Chaotic behavior in Lotka-Volterra and May-Leonard models of biodiversity.

Quantification of chaos is a challenging issue in complex dynamical systems. In this paper, we discuss the chaotic properties of generalized Lotka-Volterra and May-Leonard models of biodiversity, via the Hamming distance density. We identified chaotic behavior for different scenarios via the specific features of the Hamming distance and the method of q-exponential fitting. We also investigated the spatial autocorrelation length to find the corresponding characteristic length in terms of the number of species in each system. In particular, the results concerning the characteristic length are in good accordance with the study of the chaotic behavior implemented in this work.

Lotka-Volterra versus May-Leonard formulations of the spatial stochastic rock-paper-scissors model: The missing link.

The rock-paper-scissors (RPS) model successfully reproduces some of the main features of simple cyclic predator-prey systems with interspecific competition observed in nature. Still, lattice-based simulations of the spatial stochastic RPS model are known to give rise to significantly different results, depending on whether the three-state Lotka-Volterra or the four-state May-Leonard formulation is employed. This is true independently of the values of the model parameters and of the use of either a von Neumann or a Moore neighborhood. In this paper, we introduce a simple modification to the standard spatial stochastic RPS model in which the range of the search of the nearest neighbor may be extended up to a maximum Euclidean radius R. We show that, with this adjustment, the Lotka-Volterra and May-Leonard formulations can be designed to produce similar results, both in terms of dynamical properties and spatial features, by means of an appropriate parameter choice. In particular, we show that this modified spatial stochastic RPS model naturally leads to the emergence of spiral patterns in both its three- and four-state formulations.

Environment driven oscillation in an off-lattice May-Leonard model.

Cyclic dominance of competing species is an intensively used working hypothesis to explain biodiversity in certain living systems, where the evolutionary selection principle would dictate a single victor otherwise. Technically the May-Leonard models offer a mathematical framework to describe the mentioned non-transitive interaction of competing species when individual movement is also considered in a spatial system. Emerging rotating spirals composed by the competing species are frequently observed character of the resulting patterns. But how do these spiraling patterns change when we vary the external environment which affects the general vitality of individuals? Motivated by this question we suggest an off-lattice version of the tradition May-Leonard model which allows us to change the actual state of the environment gradually. This can be done by introducing a local carrying capacity parameter which value can be varied gently in an off-lattice environment. Our results support a previous analysis obtained in a more intricate metapopulation model and we show that the well-known rotating spirals become evident in a benign environment when the general density of the population is high. The accompanying time-dependent oscillation of competing species can also be detected where the amplitude and the frequency show a scaling law of the parameter that characterizes the state of the environment. These observations highlight that the assumed non-transitive interaction alone is insufficient condition to maintain biodiversity safely, but the actual state of the environment, which characterizes the general living conditions, also plays a decisive role on the evolution of related systems.

Performance of weak species in the simplest generalization of the rock-paper-scissors model to four species.

We investigate the problem of the predominance and survival of "weak" species in the context of the simplest generalization of the spatial stochastic rock-paper-scissors model to four species by considering models in which one, two, or three species have a reduced predation probability. We show, using lattice based spatial stochastic simulations with random initial conditions, that if only one of the four species has its probability reduced, then the most abundant species is the prey of the "weakest" (assuming that the simulations are large enough for coexistence to prevail). Also, among the remaining cases, we present examples in which "weak" and "strong" species have similar average abundances and others in which either of them dominates-the most abundant species being always a prey of a weak species with which it maintains a unidirectional predator-prey interaction. However, in contrast to the three-species model, we find no systematic difference in the global performance of weak and strong species, and we conjecture that a similar result will hold if the number of species is further increased. We also determine the probability of single species survival and coexistence as a function of the lattice size, discussing its dependence on initial conditions and on the change to the dynamics of the model which results from the extinction of one of the species.

Predominance of the weakest species in Lotka-Volterra and May-Leonard formulations of the rock-paper-scissors model.

We revisit the problem of the predominance of the "weakest" species in the context of Lotka-Volterra and May-Leonard formulations of a spatial stochastic rock-paper-scissors model in which one of the species has its predation probability reduced by 0

Invasion-controlled pattern formation in a generalized multispecies predator-prey system.

