survival probability of a mobile particle in a fluctuating field we study the the survival probability p t upto time t of a test particle moving in a fluctuating external field the particle moves according to some prescribed deterministic or stochastic rules and survives as long as the external field that it sees at its own location does not change sign this is a natural generalization of the static persistence when the particle is at rest that has generated considerable recent interests two types of motions of the particle are considered in one case the particle adopts a strategy to live longer and in the other it just diffuses randomly three different external fields were considered i the solution of diffusion equation ii the colour profile of the q state potts model undergoing zero temperature coarsening dynamics and iii spatially uncorrelated brownian signals in most cases studied p t sim t theta m for large t the exponent theta m is calculated via numerically analytically by approximate methods and in some cases exactly it is shown in some special cases that the survival probability of the mobile particle is related to the persistence of special patterns present in the initial configuration of a phase ordering system

central finite volume schemes on nonuniform grids and applications we propose a new one dimensional unstaggered central scheme on nonuniform grids for the numerical solution of homogeneous hyperbolic systems of conservation laws with applications in two phase flows and in hydrodynamics with and without gravitational effect the numerical base scheme is a generalization of the original lax friedrichs scheme and an extension of the nessyahu and tadmor central scheme to the case of nonuniform irregular grids the main feature that characterizes the proposed scheme is its simplicity and versatility in fact the developed scheme evolves a piecewise linear numerical solution defined at the cell centers of a nonuniform grid and avoids the resolution of the riemann problems arising at the cell interfaces thanks to a layer of staggered cells used intermediately spurious oscillations are avoided using a slopes limiting procedure the developed scheme is then validated and used to solve classical problems arising in gas solid two phase flow problems the proposed scheme is then extended to the case of non homogenous hyperbolic systems with a source term in particular to the case of euler equations with a gravitational source term the obtained numerical results are in perfect agreement with corresponding ones appearing in the recent literature thus confirming the efficiency and potential of the proposed method to handle both homogeneous and non homogeneous hyperbolic systems

a musta scheme for a nonconservative two fluid model we present a multistage centered scheme of the kind proposed by toro appl numer math 56 2006 pp 1464 1479 for numerically resolving the simultaneous flow of two fluids through a transport pipeline this model contains nonconservative terms in both the temporal and spatial derivatives and an extension of the standard numerical framework for conservation laws is needed in this paper we rewrite the model in an equivalent mathematical form eliminating the nonconservative time derivatives this allows us to use the framework described by pares siam j numer anal 44 2006 pp 300 321 we develop force and musta type schemes which are consistent with pares formalism numerical simulations demonstrate a high degree of stability of our proposed schemes comparisons with the roe and rusanov schemes indicate that convergence to near identical solutions is obtained when the nonconservative terms are discretized with respect to the same evaluation of the path dependent integrals however if the schemes are not mutually formally path consistent in the sense of pares different converged solutions are obtained

reaction diffusion models in one dimension with disorder we study a large class of one dimensional reaction diffusion models with quenched disorder using a real space renormalization group method rsrg which yields exact results at large time particles e g of several species undergo diffusion with random local bias sinai model and may react upon meeting we obtain a detailed description of the asymptotic states i e attractive fixed points of the rsrg such as the large time decay of the density of each specie their associated universal amplitudes and the spatial distribution of particles we also derive the spectrum of nontrivial exponents which characterize the convergence towards the asymptotic states for reactions which lead to several possible asymptotic states separated by unstable fixed points we analyze the dynamical phase diagram and obtain the critical exponents characterizing the transitions we also obtain a detailed characterization of the persistence properties for single particles as well as more complex patterns we compute the decay exponents for the probability of no crossing of a given point by respectively the single particle trajectories ensuremath theta or the thermally averaged packets overline ensuremath theta the generalized persistence exponents associated to n crossings are also obtained specifying to the process a stackrel ensuremath rightarrow a ensuremath emptyset or a with probabilities r 1 ensuremath r we compute exactly the exponents ensuremath delta r and ensuremath psi r characterizing the survival up to time t of a domain without any merging or with mergings respectively and the exponents ensuremath delta a r and ensuremath psi a r characterizing the survival up to time t of a particle a without any coalescence or with coalescences respectively overline ensuremath theta ensuremath psi and ensuremath delta obey hypergeometric equations and are numerically surprisingly close to pure system exponents though associated to a completely different diffusion length the effect of additional disorder in the reaction rates as well as some open questions are also discussed

