approximate probabilistic cellular automata for the dynamics of single species populations under discrete logisticlike growth with and without weak allee effects we investigate one dimensional elementary probabilistic cellular automata pca whose dynamics in first order mean field approximation yields discrete logisticlike growth models for a single species unstructured population with nonoverlapping generations beginning with a general six parameter model we find constraints on the transition probabilities of the pca that guarantee that the ensuing approximations make sense in terms of population dynamics and classify the valid combinations thereof several possible models display a negative cubic term that can be interpreted as a weak allee factor we also investigate the conditions under which a one parameter pca derived from the more general six parameter model can generate valid population growth dynamics numerical simulations illustrate the behavior of some of the pca found

gaussian quadratures for approximate of fuzzy multiple integrals in this paper the multiple integration formulas of gaussian methods with positive coefficient for fuzzy integrations are discussed and then are followed by convergence theorem the proposed algorithms are illustrated by solving some numerical examples

role of noise in population dynamics cycles noise is an intrinsic feature of population dynamics and plays a crucial role in oscillations called phase forgetting quasicycles by converting damped into sustained oscillations this function of noise becomes evident when considering langevin equations whose deterministic part yields only damped oscillations we formulate here a consistent and systematic approach to population dynamics leading to a fokker planck equation and the associate langevin equations in accordance with this conceptual framework founded on stochastic lattice gas models that describe spatially structured predator prey systems langevin equations in the population densities and predator prey pair density are derived in two stages first a birth and death stochastic process in the space of prey and predator numbers and predator prey pair number is obtained by a contraction method that reduces the degrees of freedom second a van kampen expansion in the inverse of system size is then performed to get the fokker planck equation we also study the time correlation function the asymptotic behavior of which is used to characterize the transition from the cyclic coexistence of species to the ordinary coexistence

stochastic lattice gas model for a predator prey system we propose a stochastic lattice gas model to describe the dynamics of two animal species population one being a predator and the other a prey this model comprehends the mechanisms of the lotka volterra model our analysis was performed by using a dynamical mean field approximation and computer simulations our results show that the system exhibits an oscillatory behavior of the population densities of prey and predators for the sets of parameters used in our computer simulations these oscillations occur at a local level mean field results predict synchronized collective oscillations

cyclical interactions with alliance specific heterogeneous invasion rates we study a six species lotka volterra type system on different two dimensional lattices when each species has two superior and two inferior partners the invasion rates from predator sites to a randomly chosen neighboring prey site depend on the predator prey pair whereby cyclic symmetries within the two three species defensive alliances are conserved monte carlo simulations reveal an unexpected nonmonotonous dependence of alliance survival on the difference of alliance specific invasion rates this behavior is qualitatively reproduced by a four point mean field approximation the study addresses fundamental problems of stability for the competition of two defensive alliances and thus has important implications in natural and social sciences

phase transitions and oscillations in a lattice prey predator model a coarse grained description of a two dimensional prey predator system is given in terms of a simple three state lattice model containing two control parameters the spreading rates of prey and predator the properties of the model are investigated by dynamical mean field approximations and extensive numerical simulations it is shown that the stationary state phase diagram is divided into two phases a pure prey phase and a coexistence phase of prey and predator in which temporal and spatial oscillations can be present besides the usual directed percolationlike transition the system exhibits an unexpected different type of transition to the prey absorbing phase the passage from the oscillatory domain to the nonoscillatory domain of the coexistence phase is described as a crossover phenomena which persists even in the infinite size limit the importance of finite size effects are discussed and scaling relations between different quantities are established finally physical arguments based on the spatial structure of the model are given to explain the underlying mechanism leading to local and global oscillations

on the fuzzy initial value problem abstract the initial value problem x t f t x t x 0 x 0 with fuzzy initial value and with deterministic or fuzzy function f is considered two different approaches viz the extension principle and the use of extremal solutions of deterministic initial value problems are applied generalizations to fuzzy integral equations and fuzzy functional differential equations are indicated

von neummann s and related scaling laws in rock paper scissors type games we introduce a family of rock paper scissors type models with z n symmetry n is the number of species and we show that it has a very rich structure with many completely different phases we study realizations that lead to the formation of domains where individuals of one or more species coexist separated by interfaces whose average dynamics is curvature driven this type of behavior which might be relevant for the development of biological complexity leads to an interface network evolution and pattern formation similar to the ones of several other nonlinear systems in condensed matter and cosmology

