zero one survival behavior of cyclically competing species the coexistence of competing species is due to unavoidable fluctuations always transient in this letter we investigate the ultimate survival probabilities characterizing different species in cyclic competition we show that they often obey a surprisingly simple though nontrivial behavior within a model where coexistence is neutrally stable we demonstrate a robust zero one law when the interactions between the three species are generically asymmetric the weakest species survives at a probability that tends to one for large population sizes while the other two are guaranteed to extinction we rationalize our findings from stochastic simulations by an analytic approach

survival behavior in the cyclic lotka volterra model with a randomly switching reaction rate we study the influence of a randomly switching reproduction predation rate on the survival behavior of the nonspatial cyclic lotka volterra model also known as the zero sum rock paper scissors game used to metaphorically describe the cyclic competition between three species in large and finite populations demographic fluctuations internal noise drive two species to extinction in a finite time while the species with the smallest reproduction predation rate is the most likely to be the surviving one law of the weakest here we model environmental external noise by assuming that the reproduction predation rate of the strongest species the fastest to reproduce and predate in a given static environment randomly switches between two values corresponding to more and less favorable external conditions we study the joint effect of environmental and demographic noise on the species survival probabilities and on the mean extinction time in particular we investigate whether the survival probabilities follow the law of the weakest and analyze their dependence on the external noise intensity and switching rate remarkably when on average there is a finite number of switches prior to extinction the survival probability of the predator of the species whose reaction rate switches typically varies nonmonotonically with the external noise intensity with optimal survival about a critical noise strength we also outline the relationship with the case where all reaction rates switch on markedly different time scales

saddles arrows and spirals deterministic trajectories in cyclic competition of four species population dynamics in systems composed of cyclically competing species has been of increasing interest recently here we investigate a system with four or more species using mean field theory we study in detail the trajectories in configuration space of the population fractions we discover a variety of orbits shaped like saddles spirals and straight lines many of their properties are found explicitly most remarkably we identify a collective variable that evolves simply as an exponential q e t where is a function of the reaction rates it provides information on the state of the system for late times as well as for t we discuss implications of these results for the evolution of a finite stochastic system a generalization to an arbitrary number of cyclically competing species yields valuable insights into universal properties of such systems

phase transitions and spatio temporal fluctuations in stochastic lattice lotka volterra models we study the general properties of stochastic two species models for predator prey competition and coexistence with lotka volterra type interactions defined on a d dimensional lattice introducing spatial degrees of freedom and allowing for stochastic fluctuations generically invalidates the classical deterministic mean field picture already within mean field theory however spatial constraints modeling locally limited resources lead to the emergence of a continuous active to absorbing state phase transition field theoretic arguments supported by monte carlo simulation results indicate that this transition which represents an extinction threshold for the predator population is governed by the directed percolation universality class in the active state where predators and prey coexist the classical center singularities with associated population cycles are replaced by either nodes or foci in the vicinity of the stable nodes the system is characterized by essentially stationary localized clusters of predators in a sea of prey near the stable foci however the stochastic lattice lotka volterra system displays complex correlated spatio temporal patterns of competing activity fronts correspondingly the population densities in our numerical simulations turn out to oscillate irregularly in time with amplitudes that tend to zero in the thermodynamic limit yet in finite systems these oscillatory fluctuations are quite persistent and their features are determined by the intrinsic interaction rates rather than the initial conditions we emphasize the robustness of this scenario with respect to various model perturbations

existence results for quasilinear parabolic hemivariational inequalities abstract this paper is devoted to the periodic problem for quasilinear parabolic hemivariational inequalities at resonance as well as at nonresonance by use of the theory of multi valued pseudomonotone operators the notion of generalized gradient of clarke and the property of the first eigenfunction we build a landesman lazer theory in the nonsmooth framework of quasilinear parabolic hemivariational inequalities

coexistence versus extinction in the stochastic cyclic lotka volterra model cyclic dominance of species has been identified as a potential mechanism to maintain biodiversity see e g b kerr m a riley m w feldman and b j m bohannan nature 418 171 2002 and b kirkup and m a riley nature 428 412 2004 through analytical methods supported by numerical simulations we address this issue by studying the properties of a paradigmatic non spatial three species stochastic system namely the rock paper scissors or cyclic lotka volterra model while the deterministic approach rate equations predicts the coexistence of the species resulting in regular yet neutrally stable oscillations of the population densities we demonstrate that fluctuations arising in the system with a finite number of agents drastically alter this picture and are responsible for extinction after long enough time two of the three species die out as main findings we provide analytic estimates and numerical computation of the extinction probability at a given time we also discuss the implications of our results for a broad class of competing population systems