stable distributions in stochastic fragmentation we investigate a class of stochastic fragmentation processes involving stable and unstable fragments we solve analytically for the fragment length density and find that a generic algebraic divergence characterizes its small size tail furthermore the entire range of acceptable values of decay exponent consistent with length conservation can be realized we show that the stochastic fragmentation process is non self averaging as moments exhibit significant sample to sample fluctuations additionally we find that the distributions of the moments and of extremal characteristics possess an infinite set of progressively weaker singularities

positive radial solutions for quasilinear biharmonic equations abstract we study the quasilinear biharmonic problem u p 2 u f x u g x x b 1 u u 0 x b 1 where b 1 x r n x 1 with n 2 criteria are established for the existence of at least one positive radial solution of the problem explicit lower and upper bounds of the solutions are obtained in our results

multiplicity of positive radially symmetric solutions for a quasilinear biharmonic equation in the plane abstract this paper is concerned with the multiplicity of positive radially symmetric solutions of the dirichlet boundary value problem for the following two dimensional quasilinear biharmonic equation u p 2 u g x f u x b 1 where b 1 is the unit ball in the plane we apply the fixed point index theory and the upper and lower solutions method to investigate the multiplicity of positive radially symmetric solutions we have found that there exists a threshold such that if then the problem has no positive radially symmetric solution while if 0 then the problem admits at least one positive radially symmetric solution especially there exist at least two positive radially symmetric solutions for 0

the existence of positive entire solutions to singular nonlinear polyharmonic equations in rn in this paper the existence of positive radially symmetric entire solutions for the equation d mu f x u u b m 2 on r n n 3 is proved some properties of the solutions are obtained the results of this paper may be regarded of results in math z 67 1957 32 arch math 9 1958 308 proc royal soc edinburgh 82a 1979 189 hiroshima math j 1989 13

domain number distribution in the nonequilibrium ising model we study domain distributions in the one dimensional ising model subject to zero temperature glauber and kawasaki dynamics the survival probability of a domain s t t and an unreacted domain q1 t t are characterized by two independent nontrivial exponents we develop an independent interval approximation that provides close estimates for many characteristics of the domain length and number distributions including the scaling exponents

noise removal using fourth order partial differential equation with applications to medical magnetic resonance images in space and time we introduce a new method for image smoothing based on a fourth order pde model the method is tested on a broad range of real medical magnetic resonance images both in space and time as well as on nonmedical synthesized test images our algorithm demonstrates good noise suppression without destruction of important anatomical or functional detail even at poor signal to noise ratio we have also compared our method with related pde models

computations of two fluid models based on a simple and robust hybrid primitive variable riemann solver with ausmd this paper is to continue our previous work in 2008 on solving a two fluid model for compressible liquid gas flows we proposed a pressure velocity based diffusion term original derived from ausmd scheme of wada and liou in 1997 to enhance its robustness the proposed ausmd schemes have been applied to gas and liquid fluids universally to capture fluid discontinuities such as the fluid interfaces and shock waves accurately for the ransom s faucet problem air water shock tube problems and 2d shock water liquid interaction problems however the proposed scheme failed at computing liquid gas interfaces in problems under large ratios of pressure density and volume of fraction the numerical instability has been remedied by chang and liou in 2007 using the exact riemann solver to enhance the accuracy and stability of numerical flux across the liquid gas interface here instead of the exact riemann solver we propose a simple ausmd type primitive variable riemann solver pvrs which can successfully solve 1d stiffened water air shock tube and 2d shock gas interaction problems under large ratios of pressure density and volume of fraction without the expensive cost of tedious computer time in addition the proposed approach is shown to deliver a good resolution of the shock front rarefaction and cavitation inside the evolution of high speed droplet impact on the wall

