the not a knot piecewise interpolatory cubic polynomial in this paper the main idea of the not a knot cubic spline of de boor 8 will be extended to all the interior points knots of the spline interval to obtain a piecewise interpolatory cubic polynomial instead of the original and conventional not a knot cubic spline to construct this piecewise polynomial we do not need to solve a tridiagonal system of linear equations and this provides a shortcut which helps us save the time of doing more computations the order of convergence of this piecewise cubic polynomial is o h 4

handling solid fluid interfaces for viscous flows explicit jump approximation vs ghost cell approaches the ghost cell approaches gca for handling stationary solid boundaries regular or irregular are first investigated theoretically and numerically for the diffusion equation with dirichlet boundary conditions the main conclusion of this part of investigation is that the approximation for the diffusion term has to be second order accurate everywhere in order for the numerical solution to be rigorously second order accurate violating this principle the linear and quadratic gcas have the following shortcomings 1 restrictive constraints on grid size when the viscosity is small 2 susceptibleness to instability of a time explicit formulation for strongly transient flows 3 convergence deterioration to zeroth or first order for solutions with high frequency modes therefore the widely used linear extrapolation for enforcing no slip boundary conditions should be avoided even for regular solid boundaries as a remedy a simple method based on explicit jump approximation eja is proposed eja hinges on the idea that a velocity derivative jump at the boundary reduces to the value of the velocity derivative at the fluid side because the velocity of the stationary boundary is zero although the time marching linear system of eja is not symmetric it is strictly diagonal dominant with positive diagonal entries numerical results show that over a large range of viscosity and grid sizes eja performs much better than gcas in terms of stability and accuracy furthermore the second order convergence of eja does not depend on viscosity and the spectrum of the solution as those of gcas do this paper is written with enough details so that one can reproduce the numerical results

a new interface tracking method the polygonal area mapping method we present a new method the polygonal area mapping pam method for tracking a non diffusive immiscible material interface between two materials in two dimensional incompressible flows this method represents material areas explicitly as piecewise polygons traces characteristic points on polygon boundaries along pathlines and calculates new material areas inside interface cells via polygon clippings in a discrete manner the new method has very little spatial numerical diffusion and tracks the interface singularities naturally and accurately in addition to high accuracy the pam method can be directly used on either a structured rectangular mesh or an unstructured mesh without any modifications the mass conservation is enforced by heuristic algorithms adjusting the volume of material polygons the results from a set of widely used benchmark tests show that the pam method is superior to existing volume of fluid vof methods

the importance of being discrete life always wins on the surface many systems in chemistry biology finance and social sciences present emerging features that are not easy to guess from the elementary interactions of their microscopic individual components in the past the macroscopic behavior of such systems was modeled by assuming that the collective dynamics of microscopic components can be effectively described collectively by equations acting on spatially continuous density distributions it turns out that to the contrary taking into account the actual individual discrete character of the microscopic components of these systems is crucial for explaining their macroscopic behavior in fact we find that in conditions in which the continuum approach would predict the extinction of all of the population respectively the vanishing of the invested capital or the concentration of a chemical substance etc the microscopic granularity insures the emergence of macroscopic localized subpopulations with collective adaptive properties that allow their survival and development in particular it is found that in two dimensions life the localized proliferating phase always prevails

replicator dynamics with turnover of players we study adaptive dynamics in games where players abandon the population at a given rate and are replaced by naive players characterized by a prior distribution over the admitted strategies we demonstrate how such a process leads macroscopically to a variant of the replicator equation with an additional term accounting for player turnover we study how nash equilibria and the dynamics of the system are modified by this additional term for prototypical examples such as the rock paper scissors game and different classes of two action games played between two distinct populations we conclude by showing how player turnover can account for nontrivial departures from nash equilibria observed in data from lowest unique bid auctions

a wavelet mra based adaptive semi lagrangian method for the relativistic vlasov maxwell system in this paper we present a new method for the numerical solution of the relativistic vlasov maxwell system on a phase space grid using an adaptive semi lagrangian method the adaptivity is performed through a wavelet multiresolution analysis which gives a powerful and natural refinement criterion based on the local measurement of the approximation error and regularity of the distribution function therefore the multiscale expansion of the distribution function allows to get a sparse representation of the data and thus save memory space and cpu time we apply this numerical scheme to reduced vlasov maxwell systems arising in laser plasma physics interaction of relativistically strong laser pulses with overdense plasma slabs is investigated these vlasov simulations revealed a rich variety of phenomena associated with the fast particle dynamics induced by electromagnetic waves as electron trapping particle acceleration and electron plasma wavebreaking however the wavelet based adaptive method that we developed here does not yield significant improvements compared to vlasov solvers on a uniform mesh due to the substantial overhead that the method introduces nonetheless they might be a first step towards more efficient adaptive solvers based on different ideas for the grid refinement or on a more efficient implementation here the vlasov simulations are performed in a two dimensional phase space where the development of thin filaments strongly amplified by relativistic effects requires an important increase of the total number of points of the phase space grid as they get finer as time goes on the adaptive method could be more useful in cases where these thin filaments that need to be resolved are a very small fraction of the hyper volume which arises in higher dimensions because of the surface to volume scaling and the essentially one dimensional structure of the filaments moreover the main way to improve the efficiency of the adaptive method is to increase the local character in phase space of the numerical scheme by considering multiscale reconstruction with more compact support and by replacing the semi lagrangian method with more local in space numerical scheme as compact finite difference schemes discontinuous galerkin method or finite element residual schemes which are well suited for parallel domain decomposition techniques