coevolution of nodes and links diversity driven coexistence in cyclic competition of three species when three species compete cyclically in a well mixed stochastic system of n individuals extinction is known to typically occur at times scaling as the system size n this happens for example in rock paper scissors games or conserved lotka volterra models in which every pair of individuals can interact on a complete graph here we show that if the competing individuals also have a social temperament to be either introverted or extroverted leading them to cut or add links respectively then long living states in which all species coexist can occur these nonequilibrium quasisteady states only occur when both introverts and extroverts are present thus showing that diversity can lead to stability in complex systems in this case it enables a subtle balance between species competition and network dynamics to be maintained

predator prey cycles from resonant amplification of demographic stochasticity we present the simplest individual level model of predator prey dynamics and show via direct calculation that it exhibits cycling behavior the deterministic analogue of our model recovered when the number of individuals is infinitely large is the volterra system with density dependent prey reproduction which is well known to fail to predict cycles this difference in behavior can be traced to a resonant amplification of demographic fluctuations which disappears only when the number of individuals is strictly infinite our results indicate that additional biological mechanisms such as predator satiation may not be necessary to explain observed predator prey cycles in real finite populations

optimal control of feedback control systems governed by hemivariational inequalities this paper is mainly concerned with the feedback control systems governed by evolution hemivariational inequalities by using the properties of multimaps and clarke s subdifferential we formulate some sufficient conditions to guarantee the existence result of feasible pairs of the feedback control systems we also present an existence result of optimal control pairs for an optimal control problem we emphasize that our results cannot be obtained straightforwardly from the previous works

extinction in the lotka volterra model birth death processes often exhibit an oscillatory behavior we investigate a particular case where the oscillation cycles are marginally stable on the mean field level an iconic example of such a system is the lotka volterra model of predator prey interaction fluctuation effects due to discreteness of the populations destroy the mean field stability and eventually drive the system toward extinction of one or both species we show that the corresponding extinction time scales as a certain power law of the population sizes this behavior should be contrasted with the extinction of models stable in the mean field approximation in the latter case the extinction time scales exponentially with size

coexistence and survival in conservative lotka volterra networks analyzing coexistence and survival scenarios of lotka volterra lv networks in which the total biomass is conserved is of vital importance for the characterization of long term dynamics of ecological communities here we introduce a classification scheme for coexistence scenarios in these conservative lv models and quantify the extinction process by employing the pfaffian of the network s interaction matrix we illustrate our findings on global stability properties for general systems of four and five species and find a generalized scaling law for the extinction time

stable and unstable attractors in boolean networks boolean networks at the critical point have been a matter of debate for many years as e g the scaling of numbers of attractors with system size recently it was found that this number scales superpolynomially with system size contrary to a common earlier expectation of sublinear scaling we point out here that these results are obtained using deterministic parallel update where a large fraction of attractors are an artifact of the updating scheme this limits the significance of these results for biological systems where noise is omnipresent here we take a fresh look at attractors in boolean networks with the original motivation of simplified models for biological systems in mind we test the stability of attractors with respect to infinitesimal deviations from synchronous update and find that most attractors are artifacts arising from synchronous clocking the remaining fraction of attractors are stable against fluctuating delays the average number of these stable attractors grows sublinearly with system size in the numerically tractable range

evolutionary stability of ecological hierarchy a self similar hierarchical solution that is both dynamically and evolutionarily stable is found for the multidimensional lotka volterra equation with a single chain of prey predator relations this gives a simple and natural explanation to the key features of hierarchical ecosystems such as its ubiquity pyramidal population distribution and higher aggressiveness among higher trophic levels

spirals and coarsening patterns in the competition of many species a complex ginzburg landau approach in order to model real ecological systems one has to consider many species that interact in complex ways however most of the recent theoretical studies have been restricted to few species systems with rather trivial interactions the few studies dealing with larger number of species and or more complex interaction schemes are mostly restricted to numerical explorations in this paper we determine starting from the deterministic mean field rate equations for large classes of systems the space of coexistence fixed points at which biodiversity is maximal for systems with a single coexistence fixed point we derive complex ginzburg landau equations that allow to describe space time pattern realized in two space dimensions for selected cases we compare the theoretical predictions with the pattern observed in numerical simulations