Rock-scissors-paper game, as the simplest model of intransitive relation between competing agents, is a frequently quoted model to explain the stable diversity of competitors in the race of surviving. When increasing the number of competitors we may face a novel situation because beside the mentioned unidirectional predator-prey-like dominance a balanced or peer relation can emerge between some competitors. By utilizing this possibility in the present work we generalize a four-state predator-prey-type model where we establish two groups of species labeled by even and odd numbers. In particular, we introduce different invasion probabilities between and within these groups, which results in a tunable intensity of bidirectional invasion among peer species. Our study reveals an exceptional richness of pattern formations where five quantitatively different phases are observed by varying solely the strength of the mentioned inner invasion. The related transition points can be identified with the help of appropriate order parameters based on the spatial autocorrelation decay, on the fraction of empty sites, and on the variance of the species density. Furthermore, the application of diverse, alliance-specific inner invasion rates for different groups may result in the extinction of the pair of species where this inner invasion is moderate. These observations highlight that beyond the well-known and intensively studied cyclic dominance there is an additional source of complexity of pattern formation that has not been explored earlier.

Expanding spatial domains and transient scaling regimes in populations with local cyclic competition.

We investigate a six-species class of May-Leonard models leading to the formation of two types of competing spatial domains, each one inhabited by three species with their own internal cyclic rock-paper-scissors dynamics. We study the resulting population dynamics using stochastic numerical simulations in two-dimensional space. We find that as three-species domains shrink, there is an increasing probability of extinction of two of the species inhabiting the domain, with the consequent creation of one-species domains. We determine the critical initial radius beyond which these one-species spatial domains are expected to expand. We further show that a transient scaling regime, with a slower average growth rate of the characteristic length scale L of the spatial domains with time t, takes place before the transition to a standard L∝t^{1/2} scaling law, resulting in an extended period of coexistence.

How directional mobility affects coexistence in rock-paper-scissors models.

This work deals with a system of three distinct species that changes in time under the presence of mobility, selection, and reproduction, as in the popular rock-paper-scissors game. The novelty of the current study is the modification of the mobility rule to the case of directional mobility, in which the species move following the direction associated to a larger (averaged) number density of selection targets in the surrounding neighborhood. Directional mobility can be used to simulate eyes that see or a nose that smells, and we show how it may contribute to reduce the probability of coexistence.

Stress-induced birefringence in the isotropic phases of lyotropic mixtures.

In this work, the frequency dependence of the known mechano-optical effect which occurs in the micellar isotropic phases (I) of mixtures of potassium laurate (KL), decanol (DeOH), and water is investigated in the range from 200mHz to 200Hz. In order to fit the experimental data, a model of superimposed damped harmonic oscillators is proposed. In this phenomenological approach, the micelles (microscopic oscillators) interact very weakly with their neighbors. Due to shape anisotropy of the basic structures, each oscillator i (i=1,2,3,...,N) remains in its natural oscillatory rotational movement around its axes of symmetry with a frequency ω_{0i}. The system will be in the resonance state when the frequency of the driving force ω reaches a value near ω_{0i}. This phenomenological approach shows excellent agreement with the experimental data. One can find f∼2.5, 9.0, and 4.0Hz as fundamental frequencies of the micellar isotropic phases I, I_{1}, and I_{2}, respectively. The different micellar isotropic phases I, I_{1}, and I_{2} that we find in the phase diagram of the KL-DeOH-water mixture are a consequence of possible differences in the intermicellar correlation lengths. This work reinforces the possibilities of technological applications of these phases in devices such as mechanical vibration sensors.

A novel procedure for the identification of chaos in complex biological systems.

We demonstrate the presence of chaos in stochastic simulations that are widely used to study biodiversity in nature. The investigation deals with a set of three distinct species that evolve according to the standard rules of mobility, reproduction and predation, with predation following the cyclic rules of the popular rock, paper and scissors game. The study uncovers the possibility to distinguish between time evolutions that start from slightly different initial states, guided by the Hamming distance which heuristically unveils the chaotic behavior. The finding opens up a quantitative approach that relates the correlation length to the average density of maxima of a typical species, and an ensemble of stochastic simulations is implemented to support the procedure. The main result of the work shows how a single and simple experimental realization that counts the density of maxima associated with the chaotic evolution of the species serves to infer its correlation length. We use the result to investigate others distinct complex systems, one dealing with a set of differential equations that can be used to model a diversity of natural and artificial chaotic systems, and another one, focusing on the ocean water level.