hybrid flux splitting schemes for a two phase flow model in this paper we deal with the construction of hybrid flux vector splitting fvs schemes and flux difference splitting fds schemes for a two phase model for one dimensional flow the model consists of two mass conservation equations one for each phase and a common momentum equation the complexity of this model as far as numerical computation is concerned is related to the fact that the flux cannot be expressed in terms of its conservative variables this is the motivation for studying numerical schemes which are not based on approximate riemann solvers and or calculations of jacobian matrix this work concerns the extension of an fvs type scheme a van leer type scheme and an advection upstream splitting method ausm type scheme to the current two phase model our schemes are obtained through natural extensions of corresponding schemes studied by y wada and m s liou 1997 siam j sci comput 18 633 657 for euler equations we explore the various schemes for flow cases which involve both fast and slow transients in particular we demonstrate that the fvs scheme is able to capture fast propagating acoustic waves in a monotone way while it introduces an excessive numerical dissipation at volume fraction contact steady and moving discontinuities on the other hand the ausm scheme gives accurate resolution of contact discontinuities but produces oscillatory approximations of acoustic waves this motivates us to propose other hybrid fvs fds schemes obtained by removing numerical dissipation at contact discontinuities in the fvs and van leer schemes

unified graph based multi fluid model for gas liquid pipeline flows we present a novel model of transient multiphase flow for gas liquid mixtures in long pipelines the study develops a general flexible formulation of a one dimensional flow for complex multiphase mixtures in the presence of slip and mass exchange multi fluid and drift flux approaches are combined the multi fluid model governs the momentum balance for the fluids and drift flux correlations are used for modelling of the slip between individual components a tree like graph is introduced to describe the hierarchy of fluids and components within the mixture the numerical implementation for an arbitrary number of components is based on the extension of the semi implicit method for pressure linked equations imbalances of equations are selected as criteria for iteration convergence specific corrections are discussed including those proposed for the equilibrium gas release model

fourth order partial differential equations for noise removal a class of fourth order partial differential equations pdes are proposed to optimize the trade off between noise removal and edge preservation the time evolution of these pdes seeks to minimize a cost functional which is an increasing function of the absolute value of the laplacian of the image intensity function since the laplacian of an image at a pixel is zero if the image is planar in its neighborhood these pdes attempt to remove noise and preserve edges by approximating an observed image with a piecewise planar image piecewise planar images look more natural than step images which anisotropic diffusion second order pdes uses to approximate an observed image so the proposed pdes are able to avoid the blocky effects widely seen in images processed by anisotropic diffusion while achieving the degree of noise removal and edge preservation comparable to anisotropic diffusion although both approaches seem to be comparable in removing speckles in the observed images speckles are more visible in images processed by the proposed pdes because piecewise planar images are less likely to mask speckles than step images and anisotropic diffusion tends to generate multiple false edges speckles can be easily removed by simple algorithms such as the one presented in this paper

multiplicity results for solutions of p biharmonic problems abstract in this paper we deal with the multiplicity existence of solutions for the following p biharmonic equation p 2 u u p 2 u u q 2 u in u u 0 on where is a bounded domain in r n p 2 u u p 2 u p q p n p n 2 p r is a parameter when p q p we prove that the above problem possesses infinitely many solutions while when q p a multiplicity existence result is obtained

constraint consistent runge kutta methods for one dimensional incompressible multiphase flow new time integration methods are proposed for simulating incompressible multiphase flow in pipelines described by the one dimensional two fluid model the methodology is based on halfexplicit runge kutta methods being explicit for the mass and momentum equations and implicit for the volume constraint these half explicit methods are constraint consistent i e they satisfy the hidden constraints of the two fluid model namely the volumetric flow incompressibility constraint and the poisson equation for the pressure a novel analysis shows that these hidden constraints are present in the continuous semi discrete and fully discrete equations next to constraint consistency the new methods are conservative the original mass and momentum equations are solved and the proper shock conditions are satisfied efficient the implicit constraint is rewritten into a pressure poisson equation and the time step for the explicit part is restricted by a cfl condition based on the convective wave speeds and accurate achieving high order temporal accuracy for all solution components masses velocities and pressure highorder accuracy is obtained by constructing a new third order runge kutta method that satisfies the additional order conditions arising from the presence of the constraint in combination with time dependent boundary conditions two test cases kelvin helmholtz instabilities in a pipeline and liquid sloshing in a cylindrical tank show that for time independent boundary conditions the half explicit formulation with a classic fourth order runge kutta method accurately integrates the two fluid model equations in time while preserving all constraints a third test case ramp up of gas production in a multiphase pipeline shows that our new third order method is preferred for cases featuring time dependent boundary conditions