coexistence and critical behavior in a lattice model of competing species in the present paper we study a lattice model of two species competing for the same resources monte carlo simulations for d 1 2 and 3 show that when resources are easily available both species coexist however when the supply of resources is on an intermediate level the species with slower metabolism becomes extinct on the other hand when resources are scarce it is the species with faster metabolism that becomes extinct the range of coexistence of the two species increases with dimension we suggest that our model might describe some aspects of the competition between normal and tumor cells with such an interpretation examples of tumor remission recurrence and different morphologies are presented in the d 1 and d 2 models we analyze the nature of phase transitions they are either discontinuous or belong to the directed percolation universality class and in some cases they have an active subcritical phase in the d 2 case one of the transitions seems to be characterized by critical exponents that differ from directed percolation ones but this transition could be also weakly discontinuous in the d 3 version monte carlo simulations are in a good agreement with the solution of the mean field approximation this approximation predicts that oscillatory behavior occurs in the present model but only for d 2 for d 2 a steady state depends on the initial configuration in some cases

a fourth order accurate finite volume method with structured adaptive mesh refinement for solving the advection diffusion equation we present a fourth order accurate algorithm for solving poisson s equation the heat equation and the advection diffusion equation on a hierarchy of block structured adaptively refined grids for spatial discretization finite volume stencils are derived for the divergence operator and laplacian operator in the context of structured adaptive mesh refinement and a variety of boundary conditions the resulting linear system is solved with a multigrid algorithm for time integration we couple the elliptic solver to a fourth order accurate runge kutta method introduced by kennedy and carpenter appl numer math 44 2003 pp 139 181 which enables us to treat the nonstiff advection term explicitly and the stiff diffusion term implicitly we demonstrate the spatial and temporal accuracy by comparing results with analytical solutions because of the general formulation of the approach the algorithm is easily extensible to more complex physical systems

deterministic evolutionary game dynamics in finite populations evolutionary game dynamics describes the spreading of successful strategies in a population of reproducing individuals typically the microscopic definition of strategy spreading is stochastic such that the dynamics becomes deterministic only in infinitely large populations here we present a microscopic birth death process that has a fully deterministic strong selection limit in well mixed populations of any size additionally under weak selection from this process the frequency dependent moran process is recovered this makes it a natural extension of the usual evolutionary dynamics under weak selection we find simple expressions for the fixation probabilities and average fixation times of the process in evolutionary games with two players and two strategies for cyclic games with two players and three strategies we show that the resulting deterministic dynamics crucially depends on the initial condition in a nontrivial way

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

evolutionary games on minimally structured populations population structure induced by both spatial embedding and more general networks of interaction such as model social networks have been shown to have a fundamental effect on the dynamics and outcome of evolutionary games these effects have however proved to be sensitive to the details of the underlying topology and dynamics here we introduce a minimal population structure that is described by two distinct hierarchical levels of interaction similar to the structured metapopulation concept of ecology and island models in population genetics we believe this model is able to identify effects of spatial structure that do not depend on the details of the topology while effects depending on such details clearly lie outside the scope of our approach we expect that those we are able to reproduce should be generally applicable to a wide range of models we derive the dynamics governing the evolution of a system starting from fundamental individual level stochastic processes through two successive mean field approximations in our model of population structure the topology of interactions is described by only two parameters the effective population size at the local scale and the relative strength of local dynamics to global mixing we demonstrate for example the existence of a continuous transition leading to the dominance of cooperation in populations with hierarchical levels of unstructured mixing as the benefit to cost ratio becomes smaller then the local population size applying our model of spatial structure to the repeated prisoner s dilemma we uncover a counterintuitive mechanism by which the constant influx of defectors sustains cooperation further exploring the phase space of the repeated prisoner s dilemma and also of the rock paper scissor game we find indications of rich structure and are able to reproduce several effects observed in other models with explicit spatial embedding such as the maintenance of biodiversity and the emergence of global oscillations