Interfaces with internal structures in generalized rock-paper-scissors models.

In this work we investigate the development of stable dynamical structures along interfaces separating domains belonging to enemy partnerships in the context of cyclic predator-prey models with an even number of species N≥8. We use both stochastic and field theory simulations in one and two spatial dimensions, as well as analytical arguments, to describe the association at the interfaces of mutually neutral individuals belonging to enemy partnerships and to probe their role in the development of the dynamical structures at the interfaces. We identify an interesting behavior associated with the symmetric or asymmetric evolution of the interface profiles depending on whether N/2 is odd or even, respectively. We also show that the macroscopic evolution of the interface network is not very sensitive to the internal structure of the interfaces. Although this work focuses on cyclic predator-prey models with an even number of species, we argue that the results are expected to be quite generic in the context of spatial stochastic May-Leonard models.

Junctions and spiral patterns in generalized rock-paper-scissors models.

We investigate the population dynamics in generalized rock-paper-scissors models with an arbitrary number of species N. We show that spiral patterns with N arms may develop both for odd and even N, in particular in models where a bidirectional predation interaction of equal strength between all species is modified to include one N-cyclic predator-prey rule. While the former case gives rise to an interface network with Y-type junctions obeying the scaling law L∝t1/2, where L is the characteristic length of the network and t is the time, the latter can lead to a population network with N-armed spiral patterns, having a roughly constant characteristic length scale. We explicitly demonstrate the connection between interface junctions and spiral patterns in these models and compute the corresponding scaling laws. This work significantly extends the results of previous studies of population dynamics and could have profound implications for the understanding of biological complexity in systems with a large number of species.

Studying the genetic basis of drought tolerance in sorghum by managed stress trials and adjustments for phenological and plant height differences.

Managed environments in the form of well watered and water stressed trials were performed to study the genetic basis of grain yield and stay green in sorghum with the objective of validating previously detected QTL. As variations in phenology and plant height may influence QTL detection for the target traits, QTL for flowering time and plant height were introduced as cofactors in QTL analyses for yield and stay green. All but one of the flowering time QTL were detected near yield and stay green QTL. Similar co-localization was observed for two plant height QTL. QTL analysis for yield, using flowering time/plant height cofactors, led to yield QTL on chromosomes 2, 3, 6, 8 and 10. For stay green, QTL on chromosomes 3, 4, 8 and 10 were not related to differences in flowering time/plant height. The physical positions for markers in QTL regions projected on the sorghum genome suggest that the previously detected plant height QTL, Sb-HT9-1, and Dw2, in addition to the maturity gene, Ma5, had a major confounding impact on the expression of yield and stay green QTL. Co-localization between an apparently novel stay green QTL and a yield QTL on chromosome 3 suggests there is potential for indirect selection based on stay green to improve drought tolerance in sorghum. Our QTL study was carried out with a moderately sized population and spanned a limited geographic range, but still the results strongly emphasize the necessity of corrections for phenology in QTL mapping for drought tolerance traits in sorghum.

Nematic liquid crystal dynamics under applied electric fields.

In this paper we investigate the coarsening dynamics of liquid crystal textures in a two-dimensional nematic under applied electric fields, using numerical simulations performed using a publicly available liquid crystal algorithm developed by the authors. We consider both positive and negative dielectric anisotropies and two different possibilities for the orientation of the electric field (parallel and perpendicular to the two-dimensional lattice). We determine the effect of an applied electric field pulse on the evolution of the characteristic length scale and other properties of the liquid crystal texture network. In particular, we show that different types of defects are produced after the electric field is switched on, depending on the orientation of the electric field and the sign of the dielectric anisotropy.

Defect-antidefect correlations in a lyotropic liquid crystal from a cosmological point of view.

In this work, we analyze the defect and antidefect distribution in the nematic calamitic phase of a lyotropic liquid crystal [the ternary mixture formed by potassium laurate (KL), decanol (DeOH), and water]. We obtain defects with wedge disclinations of strength +/-1/2, and the scaling exponent determined by the defect-antidefect correlation was 0.29+/-0.07. This value is in good agreement with the theoretical value of 14 obtained by the Kibble mechanism. The constant of the scaling relation of the defect and antidefect distribution is also discussed. We compare our results with the values obtained by Digal [Phys. Rev. Lett. 83, 5030 (1999)] who used a thermotropic liquid crystal.