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

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

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

equivalence between poisson representation and fock space formalism for birth death processes the equivalence between the poisson representation and the fock space formalism both used to derive coarse grained equations from the master equation for non equilibrium systems is explicitly demonstrated

reconstruction of an impedance cylinder at oblique incidence from the far field data consider scattering of obliquely incident plane electromagnetic waves by an impedance cylinder embedded in a homogeneous background medium under some assumptions the direct scattering problem is governed by a coupled system of the two dimensional helmholtz equations for x 3 components of the electric field and the magnetic field in this paper we study a corresponding inverse scattering problem which is to reconstruct an unknown scattering object from measurement data of the x 3 component of only the electric field a reconstruction scheme called the linear sampling method is established with rigorous mathematical analysis due to the coupled oblique derivative boundary condition some new characteristics appear in our analysis numerical examples are also presented to illustrate the feasibility of the reconstruction algorithm

host parasite models on graphs the behavior of two interacting populations hosts and parasites is investigated on cayley trees and scale free networks in the former case analytical and numerical arguments elucidate a phase diagram for the susceptible infected susceptible model whose most interesting feature is the absence of a tricritical point as a function of the two independent spreading parameters for scale free graphs the parasite population can be described effectively by its dynamics in a host background this is shown both by considering the appropriate dynamical equations and by numerical simulations on barab asi albert networks with the major implication that in the thermodynamic limit the critical parasite spreading parameter vanishes some implications and generalizations are discussed

evolutionary prisoner s dilemma game on newman watts networks maintenance of cooperation was studied for a two strategy evolutionary prisoner s dilemma game where the players are located on a one dimensional chain and their payoff comes from games with the nearest and next nearest neighbor interactions the applied host geometry makes it possible to study the impacts of two conflicting topological features the evolutionary rule involves some noise affecting the strategy adoptions between the interacting players using monte carlo simulations and the extended versions of dynamical mean field theory we determined the phase diagram as a function of noise level and a payoff parameter the peculiar feature of the diagram is changed significantly when the connectivity structure is extended by extra links as suggested by newman and watts

stochastic dynamics for two biological species and ecological niches abstract we consider an ecological system in which two species interact with two niches to this end we introduce a stochastic model with four states our analysis is founded in three approaches monte carlo simulations of the model on a square lattice mean field approximation and birth and death master equation from this last approach we obtain a description in terms of langevin equations which show in an explicit way the role of noise in population biology we focus mainly on the description of time oscillations of the species population and the alternating dominance between them the model treated here may provide insights on these properties

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

self organized packs selection in predator prey ecosystems received 19 july 2005 revised manuscript received 3 october 2005 published 23 february 2006we present a lattice model of a system of predators of ve kinds competing for prey the predators aregrouped in packs and characterized by two parameters the energy spent on hunting and energy gained by thekill the success of hunting depends on the actual competition among predators found near a prey wedetermine via monte carlo simulations the numbers of predators of each kind as a function of time and thedistribution of the size of their packs we show that the ratio of the energy spent by the competing predatorsdetermines their fate the energy gain plays only a secondary role we show also that the system self organizesitself into groups of predators living in well de ned packs which size depends on the energy spent the mostpreferred size dependence on the energy spent follows a very simple power law we present also a mean eld type approach to the problem and we discuss the differences in the results obtained by the two methods showing in particular that the simulation approach produces more reliable results doi 10 1103 physreve 73 021913 pacs number s 87 23 cc 87 23 kg 05 10 ln 87 23 n