on donating regions lagrangian flux through a fixed curve lagrangian flux through a fixed curve segment naturally gives rise to the notion of a donating region a compact point set in which each particle will pass through the curve and contribute to the flux a precise geometric determination of the donating region has not been available until now the author proposes an explicit constructive and analytical definition of the donating region based on two characteristic curves of the flow field viz streaklines and timelines it is also shown that within a time interval of finite size each particle in the proposed donating region has a net effect of going across the fixed curve once the presented analysis might be potentially useful for flow topology studies lagrangian flux calculation and explicit interface tracking

on competitive lotka volterra model in random environments focusing on competitive lotka volterra model in random environments this paper uses regime switching diffusions to model the dynamics of the population sizes of n different species in an ecosystem subject to the random changes of the external environment it is demonstrated that the growth rates of the population sizes of the species are bounded above moreover certain long run average limits of the solution are examined from several angles a partial stochastic principle of competitive exclusion is also derived finally simple examples are used to demonstrate our findings

infection promotes species coexistence rock paper scissors game with epidemic on graphs abstract we combine the rock paper scissors rps game with sis epidemic model we study the effect of infection on population dynamics of rps games with epidemic by using the metapopulation dynamic model on graphs via infection an individual r changes to i with infection rate an infected individual i returns to r with recovery rate all agents move by random walk between patches subpopulations and the rps game with epidemic is performed in each patch the dynamics are represented by the reaction diffusion equations in diffusively coupled reactors we obtain the solutions of the reaction diffusion equations for the mean field dynamics in a single patch analytically the numerical solutions are obtained for metapopulation dynamics on graphs with three nodes the analytical solutions are also obtained for metapopulation dynamics on complete graph when the recovery rate is lower than the critical value infected individuals survive the dynamics exhibit stable focuses and all species coexist infection promotes the coexistence of species

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

evolutionary dynamics intrinsic noise and cycles of cooperation we use analytical techniques based on an expansion in the inverse system size to study the stochastic evolutionary dynamics of finite populations of players interacting in a repeated prisoner s dilemma game we show that a mechanism of amplification of demographic noise can give rise to coherent oscillations in parameter regimes where deterministic descriptions converge to fixed points with complex eigenvalues these quasicycles between cooperation and defection have previously been observed in computer simulations here we provide a systematic and comprehensive analytical characterization of their properties we are able to predict their power spectra as a function of the mutation rate and other model parameters and to compare the relative magnitude of the cycles induced by different types of underlying microscopic dynamics we also extend our analysis to the iterated prisoner s dilemma game with a win stay lose shift strategy appropriate in situations where players are subject to errors of the trembling hand type

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

evolution equation of phenotype distribution general formulation and application to error catastrophe an equation describing the evolution of phenotypic distribution is derived using methods developed in statistical physics the equation is solved by using the singular perturbation method and assuming that the number of bases in the genetic sequence is large applying the equation to the mutation selection model by eigen provides the critical mutation rate for the error catastrophe phenotypic fluctuation of clones individuals sharing the same gene is introduced into this evolution equation with this formalism it is found that the critical mutation rate is sometimes increased by the phenotypic fluctuations i e noise can enhance robustness of a fitted state to mutation our formalism is systematic and general while approximations to derive more tractable evolution equations are also discussed

integration of polynomials over linear polyhedra in euclidean three dimensional space abstract this paper is concerned with explicit formulas and algorithms for computing integrals of polynomials over a linear polyhedron in euclidean three dimensional space symbolic formulas for surface and volume integration are given two different approaches are discussed the first algorithm is obtained by transforming a volume integral into a surface integral and then into a parametric line integral while the second algorithm is obtained by transforming a volume integral into a surface integral and then into a parametric double integral these algorithms and formulas are followed by an application example for which we have explained the detailed computational scheme the symbolic results presented in this paper may lead to an easy incorporation of global geometric properties of solid objects for example the volume centre of mass moments of inertia required in the engineering design process

a fully mass and volume conserving implementation of a characteristic method for transport problems the characteristics mixed method considers the transport not of a single point or fluid particle but rather the mass in an entire region of fluid this mass is transported along the characteristic curves of the hyperbolic part of the transport equation and the scheme thereby produces very little numerical dispersion conserves mass locally and can use long time steps however since the shape of a characteristic trace back region must be approximated in numerical implementation its volume may be incorrect resulting in inaccurate concentration densities and further inaccurate reaction dynamics we present a simple modification to the characteristics mixed method that conserves both mass and volume of the transported fluid regions our algorithm also handles boundary conditions through a space time change of variables in the trace back routines which allows the boundary to be treated as if it were interior to the domain nearly point sources such as wells present special difficulties since characteristic trace back curves converge in their vicinity we also present techniques that allow one to conservatively implement wells the techniques are illustrated in five numerical examples