evolutionary game theory theoretical concepts and applications to microbial communities ecological systems are complex assemblies of large numbers of individuals interacting competitively under multifaceted environmental conditions recent studies using microbial laboratory communities have revealed some of the self organization principles underneath the complexity of these systems a major role of the inherent stochasticity of its dynamics and the spatial segregation of different interacting species into distinct patterns has thereby been established it ensures the viability of microbial colonies by allowing for species diversity cooperative behavior and other kinds of social behavior a synthesis of evolutionary game theory nonlinear dynamics and the theory of stochastic processes provides the mathematical tools and a conceptual framework for a deeper understanding of these ecological systems we give an introduction into the modern formulation of these theories and illustrate their effectiveness focussing on selected examples of microbial systems intrinsic fluctuations stemming from the discreteness of individuals are ubiquitous and can have an important impact on the stability of ecosystems in the absence of speciation extinction of species is unavoidable it may however take very long times we provide a general concept for defining survival and extinction on ecological time scales spatial degrees of freedom come with a certain mobility of individuals when the latter is sufficiently high bacterial community structures can be understood through mapping individual based models in a continuum approach onto stochastic partial differential equations these allow progress using methods of nonlinear dynamics such as bifurcation analysis and invariant manifolds we conclude with a perspective on the current challenges in quantifying bacterial pattern formation and how this might have an impact on fundamental research in non equilibrium physics

coevolutionary dynamics from finite to infinite populations traditionally frequency dependent evolutionary dynamics is described by deterministic replicator dynamics assuming implicitly infinite population sizes only recently have stochastic processes been introduced to study evolutionary dynamics in finite populations however the relationship between deterministic and stochastic approaches remained unclear here we solve this problem by explicitly considering large populations in particular we identify different microscopic stochastic processes that lead to the standard or the adjusted replicator dynamics moreover differences on the individual level can lead to qualitatively different dynamics in asymmetric conflicts and depending on the population size can even invert the direction of the evolutionary process

self organization of five species in a cyclic competition game abstract cyclic competition game models particularly the rock paper scissors model play important roles in exploring the problem of multi species coexistence in spatially ecological systems we propose an extended rock paper scissors game to model cyclic interactions among five species and find that two of the five can coexistent when biodiversity disappears which is different from the rock paper scissors game as the number of fingers is five we named the new model the fingers game where the thumb forefinger middle finger ring finger and little finger cyclically dominate their subsequent species and are dominated by their former species we investigate the fingers model in two ways direct simulations and nonlinear partial differential equations an important finding is that the number of species in a cyclic competition game has an influence on the emergence of biodiversity to be specific the rock paper scissors model is in favor of maintaining biodiversity in comparison with the fingers model when the variables population size reproduction rate selection rate and migration rate are the same it is also shown that the mobility and reproduction rate can promote or jeopardize biodiversity

topologically robust zero sum games and pfaffian orientation how network topology determines the long time dynamics of the antisymmetric lotka volterra equation to explore how the topology of interaction networks determines the robustness of dynamical systems we study the antisymmetric lotka volterra equation alve the alve is the replicator equation of zero sum games in evolutionary game theory in which the strengths of pairwise interactions between strategies are defined by an antisymmetric matrix such that typically some strategies go extinct over time here we show that there also exist topologically robust zero sum games such as the rock paper scissors game for which all strategies coexist for all choices of interaction strengths we refer to such zero sum games as coexistence networks and construct coexistence networks with an arbitrary number of strategies by mapping the long time dynamics of the alve to the algebra of antisymmetric matrices we identify simple graph theoretical rules by which coexistence networks are constructed examples are triangulations of cycles characterized by the golden ratio phi 1 6180 cycles with complete subnetworks and non hamiltonian networks in graph theoretical terms we extend the concept of a pfaffian orientation from even sized to odd sized networks our results show that the topology of interaction networks alone can determine the long time behavior of nonlinear dynamical systems and may help to identify robust network motifs arising for example in ecology

