov-Zhabotinsky reaction synchronize throughout the extended fluid system only if transport in the flow is superdiffusive, with tracers in the flow undergoing rapid, distant jumps called Lévy flights.

Competitive exclusion and limiting similarity: A unified theory

Robustness of coexistence against changes of parameters is investigated in a model-independent manner by analyzing the feedback loop of population regulation. We define coexistence as a fixed point of the community dynamics with no population having zero size. It is demonstrated that the parameter range allowing coexistence shrinks and disappears when the Jacobian of the dynamics decreases to zero. A general notion of regulating factors/variables is introduced. For each population, its impact and sensitivity niches are defined as the differential impact on, and the differential sensitivity towards, the regulating variables, respectively. Either the similarity of the impact niches or the similarity of the sensitivity niches results in a small Jacobian and in a reduced likelihood of coexistence. For the case of a resource continuum, this result reduces to the usual "limited niche overlap" picture for both kinds of niche. As an extension of these ideas to the coexistence of infinitely many species, we demonstrate that Roughgarden's example for coexistence of a continuum of populations is structurally unstable.

Rock-scissors-paper game in a chaotic flow: The effect of dispersion on the cyclic competition of microorganisms

Laboratory experiments and numerical simulations have shown that the outcome of cyclic competition is significantly affected by the spatial distribution of the competitors. Short-range interaction and limited dispersion allows for coexistence of competing species that cannot coexist in a well-mixed environment. In order to elucidate the mechanisms that destroy species diversity we study the intermediate situation of imperfect mixing, typical in aquatic media, in a model of cyclic competition between toxin producing, sensitive and resistant phenotypes. It is found, that chaotic mixing, by changing the character of the spatial distribution, induces coherent oscillations in the populations. The magnitude of the oscillations increases with the strength of mixing, leading to the extinction of some species beyond a critical mixing rate. When mixing is non-uniform in space, coexistence can be sustained at much stronger mixing by the formation of partially isolated regions, that prevent global extinction. The heterogeneity of mixing may enable toxin producing and sensitive strains to coexist for very long time at strong mixing.

Discrete time model for chemical or biological decay in chaotic flows: reentrance phase transitions

We consider a discrete time model of advection, reaction, and diffusion on a lattice to investigate the steady-state spatial structure of chemically decaying substances. The time discretization of the dynamics has a considerable impact on these structures. Additional smooth-filamental phase transitions, nonexistent in the continuous-time description, appear. We show how these structures and their scaling properties depend on the time step of the discrete dynamics. Exploiting the analogies of this discrete model with the logistic map, some general features are discussed.

Fractal scaling of microbial colonies affects growth

The growth dynamics of filamentary microbial colonies is investigated. Fractality of the fungal or actinomycetes colonies is shown both theoretically and in numerical experiments to play an important role. The growth observed in real colonies is described by the assumption of time-dependent fractality related to the different ages of various parts of the colony. The theoretical results are compared to a simulation based on branching random walks.

Multifractal spectra of chemical fields in fluid flows

In the filamental phase of reactions embedded in fluid flows, where the concentration distribution is strongly fluctuating, we show that a chemical measure can be defined based on the absolute value of the concentration gradients. We express the generalized dimensions in terms of the roughness exponents of the structure functions as well as of the cancellation exponents of the chemical concentration. This measure is of basically different character than the natural distribution of the passive advection. It is similar to the SRB measures of dissipative systems, although the advection problem is area preserving. This approach is shown to be a useful tool in analyzing sea surface temperature anomalies.

Experimental studies of pattern formation in a reaction-advection-diffusion system

Experiments are presented on pattern formation in the Belousov-Zhabotinsky (BZ) reaction in a blinking vortex flow. Mixing in this flow is chaotic, with nearby tracers separating exponentially with time. The patterns that form in this flow with the BZ reaction mimic chaotic mixing structures seen in passive transport. The behavior is analyzed in terms of a mixing time taum and a characteristic decorrelation time TBZ for the BZ system. Flows with taum comparable to or smaller than TBZ generate large-scale patterns whose features are captured by simulations of mixing fields for the flow.