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

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

on generalized donating regions classifying lagrangian fluxing particles through a fixed curve in the plane abstract for a fixed simple curve in a nonautonomous flow the fluxing index of a passively advected lagrangian particle is the total number of times it goes across the curve within a given time interval such indices naturally induce flux sets equivalence classes of the particles at the initial time this line of research mainly concerns donating regions the explicit construction of the flux sets from a sufficiently continuous velocity field in the author s previous paper zhang 2013 13 the flux sets with indices 1 were constructed from characteristic curves of the flow field this work generalizes the notion of donating regions and shows that flux sets are index by index equivalent to the generalized donating regions for any finite integer index provided that the two backward streaklines seeded at the two endpoints of the simple curve neither intersect nor self intersect all lagrangian particles marked by their initial positions are thus classified by their fluxing indices

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

a comparison of thee 3 and not a knot cubic splines abstract the algorithms and schemes of the cubic spline interpolation with two end conditions which do not require thederivative information at the end points are of great practical importance and have been included in several general purpose software libraries the cubic spline with not a knot end conditions is one of them and it is implemented in the widely used imsl and nag software libraries in this paper we compare thee 3 cubic spline with the not a knot cubic spline and we will show that thee 3 cubic spline is more accurate than the second one and also it has superconvergence properties which the not a knot cubic spline does not have these superconvergence properties of thee 3 cubic spline can be used in different fields for better approximation

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

an algebraic cubature formula on curvilinear polygons abstract we implement in matlab a gauss like cubature formula on bivariate domains whose boundary is a piecewise smooth jordan curve curvilinear polygons the key tools are green s integral formula together with the recent software package chebfun to approximate the boundary curve close to machine precision by piecewise chebyshev interpolation several tests are presented including some comparisons of this new routine chebfungauss with the recent splinegauss that approximates the boundary by splines

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

bursts in three strategy evolutionary ordinal potential games on a square lattice abstract evolutionary potential games provide an analogy for equilibrium statistical physics under suitable conditions that is broken if cyclic components are added using monte carlo simulations we study the effect of weak cyclic perturbations on the equilibrium in a system of players located on a square lattice while the dynamics is controlled by a logit rule the pair interaction is composed of a symmetric three strategy coordination game of unit strength a weak self dependent term and a cyclic rock paper scissors component the self dependent component favors the first rock strategy which dominates the system behavior at low noises if the strength of the cyclic component is below a threshold value in the opposite case the predator of the first strategy paper rules the behavior at low noises while the first strategy can occasionally form growing domains bursts diminished after the appearance of the third one the variation of noise modifies the invasion velocities as well as the rate of nucleation processes and yields drastic changes in the strategy frequencies fluctuations average payoffs and spatio temporal patterns

highly accurate lagrangian flux calculation via algebraic quadratures on spline approximated donating regions abstract lagrangian flux through a fixed curve segment within a time interval t n t n k can be formulated as an integral at the initial time t n over a compact point set called donating region in which each particle will pass through the curve during the time interval and contribute to the flux based an explicit constructive and analytical solution of the donating region the author proposes algorithms of lagrangian flux calculation lfc as algebraic quadratures over the spline approximated donating regions this generic formulation yields high accuracies up to the 8th order both in time and space as another benefit lfc leads to a conservative semi lagrangian method whose time step size is free of the eulerian stability constraint c r 1 the high accuracy and robustness of the proposed lfc algorithms are demonstrated by various numerical tests with courant numbers ranging from 1 to 10 000 and with accuracies from the 2nd order to the 8th order lfc might also be useful in designing hybrid eulerian methods for complex geometries

predicting population extinction from early observations of the lotka volterra system abstract population extinction is one of the central themes in population biology we propose a statistical algorithm for long term prediction of an extinction event in the paradigmatic predator prey model the algorithm is based on noisy and sporadic observations of the lotka volterra lv system at the early stages of its evolution when the system is still very far from extinction there are two stages in the algorithm first the unknown parameters reaction rates of the lv system are estimated using the approximate bayesian computation method then an analytical expression for the time scale of extinction which involves the estimated parameters is applied to compute the probability density function of extinction time the proposed algorithm is validated by numerical simulations for the case of a stochastic lv system specified by the birth death rate equations the algorithm can be seen as an initial step in the quest for long term prediction of rare catastrophic events in complex stochastic dynamic systems epidemics host parasite dynamics enzyme kinetics dynamic trading etc

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

clenshaw curtis and gauss legendre quadrature for certain boundary element integrals following a recent article by trefethen siam review 50 2008 pp 67 87 the use of clenshaw curtis quadrature rather than gauss legendre quadrature for nearly singular integrals which arise in the boundary element method has been investigated when these quadrature rules are used in association with the sinh transformation the authors have concluded after considering asymptotic estimates of the truncation errors for certain proto type functions arising in this context that gauss legendre quadrature should continue to be the preferred quadrature rule

a golden point rule in rock paper scissors lizard spock game abstract we study a novel five species system on two dimensional lattices when each species have two superior and two inferior partners here we simplify the huge parameter space of predation probability to only two parameters both of monte carlo simulation and mean field theory reveal that two of strategies may die out when the ratio of the two parameters is close to the golden point 0 618 and the remaining three strategies are provided a cyclic dominance system