romberg integration for fuzzy functions in this paper the integration formulas of newton cot s methods with positive coefficient for fuzzy integrations in t allahviranloo newton cot s methods for integration of fuzzy functions in press are considered and then are followed by romberg integration to obtain improvements of the approximations of fuzzy integrations the proposed algorithm is illustrated by solving some numerical examples

boundary effects on population dynamics in stochastic lattice lotka volterra models abstract we investigate spatially inhomogeneous versions of the stochastic lotka volterra model for predator prey competition and coexistence by means of monte carlo simulations on a two dimensional lattice with periodic boundary conditions to study boundary effects for this paradigmatic population dynamics system we employ a simulation domain split into two patches upon setting the predation rates at two distinct values one half of the system resides in an absorbing state where only the prey survives while the other half attains a stable coexistence state wherein both species remain active at the domain boundary we observe a marked enhancement of the predator population density the predator correlation length displays a minimum at the boundary before reaching its asymptotic constant value deep in the active region the frequency of the population oscillations appears only very weakly affected by the existence of two distinct domains in contrast to their attenuation rate which assumes its largest value there we also observe that boundary effects become less prominent as the system is successively divided into subdomains in a checkerboard pattern with two different reaction rates assigned to neighboring patches when the domain size becomes reduced to the scale of the correlation length the mean population densities attain values that are very similar to those in a disordered system with randomly assigned reaction rates drawn from a bimodal distribution

detecting buried wave penetrable scatterers in a two layered medium abstract in a two layered medium we prove that a buried inhomogeneous scatterer is uniquely determined from the wave field data measured in the upper half space with respect to many incident point sources moreover we extend the multilevel sampling method in liu and zou 2013 to numerically reconstruct the buried scatterer applying only few incident fields and partial scattered data the extended recovery scheme only involves matrix vector operations and does not need to solve any large scale ill posed linear systems or any optimization process it is feasible to deal with the scatterers of different features and easy to implement highly tolerant to noise and computationally quite cheap we can regard it as an effective yet simple computational method to provide a reliable initial guess for the implementation in existing more accurate and refined optimization type reconstruction algorithms

a multilevel sampling method for detecting sources in a stratified ocean waveguide in the reconstruction process of sound waves in a 3d stratified waveguide a key technique is to effectively reduce the huge computational demand in this work we propose an efficient and simple multilevel reconstruction method to locate the accurate position of a point source in a stratified ocean the proposed method can be viewed as a direct sampling method since no solutions of optimizations or linear systems are involved the novel method exhibits several strengths fast convergence robustness against noise advantages in computational complexity and applicability for a very small number of receivers

reconstruction of a small inclusion in a two dimensional open waveguide we consider wave propagation in a perturbed open waveguide we provide a new asymptotic expansion for the scattered wave when the inclusion is of small diameter we design a music multiple signal classification type of algorithm for locating the inclusion and illustrate its viability in numerical examples

population dynamics with or without evolution a physicist s approach abstract modeling the dynamics of interacting species or populations is a long standing problem in sciences which in the recent years has attracted a lot of physicists working in statistical physics the similarities and differences between models of dynamics of population and usual statistical mechanics problems on a lattice are discussed first the question of the appropriate level of description ordinary differential equations reaction diffusion equations patches models or individual based models for extended systems is considered second the role of the internal degrees of freedom associated to the phenotype of the individuals on the dynamics is discussed

coexistence in a predator prey system we propose a lattice model of two populations predators and prey the model is solved via monte carlo simulations each species moves randomly on the lattice and can live only a certain time without eating the lattice cells are either grass eaten by prey or tree giving cover for prey each animal has a reserve of food that is increased by eating prey or grass and decreased after each monte carlo step to breed a pair of animals must be adjacent and have a certain minimum of food supply the number of offspring produced depends on the number of available empty sites we show that such a predator prey system may finally reach one of the following three steady states coexisting with predators and prey pure prey or an empty one in which both populations become extinct we demonstrate that the probability of arriving at one of the above states depends on the initial densities of the prey and predator populations the amount of cover and the way it is spatially distributed