existence of positive solutions of singular elliptic boundary value problems in a ball this paper deals with existence theorems of positive solutions for singular elliptic boundary value problems in a ball the results of this paper are generalizations of those proved by authors cited in the references

finite volume algorithm to compute dense compressible gas solid flows a finite volume scheme is presented that allows the performance of numerical simulations of a four equation compressible two fluid model on unstructured meshes the convective part of the whole set of governing equations is a nonconservative conditionally hyperbolic system to solve the latter a fractional step method is presented which is based on a splitting of convective fluxes approximate roe type riemann solvers are used to compute numerical convective fluxes some computations of shock tube test cases are presented and compared with numerical results of a three equation model which assumes equal velocities within each phase a steady computation of a gas solid flow in a simple nozzle is described and an unsteady computation of a dense fluidized bed is also presented particular emphasis is given to the preservation of the maximum principle for the volumetric fraction

hyperbolic two pressure models for two phase flow revisited this paper presents correction to a previous paper j comput phys 53 1984 124 on stability analyses of one pressure models for two phase flow it also presents some extensions and generalization of previous work on two pressure models the extensions allow both the slopes to the interfaces and the rates of mass transfer through the interfaces to be non negligible these enlargements of the domain of applicability of the two pressure models introduce extra derivative terms into the models these extra derivative terms do not change the basic character of the model the two pressure models remain stable in the sense of von neumann a e in state space with the more complete modeling of the interface copyright 1988 academic press inc

emergence of fractals in aggregation with stochastic self replication we propose and investigate a simple model which describes the kinetics of aggregation of brownian particles with stochastic self replication an exact solution and the scaling theory are presented alongside numerical simulation which fully support all theoretical findings in particular we show analytically that the particle size distribution function exhibits dynamic scaling and we verify it numerically using the idea of data collapse furthermore the conditions under which the resulting system emerges as a fractal are found the fractal dimension of the system is given and the relationship between this fractal dimension and a conserved quantity is pointed out

nonlinear generalized functions and jump conditions for a standard one pressure liquid gas model abstract the mixture of a liquid and a gas is classically represented by one pressure models these models are a system of pdes in nonconservative form and shock wave solutions do not make sense within the theory of distributions they give rise to products of distributions that are not defined within distribution theory but they make sense by applying a theory of nonlinear generalized functions to these equations in contrast to the familiar case of conservative systems the jump conditions cannot be calculated a priori jump conditions for these nonconservative systems can be obtained using the theory of nonlinear generalized functions by inserting some adequate physical information into the equations the physical information that we propose to insert for the one pressure models of a mixture of a liquid and a gas is a natural mathematical expression in the theory of nonlinear generalized functions of the fact that liquids are practically incompressible while gases are very compressible and so they do not satisfy equally well their respective state laws on the shock waves this modelization gives well defined explicit jump conditions the great numerical difficulty for solving numerically nonconservative systems is due to the fact that slightly different numerical schemes can give significantly different results the jump conditions obtained above permit to select the numerical schemes and validate those that give numerical solutions that satisfy these jump conditions which can be an important piece of information in the absence of other explicit discontinuous solutions and of precise observational results we expose with care the mathematical originality of the theory of nonlinear generalized functions an original abstract analysis issued by the leopoldo nachbin team on infinite dimensional holomorphy that permits to state mathematically physical facts that cannot be formulated within distribution theory and are the key for the removal of ambiguities that classically appear when one tries to calculate on multiplications of distributions that occur in the differential equations of physics

domain statistics in coarsening systems we study the domain number and size distributions in the one dimensional ising and q state potts models subject to zero temperature glauber dynamics the survival probability of a domain s t t 2c and an unreacted domain q1 t t 2d are characterized by two independent nontrivial exponents for the ising case we find c50 126 and d51 27 using numerical simulations we develop an independent interval approximation that predicts the qualitative behavior of the domain distribution and provides good estimates for the exponents exact results for the domain distribution are also obtained in several solvable cases s1063 651x 97 00110 4