pattern formation in individual based systems with time varying parameters we study the patterns generated in finite time sweeps across symmetry breaking bifurcations in individual based models similar to the well known kibble zurek scenario of defect formation large scale patterns are generated when model parameters are varied slowly whereas fast sweeps produce a large number of small domains the symmetry breaking is triggered by intrinsic noise originating from the discrete dynamics at the microlevel based on a linear noise approximation we calculate the characteristic length scale of these patterns we demonstrate the applicability of this approach in a simple model of opinion dynamics a model in evolutionary game theory with a time dependent fitness structure and a model of cell differentiation our theoretical estimates are confirmed in simulations in further numerical work we observe a similar phenomenon when the symmetry breaking bifurcation is triggered by population growth

expectation propagation for continuous time stochastic processes we consider the inverse problem of reconstructing the posterior measure over the trajectories of a diffusion process from discrete time observations and continuous time constraints we cast the problem in a bayesian framework and derive approximations to the posterior distributions of single time marginals using variational approximate inference giving rise to an expectation propagation type algorithm for non linear diffusion processes this is achieved by leveraging moment closure approximations we then show how the approximation can be extended to a wide class of discrete state markov jump processes by making use of the chemical langevin equation our empirical results show that the proposed method is computationally efficient and provides good approximations for these classes of inverse problems

cellwise conservative unsplit advection for the volume of fluid method we present a cellwise conservative unsplit ccu advection scheme for the volume of fluid method vof in 2d contrary to other schemes based on explicit calculations of the flux balances the ccu advection adopts a cellwise approach where the pre images of the control volumes are traced backwards through the flow map the donating regions of the fluxes are calculated via the streaklines of the grid intersections represented as polygonal chains whose vertices are determined by backward tracing of particles injected in the flow at different times high order accuracy is obtained from the fourth order runge kutta method where intermediate velocities along pathlines are determined with quadratic temporal and bicubic spatial interpolations the volumes of the donating regions are corrected in order to fulfill the discrete continuity of incompressible flows consequently the calculation produces non overlapping donating regions and pre images with conforming edges to their neighbors resulting in the conservativeness and the boundedness liquid volume fraction inside the interval 0 1 of the ccu advection scheme finally the update of the liquid volume fractions is computed from the intersections of the pre image polygons with the reconstructed interfaces the ccu scheme is tested on several benchmark tests for the vof advection together with the standard piecewise linear interface calculation plic the geometrical errors of the ccu compare favorably with other unsplit vof plic schemes finally potential improvements of the vof method with the use of more precise interface representation techniques and the future extension of the ccu scheme to 3d are discussed cellwise vof update through backward tracing of the cell s pre image accuracy enhanced via bicubic spatial interpolation and polygonal chain streakline the ccu advection scheme is fully conservative and bounded no limitation on the courant number geometrical errors dominated by plic interface representation errors

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

gauss green cubature and moment computation over arbitrary geometries we have implemented in matlab a gauss like cubature formula over arbitrary bivariate domains with a piecewise regular boundary which is tracked by splines of maximum degree p spline curvilinear polygons the formula is exact for polynomials of degree at most 2n 1 using n cmn 2 nodes 1 c p m being the total number of points given on the boundary it does not need any decomposition of the domain but relies directly on univariate gauss legendre quadrature via green s integral formula several numerical tests are presented including computation of standard as well as orthogonal moments over a nonstandard planar region

boundary integration of polynomials over an arbitrary linear tetrahedron in euclidean three dimensional space abstract this paper is concerned with explicit integration formulas and algorithms for computing integrals of trivariate polynomials over an arbitrary linear tetrahedron in euclidean three dimensional space this basic three dimensional integral governing the problem is transformed to surface integrals by use of the divergence theorem the resulting two dimensional integrals are then transformed into convenient and computationally efficient line integrals these algorithms and explicit finite integration formulas are followed by an application example for which we have explained the detailed computational scheme the numerical result thus found is in complete agreement with previous works further it is shown that the present algorithms are much simpler and more economical as well in terms of arithmetic operations the symbolic finite integration formulas presented in this paper may lead to an easy incorporation of geometric properties of solid objects for example the centre of mass moment of inertia etc required in the engineering design process as well as several applications of numerical analysis where integration is required for example in the finite element and boundary integral equation methods

coevolutionary dynamics in large but finite populations coevolving and competing species or game theoretic strategies exhibit rich and complex dynamics for which a general theoretical framework based on finite populations is still lacking recently an explicit mean field description in the form of a fokker planck equation was derived for frequency dependent selection with two strategies in finite populations based on microscopic processes a traulsen j c claussen and c hauert phys rev lett 95 238701 2005 here we generalize this approach in a twofold way first we extend the framework to an arbitrary number of strategies and second we allow for mutations in the evolutionary process the deterministic limit of infinite population size of the frequency dependent moran process yields the adjusted replicator mutator equation which describes the combined effect of selection and mutation for finite populations we provide an extension taking random drift into account in the limit of neutral selection i e whenever the process is determined by random drift and mutations the stationary strategy distribution is derived this distribution forms the background for the coevolutionary process in particular a critical mutation rate uc is obtained separating two scenarios above uc the population predominantly consists of a mixture of strategies whereas below uc the population tends to be in homogeneous states for one of the fundamental problems in evolutionary biology the evolution of cooperation under darwinian selection we demonstrate that the analytical framework provides excellent approximations to individual based simulations even for rather small population sizes this approach complements simulation results and provides a deeper systematic understanding of coevolutionary dynamics