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

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

stable oscillations of a predator prey probabilistic cellular automaton a mean field approach we analyze a probabilistic cellular automaton describing the dynamics of coexistence of a predator prey system the individuals of each species are localized over the sites of a lattice and the local stochastic updating rules are inspired by the processes of the lotka volterra model two levels of mean field approximations are set up the simple approximation is equivalent to an extended patch model a simple metapopulation model with patches colonized by prey patches colonized by predators and empty patches this approximation is capable of describing the limited available space for species occupancy the pair approximation is moreover able to describe two types of coexistence of prey and predators one where population densities are constant in time and another displaying self sustained time oscillations of the population densities the oscillations are associated with limit cycles and arise through a hopf bifurcation they are stable against changes in the initial conditions and in this sense they differ from the lotka volterra cycles which depend on initial conditions in this respect the present model is biologically more realistic than the lotka volterra model

newton cot s methods for integration of fuzzy functions in this paper the integration formulas of newton cot s methods with positive coefficient for fuzzy integrations are discussed and then is followed by peano s error representation theorem and convergence theorem the proposed algorithms are illustrated by solving some numerical examples

the threshold of coexistence and critical behaviour of a predator prey cellular automaton we study a probabilistic cellular automaton to describe two population biology problems the threshold of species coexistence in a predator prey system and the spreading of an epidemic in a population by carrying out mean field approximations and numerical simulations we obtain the phase boundaries thresholds related to the transition between an active state where prey and predators present a stable coexistence and a prey absorbing state the numerical estimates for the critical exponents show that the transition belongs to the directed percolation universality class in the limit where the cellular automaton maps into a model for the spreading of an epidemic with immunization we observe a crossover from directed percolation class to the dynamic percolation class patterns of growing clusters related to species coexistence and spreading of epidemic are shown and discussed

junctions and spiral patterns in generalized rock paper scissors models we investigate the population dynamics in generalized rock paper scissors models with an arbitrary number of species n we show that spiral patterns with n arms may develop both for odd and even n in particular in models where a bidirectional predation interaction of equal strength between all species is modified to include one n cyclic predator prey rule while the former case gives rise to an interface network with y type junctions obeying the scaling law l ensuremath propto t 1 2 where l is the characteristic length of the network and t is the time the latter can lead to a population network with n armed spiral patterns having a roughly constant characteristic length scale we explicitly demonstrate the connection between interface junctions and spiral patterns in these models and compute the corresponding scaling laws this work significantly extends the results of previous studies of population dynamics and could have profound implications for the understanding of biological complexity in systems with a large number of species

direct recovery of wave penetrable scatterers in a stratified ocean waveguide abstract we shall derive a direct sampling method for the recovery of wave penetrable inhomogeneous scatterers in a three dimensional stratified ocean waveguide the proposed method is simple straightforward and computationally efficient since no matrix inversions optimizations or solutions of ill posed linear systems are involved moreover it is highly tolerant to noise and is applicable with a few scattered fields corresponding to only two or four incidences in addition a priori knowledge of either the number of disconnected components or the physical features of the unknown obstacles is not required by the direct sampling method a mathematical derivation is provided for its validation and several extensive numerical experiments are shown to evaluate the effectiveness and feasibility of the newly proposed method

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

fluctuations and correlations in lattice models for predator prey interaction including spatial structure and stochastic noise invalidates the classical lotka volterra picture of stable regular population cycles emerging in models for predator prey interactions growth limiting terms for the prey induce a continuous extinction threshold for the predator population whose critical properties are in the directed percolation universality class we discuss the robustness of this scenario by considering an ecologically inspired stochastic lattice predator prey model variant where the predation process includes next nearest neighbor interactions we find that the corresponding stochastic model reproduces the above scenario in dimensions 1 d or 4 in contrast with the mean field theory which predicts a first order phase transition however the mean field features are recovered upon allowing for nearest neighbor particle exchange processes provided these are sufficiently fast