asymptotic study of the initial value problem to a standard one pressure model of multifluid flows in nondivergence form abstract we construct families of approximate solutions to the initial value problem and provide complete mathematical proofs that they tend to satisfy the standard system of isothermal one pressure two fluid flows in 1 d when the data are l 1 in densities and l in velocities to this end we use a method that reduces this system of pdes to a family of systems of four odes in banach spaces whose smooth solutions are these approximate solutions this method is constructive using standard numerical methods for odes one can easily and accurately compute these approximate solutions which therefore from the mathematical proof can serve for comparison with numerical schemes one observes agreement with previously known solutions from scientific computing evje and flatten 2003 16 we show that one recovers the solutions of these authors exactly in one case with a slight difference in another case then we propose an efficient numerical scheme for the original system of two fluid flows and show it gives back exactly the same results as the theoretical solutions obtained above

positive entire solutions for a class of singular nonlinear polyharmonic equations on r2 in the paper the existence of positive radially symmetric entire solutions for a class of singular nonlinear polyharmonic equations on r 2 is proved some properties are obtained the results of this paper are generalizations of those in refs math z 67 1957 32 arch math 9 1958 308 proc r soc edinburg 82a 1979 189 hiroshima math j 17 1989 13

a nearly implicit hydrodynamic numerical scheme for two phase flows this article presents an extension of the semi implicit numerical scheme for two phase flow simulation the extension uses an implicit evaluation of the convective fluxes and thus eliminates the material courant stability restriction the new algorithm involves a two step approach for the mass and energy equations and a single fully implicit step for the momentum evaluations some analysis and accuracy considerations for the scheme are also presented

stochastic aggregation scaling properties we study the scaling properties of stochastic aggregation processes in one dimension numerical simulations for both diffusive and ballistic transport show that the mass distribution is characterized by two independent non trivial exponents corresponding to the survival probability of particles and monomers the overall behaviour agrees qualitatively with the mean field theory this theory also provides a useful approximation for the decay exponents as well as the limiting mass distribution

on the existence of radial positive entire solutions for polyharmonic systems abstract two dimensional nonlinear polyharmonic systems of the type n u j p j 2 n u j f j x u 1 u 2 u 1 u 2 x r 2 j 1 2 are considered where p j 1 is a constant and f j 0 0 2 0 2 0 is a continuous function for j 1 2 some sufficient conditions are obtained for the existence of infinitely many radial positive entire solutions of the system with the prescribed asymptotic behavior at infinity

runge kutta discontinuous galerkin method for the approximation of baer and nunziato type multiphase models a high order numerical method is developed for the computation of compressible multiphase flows the model we use is based on the baer and nunziato type systems 4 among all the other available models in the literature these systems present the advantage to be able to simulate either interface or mixture problems nevertheless they still raise some issues mainly based on their non conservative feature the numerical method we propose is a discontinuous galerkin type in this work the interior side integrals are computed thanks to 2 robustness and high order of accuracy of the method are proved on classical interface problems but also on suitably derived analytical solutions

non trivial exponents in the zero temperature dynamics of the 1d ising and potts models we consider the glauber dynamics of the q state potts model in one dimension at zero temperature starting with a random initial configuration we measure the density r t of spins which have never flipped from the beginning of the simulation until time t we find that for large t the density r t has a power law decay left r t sim t theta right where the exponent theta varies with q our simulations lead to theta simeq 37 for q 2 theta simeq 53 for q 3 and theta longrightarrow 1 as q longrightarrow infty

positive entire solutions to nonlinear biharmonic equations in the plane abstract two dimensional nonlinear biharmonic equations of the form u p 2 u f x u x r2 are considered where p 1 is a constant and f r r r is a continuous function it can be shown that any positive radially symmetric solution is unbounded and grows at least as fast as positive constant multiples of x 2 log x 1 p 1 as x in this paper sharp conditions on f are presented under which has infinitely many positive symmetric entire solutions which are asymptotic to positive constant multiples of x 2 log x 1 p 1 as x

an approximate linearized riemann solver for a two fluid model an approximate linearized riemann solver is presented for the numerical simulation of two phase flows this new solver is based on a linearization of nonconservative products and uses an extension of roe s approximate riemann solver the scheme is applied to shock tube problem and to a standard test for two fluid codes

reaction kinetics of cluster impurities we study the density of clustered immobile reactants in the diffusion controlled single species annihilation an initial state in which these impurities occupy a subspace of codimension d leads to a substantial enhancement of their survival probability the smoluchowski rate theory suggests that the codimensionality plays a crucial role in determining the long time behavior the system undergoes a transition at d 2 for d 2 above this critical codimension d 2 the subspace decays indefinitely at the critical codimension inverse logarithmic decay occurs ni t log t a d d above the critical codimension the decay is algebraic ni t t a d d in general the exponents governing the long time behavior depend on the dimension as well as the codimension