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

explicit gaussian quadrature rules for c 1 cubic splines with symmetrically stretched knot sequences we provide explicit expressions for quadrature rules on the space of c 1 cubic splines with non uniform symmetrically stretched knot sequences the quadrature nodes and weights are derived via an explicit recursion that avoids an intervention of any numerical solver and the rule is optimal that is it requires minimal number of nodes numerical experiments validating the theoretical results and the error estimates of the quadrature rules are also presented the paper provides expressions for quadrature rules on the space of c 1 cubic splines with non uniform symmetrically stretched knot sequences any function from the space is exactly integrated by the rule the quadrature nodes and weights are derived via explicit recursion formulae with no intervention of any numerical solver the rule is optimal i e it requires minimal number of nodes

dynamical behaviors determined by the lyapunov function in competitive lotka volterra systems dynamical behaviors of the competitive lotka volterra system even for 3 species are not fully understood in this paper we study this problem from the perspective of the lyapunov function we construct explicitly the lyapunov function using three examples of the competitive lotka volterra system for the whole state space 1 the general 2 species case 2 a 3 species model and 3 the model of may leonard the basins of attraction for these examples are demonstrated including cases with bistability and cyclical behavior the first two examples are the generalized gradient system where the energy dissipation may not follow the gradient of the lyapunov function in addition under a new type of stochastic interpretation the lyapunov function also leads to the boltzmann gibbs distribution on the final steady state when multiplicative noise is added

high order embedded runge kutta formulae abstract the criteria to be satisfied by embedded runge kutta pairs of formulae are reviewed two new formulae of orders 6 and 8 are presented together with tests on their efficiency relative to other high order formulae in current use

hypam a hybrid continuum particle model for incompressible free surface flows three major issues associated with numerical simulations of complex free surface flows viz interface tracking fragmentation and large physical jumps are addressed by a new hybrid continuum particle model hypam the new model consists of three parts 1 the polygonal area mapping method q zhang p l f liu a new interface tracking method the polygonal area mapping method j comput phys 227 8 2008 4063 4088 2 a new algorithm that decomposes the interested water phase into a continuum zone a buffer zone and a particle zone based on material topology and graph theory 3 a passive response assumption in which the air phase is assumed to respond passively to the continuum part of the water phase the incompressible inviscid euler equations and the equations describing the free fall of rigid bodies are used as the governing equations for the continuum buffer zone and the particle zone respectively and separately a number of examples including water droplet impact solitary wave propagation and dam break problems are simulated for the illustration and validation of hypam it is shown that hypam is more accurate and versatile than a continuum based volume of fluid model one major contribution of this work is the single phase decomposition algorithm useful for many other hybrid formulations neglecting surface tension viscosity and particle interactions hypam is currently limited to mildly fragmented free surface flows with high reynolds and weber numbers

orthogonal polynomials in matlab a suite of matlab programs has been developed as part of the book orthogonal polynomials computation and approximation oxford university press oxford 2004 by gautschi the package contains routines for generating orthogonal polynomials as well as routines dealing with applications in this paper a brief review of the first part of the package is given dealing with procedures for generating the three term recurrence relation for orthogonal polynomials and more general recurrence relations for sobolev orthogonal polynomials moment based methods and discretization methods and their implementation in matlab are among the principal topics discussed

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

on geometric interpolation by planar parametric polynomial curves in this paper the problem of geometric interpolation of planar data by parametric polynomial curves is revisited the conjecture that a parametric polynomial curve of degree n can interpolate 2n given points in r 2 is confirmed for n 5 under certain natural restrictions this conclusion also implies the optimal asymptotic approximation order more generally the optimal order 2n can be achieved as soon as the interpolating curve exists