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

dynamics of populations in a changing environment we present an individual based model of a population that lives in a changing environment the individuals forming the population are subject to mutations and selection pressure using monte carlo simulations we have shown that depending on the values of the mutation rate and selection the population may reach either an active phase it will survive or an absorbing phase it will become extinct we have determined that the transition between the two states phases is continuous we have shown that when the selection is weaker the population lives in all available space while if the selection is stronger it will move to the regions where the living conditions are better avoiding those with more difficult conditions the dependence of the mean time to extinction on the rate of mutations has been determined and discussed

susceptible infected recovered and susceptible exposed infected models two stochastic epidemic lattice models the susceptible infected recovered and the susceptible exposed infected models are studied on a cayley tree of coordination number k the spreading of the disease in the former is found to occur when the infection probability b is larger than bc k 2 k 1 in the latter which is equivalent to a dynamic site percolation model the spreading occurs when the infection probability p is greater than pc 1 k 1 we set up and solve the time evolution equations for both models and determine the final and time dependent properties including the epidemic curve we show that the two models are closely related by revealing that their relevant properties are exactly mapped into each other when p b k k 1 b these include the cluster size distribution and the density of individuals of each type quantities that have been determined in closed forms

the two dimensional direct and inverse scattering problems with generalized oblique derivative boundary condition consider the two dimensional exterior problem for the helmholtz equation with a generalized oblique derivative boundary condition which arises in some new scattering problems such as the scattering of tidal waves by islands under suitable assumptions different from the well studied scattering problems with dirichlet neumann and impedance boundary conditions the boundary condition in our new model involves a linear combination of the normal and tangential derivatives of the wave field with complex coefficient compared with the classical scattering models the tangential derivative on the obstacle boundary leads to some essential differences such as the symmetric property of the green function and the reciprocity principle of the scattering data both the direct and inverse scattering problems are studied in this paper first based on the lax phillips method we show the unique solvability of the direct scattering problem and the analyticity of the solution on the wave number second we investigate

penna model of biological aging on a lattice abstract the penna bit string model of biological aging is implemented on a square lattice hence there is no need to manipulate the population size by the verhulst factor instead the spatial lattice relations between neighboring individuals are considered mortality obtained in the lattice asexual penna model is different from the corresponding feature of the standard penna model

quasicycles in a spatial predator prey model we use spatial models of simple predator prey interactions to predict that predator and prey numbers oscillate in time and space these oscillations are not seen in the deterministic versions of the models but are due to stochastic fluctuations about the time independent solutions of the deterministic equations which are amplified due to the existence of a resonance we calculate the power spectra of the fluctuations analytically and show that they agree well with results obtained from stochastic simulations this work extends the analysis of these quasicycles from that previously developed for well mixed systems to spatial systems and shows that the ideas and methods used for nonspatial models naturally generalize to the spatial case

mean field approximation to a spatial host pathogen model we study the mean field approximation to a simple spatial host pathogen model that has been shown to display interesting evolutionary properties we show that previous derivations of the mean field equations for this model are actually only low density approximations to the true mean field limit we derive the correct equations and the corresponding equations including pair correlations the process of invasion by a mutant type of pathogen is also discussed doi 10 1103 physreve 67 047102

a simple method for detecting scatterers in a stratified ocean waveguide abstract in this paper we have designed a simple method for the inverse scattering problem of stratified ocean waveguide the simple approach applies the bisection technique in parallel to estimate the boundaries of inhomogeneous media from the received partial data in addition the algorithm only involves the matrix vector operations and possesses the optimal computation complexity in both two and three dimensions in practice it is easy to carry out robust against noise and capable of reconstructing the penetrable obstacles of different locations and shapes we can consider the simple method as a direct and straightforward process to supply satisfactory initial positions for the implementation of any existing more refined but computationally more demanding techniques to achieve accurate reconstructions of physical features of scatterers

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