entropy solutions for a fourth order nonlinear degenerate problem for noise removal abstract in this paper we establish the existence and uniqueness of entropy solutions for a fourth order nonlinear degenerate parabolic problem for noise removal in dimension 1 d 4

mathematical structure of the black oil model for petroleum reservoir simulation this model incorporates compressibility and general mass transfer between phases it consists of the conditions of thermodynamic equilibrium an equation of state for the volume balance between the fluidand the rock void darcy s law for the volumetric flow rates and component conservation equations these relations are manipulated to form a pressure equation and a modified system of component conservation equations it is shown that the pressure equation is parabolic and that in the absence of diffusive forces such as capillary pressure and mixing the component conservation equations are hyperbolic subject to technical conditions on the relative permeabilities this sequential formulation of the flow equations forms types of wave structures that occur for the system

universal distribution of masses in a one dimensional sticky gas abstract agglomeration of particles by perfectly inelastic collisions is studied in the case of a one dimensional gas particles move freely between collisions both mass and momentum are conserved at binary encounters it is shown that under a weak mean field like hypothesis admitting the existence of correlations between the velocities of colliding particles the dynamics drives the system towards a scaling regime with a universal exponential distribution of masses

high order total variation based image restoration the total variation tv denoising method is a pde based technique that preserves edges well but has the sometimes undesirable staircase effect namely the transformation of smooth regions ramps into piecewise constant regions stairs in this paper we present an improved model constructed by adding a nonlinear fourth order diffusive term to the euler lagrange equations of the variational tv model our technique substantially reduces the staircase effect while preserving sharp jump discontinuities edges we show numerical evidence of the power of resolution of this novel model with respect to the tv model in some 1d and 2d numerical examples

random walkers in one dimensional random environments exact renormalization group analysis sinai s model of diffusion in one dimension with random local bias is studied by a real space renormalization group which yields exact results at long times the effects of an additional small uniform bias force are also studied we obtain analytically the scaling form of the distribution of the position x t of a particle the probability of it not returning to the origin and the distributions of first passage times in an infinite sample as well as in the presence of a boundary and in a finite but large sample we compute the distribution of the meeting time of two particles in the same environment we also obtain a detailed analytic description of the thermally averaged trajectories by computing quantities such as the joint distribution of the number of returns and of the number of jumps forward these quantities obey multifractal scaling characterized by generalized persistence exponents ensuremath theta g which we compute in the presence of a small bias the number of returns to the origin becomes finite characterized by a universal scaling function which we obtain the full statistics of the distribution of successive times of return of thermally averaged trajectories is obtained as well as detailed analytical information about correlations between directions and times of successive jumps the two time distribution of the positions of a particle x t and x t ensuremath with tg t ensuremath is also computed exactly it is found to exhibit aging with several time regimes characterized by different behaviors in the unbiased case for t ensuremath t ensuremath ensuremath sim t ensuremath ensuremath alpha with ensuremath alpha g1 it exhibits a mathrm ln t mathrm ln t ensuremath scaling with a singularity at coinciding rescaled positions x t x t ensuremath this singularity is a novel feature and corresponds to particles that remain in a renormalized valley for closer times ensuremath alpha l1 the two time diffusion front exhibits a quasiequilibrium regime with a mathrm ln t ensuremath t ensuremath mathrm ln t ensuremath behavior which we compute the crossover to a t t ensuremath aging form in the presence of a small bias is also obtained analytically rare events corresponding to intermittent splitting of the thermal packet between separated wells which dominate some averaged observables are also characterized in detail connections with the green function of a one dimensional schr odinger problem and quantum spin chains are discussed

aggregation with multiple conservation laws aggregation processes with an arbitrary number of conserved quantities are investigated on the mean field level an exact solution for the size distribution is obtained the asymptotic form of this solution exhibits nontrivial double scaling while processes with one conserved quantity are governed by a single scale processes with multiple conservation laws exhibit an additional diffusion like scale the theory is applied to ballistic aggregation with mass and momentum conserving collisions and to diffusive aggregation with multiple species