a positivity preserving high order semi lagrangian discontinuous galerkin scheme for the vlasov poisson equations the vlasov poisson equations describe the evolution of a collisionless plasma represented through a probability density function pdf that self interacts via an electrostatic force one of the main difficulties in numerically solving this system is the severe time step restriction that arises from parts of the pdf associated with moderate to large velocities the dominant approach in the plasma physics community for removing these time step restrictions is the so called particle in cell pic method which discretizes the distribution function into a set of macro particles while the electric field is represented on a mesh several alternatives to this approach exist including fully lagrangian fully eulerian and so called semi lagrangian methods the focus of this work is the semi lagrangian approach which begins with a grid based eulerian representation of both the pdf and the electric field then evolves the pdf via lagrangian dynamics and finally projects this evolved field back onto the original eulerian mesh in particular we develop in this work a method that discretizes the 1 1 vlasov poisson system via a high order discontinuous galerkin dg method in phase space and an operator split semi lagrangian method in time second order accuracy in time is relatively easy to achieve via strang operator splitting with additional work using higher order splitting and a higher order method of characteristics we also demonstrate how to push this scheme to fourth order accuracy in time we show how to resolve all of the lagrangian dynamics in such a way that mass is exactly conserved positivity is maintained and high order accuracy is achieved the poisson equation is solved to high order via the smallest stencil local discontinuous galerkin ldg approach we test the proposed scheme on several standard test cases

numerical integration over polygons using an eight node quadrilateral spline finite element in this paper a cubature formula over polygons is proposed and analysed it is based on an eight node quadrilateral spline finite element c j li r h wang a new 8 node quadrilateral spline finite element j comp appl math 195 2006 54 65 and is exact for quadratic polynomials on arbitrary convex quadrangulations and for cubic polynomials on rectangular partitions the convergence of sequences of the above cubatures is proved for continuous integrand functions and error bounds are derived some numerical examples are given by comparisons with other known cubatures

orbits in a stochastic goodwin lotka volterra model abstract this paper examines the cycling behavior of a deterministic and a stochastic version of the economic interpretation of the lotka volterra model the goodwin model we provide a characterization of orbits in the deterministic highly non linear model we then study a stochastic version with brownian noise introduced via a heterogeneous productivity factor existence conditions for a solution to the system are provided we prove that the system produces cycles around a unique equilibrium point in finite time for general volatility levels using stochastic lyapunov techniques for recurrent domains numerical insights are provided

stochastic analysis of a pulse type prey predator model a stochastic lotka volterra model a so called pulse type model for the interaction between two species and their random natural environment is investigated the effect of a random environment is modeled as random pulse trains in the birth rate of the prey and the death rate of the predator the generalized cell mapping method is applied to calculate the probability distributions of the species populations at a state of statistical quasistationarity the time evolution of the population densities is studied and the probability of the near extinction time from an initial state to a critical state is obtained the effects on the ecosystem behaviors of the prey self competition term and of the pulse mean arrival rate are also discussed our results indicate that the proposed pulse type model shows obviously distinguishable characteristics from a gaussian type model and may confer a significant advantage for modeling the prey predator system under discrete environmental fluctuations

fractal properties of the lattice lotka volterra model the lattice lotka volterra llv model is studied using mean field analysis and monte carlo simulations while the mean field phase portrait consists of a center surrounded by an infinity of closed trajectories when the process is restricted to a two dimensional 2d square lattice local inhomogeneities fluctuations appear spontaneous local clustering is observed on lattice and homogeneous initial distributions turn into clustered structures reactions take place only at the interfaces between different species and the borders adopt locally fractal structure intercluster surface reactions are responsible for the formation of local fluctuations of the species concentrations the box counting fractal dimension of the llv dynamics on a 2d support is found to depend on the reaction constants while the upper bound of fractality determines the size of the local oscillators lacunarity analysis is used to determine the degree of clustering of homologous species besides the spontaneous clustering that takes place on a regular 2d lattice the effects of fractal supports on the dynamics of the llv are studied for supports of dimensionality d s 2 the lattice can for certain domains of the reaction constants adopt a poisoned state where only one of the species survives by appropriately selecting the fractal dimension of the substrate it is possible to direct the system into a poisoned or oscillatory steady state at will

fourth order interface tracking in two dimensions via an improved polygonal area mapping method we present an improved pam ipam method as the first fourth order interface tracking method whose convergence rates are independent of c 1 derivative discontinuities of the interface as an improved version of the polygonal area mapping pam method q zhang and p l f liu j comput phys 227 2008 pp 4063 4088 the accuracy of the ipam method is achieved via i augmenting the abstract data structure of pam to faithfully represent multiple components of material regions within a single cell ii removing restrictive assumptions of pam iii adjusting the volume of represented cell material regions via polygon ear removal and iv maintaining a relation h l r hh alpha between the eulerian grid size h and the lagrangian length scale that measures the distance between adjacent interface markers unlike volume of fluid methods and level set methods merging and separation of the tracked material is not automatically handled by the ipam method instead flexible subgrid resolutions

new interpolatory quadrature formulae with chebyshev abscissae abstract we study interpolatory quadrature formulae relative to the legendre weight function on 1 1 having as nodes the zeros of any one of the four chebyshev polynomials of degree n plus one of the points 1 or 1 in particular we derive explicit formulae for the weights and examine their positivity we determine the precise degree of exactness we obtain asymptotically optimal error bounds and we examine the definiteness f these quadrature formulae in addition we establish their convergence for riemann integrable functions on 1 1 as well as for functions having a monotonic singularity at 1 or 1

matlab program for quadrature in 2d abstract we discuss here the algorithms of twod a m atlab program for approximating integrals over generalized rectangles and sectors capabilities of the language are exploited to make twod very easy to use vectorization is exploited by a novel algorithm to make the program efficient and reliable twod provides for moderate singularities on boundaries of the region