fixation in a cyclic lotka volterra model we study a cyclic lotka volterra model of n interacting species populating a d dimensional lattice in the realm of a kirkwood approximation a critical number of species nc d above which the system fixates is determined analytically we find nc d 5 14 23 in dimensionsdd 1 2 3 in remarkably good agreement with simulation results in two dimensions a cyclic variant of the lotka volterra model of interacting populations originally introduced by vito volterra for the description of the struggle for existence among species 1 2 has then appeared in a number of apparently unrelated fields ranging from plasma physics 3 to integrable systems 4 5 recently the cyclic lotka volterra model also known as the cyclic voter model has attracted considerable interest as it was realized that introduction of the spatial structure drastically enriches the dynamics 6 11 namely if species live on a one dimensional 1d lattice a homogeneous initial state evolves into a coarsening mosaic of interacting species this heterogeneous spatial structure spontaneously develops when the number of species is sufficiently small n nc fixation occurs i e the system approaches a frozen state little is known in higher dimensions not even the existence of nc has yet been established theoretically or numerically in simulations on two dimensional 2d lattices with n 6 10 species no sign of fixation has been found and instead a reactive steady state has been observed 6 11 in this letter we investigate the cyclic lotka volterra model in the framework of a kirkwood like approximation this approach predicts a finite nc in all spatial dimensions in the following we shall use the language of the voter model 12 consider the cyclic voter model with n possible opinions each site of a d dimensional cubic lattice is occupied by a voter which has an opinion labelled by with d 1 n voters can change their opinions in a cyclic manner 1 modulo n according to the opinions of their neighbourhood specifically the following sequential dynamics is implemented i we randomly choose a site of opinion say and one of its 2d nearest neighbours of opinion ii if d 1 then the chosen site changes its opinion from to d 1 and iii otherwise opinion does not change we set the timescale so that in unit time each site of the lattice is chosen once on average when nd 2 the cyclic voter model is

synchronization and extinction in cyclic games with mixed strategies we consider cyclic lotka volterra models with three and four strategies where at every interaction agents play a strategy using a time dependent probability distribution agents learn from a loss by reducing the probability to play a losing strategy at the next interaction for that an agent is described as an urn containing balls of three and four types respectively where after a loss one of the balls corresponding to the losing strategy is replaced by a ball representing the winning strategy using both mean field rate equations and numerical simulations we investigate a range of quantities that allows us to characterize the properties of these cyclic models with time dependent probability distributions for the three strategy case in a spatial setting we observe a transition from neutrally stable to stable when changing the level of discretization of the probability distribution for large values of yielding a good approximation to a continuous distribution spatially synchronized temporal oscillations dominate the system for the four strategy game the system is always neutrally stable but different regimes emerge depending on the size of the system and the level of discretization

extinction in neutrally stable stochastic lotka volterra models populations of competing biological species exhibit a fascinating interplay between the nonlinear dynamics of evolutionary selection forces and random fluctuations arising from the stochastic nature of the interactions the processes leading to extinction of species whose understanding is a key component in the study of evolution and biodiversity are influenced by both of these factors here we investigate a class of stochastic population dynamics models based on generalized lotka volterra systems in the case of neutral stability of the underlying deterministic model the impact of intrinsic noise on the survival of species is dramatic it destroys coexistence of interacting species on a time scale proportional to the population size we introduce a new method based on stochastic averaging which allows one to understand this extinction process quantitatively by reduction to a lower dimensional effective dynamics this is performed analytically for two highly symmetrical models and can be generalized numerically to more complex situations the extinction probability distributions and other quantities of interest we obtain show excellent agreement with simulations

solution of the stability problem for a class of generalized volterra prey predator systems where ei pij and qj are real constants ui are real valued continuous functions of t for t 0 and zii dui dt both equations are required for i 1 2 m condition lb is here considered to be a restriction on the vector e ei rather than on the matrix p pii as is more usual the hypothesis 0 0 qi 0 is appropriate in applications but as far as the mathematical development is concerned it could be replaced by qiui 0 0 a discussion of historical background can be found in the papers 4 5 to which the present paper is a sequel suffice it to say here that the chief landmarks are volterra s book 6 la salle s theorem 1 21 and krikorian s analysis of the case n 3 3 references 4 5 provide useful

approximate controllability of differential inclusions in hilbert spaces abstract in this paper controllability for the system originating from semilinear functional differential equations in hilbert spaces is studied we consider the problem of approximate controllability of semilinear differential inclusion assuming that semigroup generated by the linear part of the inclusion is compact and under the assumption that the corresponding linear system is approximately controllable by using resolvent of controllability gramian operator and fixed point theorem sufficient conditions have been formulated and proved an example is presented to illustrate the utility and applicability of the proposed method