Reactions in flows with nonhyperbolic dynamics

We study the reaction dynamics of active particles that are advected passively by 2D incompressible open flows, whose motion is nonhyperbolic. This nonhyperbolicity is associated with the presence of persistent vortices near the wake, wherein fluid is trapped. We show that the fractal equilibrium distribution of the reactants is described by an effective dimension d(eff) , which is a finite resolution approximation to the fractal dimension. Furthermore, d(eff) depends on the resolution epsilon and on the reaction rate 1/tau . As tau is increased, the equilibrium distribution goes through a series of transitions where the effective dimension increases abruptly. These transitions are determined by the complex structure of Cantori surrounding the Kolmogorov-Arnold-Moser (KAM) islands.

Chemical and biological activity in three-dimensional flows

We study the dynamics of active particles advected by three-dimensional (3D) open incompressible flows, both analytically and numerically. We find that 3D reactive flows have fundamentally different dynamical features from those in 2D systems. In particular, we show that the reaction's productivity per reaction step can be enhanced, with respect to the 2D case, while the productivity per unit time in some 3D flows goes to zero in the limit of high mixing rates, in contrast to the 2D behavior, in which the productivity goes to a finite constant. These theoretical predictions are validated by numerical simulations on a generic map model.

Description of diffusive and propagative behavior on fractals

The known properties of diffusion on fractals are reviewed in order to give a general outlook of these dynamic processes. After that, we propose a description developed in the context of the intrinsic metric of fractals, which leads us to a differential equation able to describe diffusion in real fractals in the asymptotic regime. We show that our approach has a stronger physical justification than previous works on this field. The most important result we present is the introduction of a dependence on time and space for the conductivity in fractals, which is deduced by scaling arguments and supported by computer simulations. Finally, the diffusion equation is used to introduce the possibility of reaction-diffusion processes on fractals and analyze their properties. Specifically, an analytic expression for the speed of the corresponding travelling fronts, which can be of great interest for application purposes, is derived.

Spatial models of prebiotic evolution: soup before pizza?

The problem of information integration and resistance to the invasion of parasitic mutants in prebiotic replicator systems is a notorious issue of research on the origin of life. Almost all theoretical studies published so far have demonstrated that some kind of spatial structure is indispensable for the persistence and/or the parasite resistance of any feasible replicator system. Based on a detailed critical survey of spatial models on prebiotic information integration, we suggest a possible scenario for replicator system evolution leading to the emergence of the first protocells capable of independent life. We show that even the spatial versions of the hypercycle model are vulnerable to selfish parasites in heterogeneous habitats. Contrary, the metabolic system remains persistent and coexistent with its parasites both on heterogeneous surfaces and in chaotically mixing flowing media. Persistent metabolic parasites can be converted to metabolic cooperators, or they can gradually obtain replicase activity. Our simulations show that, once replicase activity emerged, a gradual and simultaneous evolutionary improvement of replicase functionality (speed and fidelity) and template efficiency is possible only on a surface that constrains the mobility of macromolecule replicators. Based on the results of the models reviewed, we suggest that open chaotic flows ('soup') and surface dynamics ('pizza') both played key roles in the sequence of evolutionary events ultimately concluding in the appearance of the first living cell on Earth.

Uniform resonant chaotic mixing in fluid flows

Laminar flows can produce particle trajectories that are chaotic, with nearby tracers separating exponentially in time. For time-periodic, two-dimensional flows and steady three-dimensional (3D) flows, enhancements in mixing due to chaotic advection are typically limited by impenetrable transport barriers that form at the boundaries between ordered and chaotic mixing regions. However, for time-dependent 3D flows, it has been proposed theoretically that completely uniform mixing is possible through a resonant mechanism called singularity-induced diffusion; this is thought to be the case even if the time-dependent and 3D perturbations are infinitesimally small. It is important to establish the conditions for which uniform mixing is possible and whether or not those conditions are met in flows that typically occur in nature. Here we report experimental and numerical studies of mixing in a laminar vortex flow that is weakly 3D and weakly time-periodic. The system is an oscillating horizontal vortex chain (produced by a magnetohydrodynamic technique) with a weak vertical secondary flow that is forced spontaneously by Ekman pumping--a mechanism common in vortical flows with rigid boundaries, occurring in many geophysical, industrial and biophysical flows. We observe completely uniform mixing, as predicted by singularity-induced diffusion, but only for oscillation periods close to typical circulation times.