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

stochastic model for tumor growth with immunization we analyze a stochastic model for tumor cell growth with both multiplicative and additive colored noises as well as nonzero cross correlations in between whereas the death rate within the logistic model is altered by a deterministic term characterizing immunization the birth rate is assumed to be stochastically changed due to biological motivated growth processes leading to a multiplicative internal noise moreover the system is subjected to an external additive noise which mimics the influence of the environment of the tumor the stationary probability distribution p s is derived depending on the finite correlation time the immunization rate and the strength of the cross correlation p s offers a maximum which becomes more pronounced for increasing immunization rate the mean first passage time is also calculated in order to find out under which conditions the tumor can suffer extinction its characteristics are again controlled by the degree of immunization and the strength of the cross correlation the behavior observed can be interpreted in terms of a biological model of tumor evolution

on a family of unsplit advection algorithms for volume of fluid methods volume of fluid vof methods are widely used for the interface tracking problem yet rigorous analyses of them are rare this paper presents such an analysis for incompressible flows by combining the theories of ordinary differential equations differential geometry and boolean algebra based on the concept of donating region dr q zhang siam rev 55 2013 pp 443 461 the author classifies the fluxing particles of a fixed control volume into four categories and derives three analytical solutions for the advection equation of the color function one edgewise solution provides a unified view of dr based advection algorithms for vof methods while another cellwise solution serves as the theoretical foundation of the recent polygonal area mapping method the well known second order convergence rates of streamline based vof advection algorithms are proved rigorously potential deterioration of this second order convergence is also discussed for several subtle issues such as exact mass conservation a

phase diagrams for three strategy evolutionary prisoner s dilemma games on regular graphs evolutionary prisoner s dilemma games are studied with players located on square lattice and random regular graph defining four neighbors for each one the players follow one of the three strategies tit for tat unconditional cooperation and defection the simplified payoff matrix is characterized by two parameters the temptation b to choose defection and the cost c of inspection reducing the income of tit for tat the strategy imitation from one of the neighbors is controlled by pairwise comparison at a fixed level of noise using monte carlo simulations and the extended versions of pair approximation we have evaluated the b c phase diagrams indicating a rich plethora of phase transitions between stationary coexistence absorbing and oscillatory states including continuous and discontinuous phase transitions by reasonable costs the tit for tat strategy prevents extinction of cooperators across the whole span of b determining the prisoner s dilemma game irrespective of the connectivity structure we also demonstrate that the system can exhibit a repetitive succession of oscillatory and stationary states upon changing a single payoff value which highlights the remarkable sensitivity of cyclical interactions on the parameters that define the strength of dominance

characterization of spiraling patterns in spatial rock paper scissors games the spatiotemporal arrangement of interacting populations often influences the maintenance of species diversity and is a subject of intense research here we study the spatiotemporal patterns arising from the cyclic competition between three species in two dimensions inspired by recent experiments we consider a generic metapopulation model comprising rock paper scissors interactions via dominance removal and replacement reproduction mutations pair exchange and hopping of individuals by combining analytical and numerical methods we obtain the model s phase diagram near its hopf bifurcation and quantitatively characterize the properties of the spiraling patterns arising in each phase the phases characterizing the cyclic competition away from the hopf bifurcation at low mutation rate are also investigated our analytical approach relies on the careful analysis of the properties of the complex ginzburg landau equation derived through a controlled perturbative multiscale expansion around the model s hopf bifurcation our results allow us to clarify when spatial rock paper scissors competition leads to stable spiral waves and under which circumstances they are influenced by nonlinear mobility

stochastic models in population biology and their deterministic analogs we introduce a class of stochastic population models based on patch dynamics the size of the patch may be varied and this allows one to quantify the departures of these stochastic models from various mean field theories which are generally valid as the patch size becomes very large these models may be used to formulate a broad range of biological processes in both spatial and nonspatial contexts here we concentrate on two species competition we present both a mathematical analysis of the patch model in which we derive the precise form of the competition mean field equations and their first order corrections in the nonspatial case and simulation results these mean field equations differ in some important ways from those which are normally written down on phenomenological grounds our general conclusion is that mean field theory is more robust for spatial models than for a single isolated patch this is due to the dilution of stochastic effects in a spatial setting resulting from repeated rescue events mediated by interpatch diffusion however discrete effects due to modest patch sizes lead to striking deviations from mean field theory even in a spatial setting