bifurcation analysis of a generalized gause model with prey harvesting and a generalized holling response function of type iii abstract in this paper we study a generalized gause model with prey harvesting and a generalized holling response function of type iii p x m x 2 a x 2 b x 1 the goal of our study is to give the bifurcation diagram of the model for this we need to study saddle node bifurcations hopf bifurcation of codimension 1 and 2 heteroclinic bifurcation and nilpotent saddle bifurcation of codimension 2 and 3 the nilpotent saddle of codimension 3 is the organizing center for the bifurcation diagram the hopf bifurcation is studied by means of a generalized lienard system and for b 0 we discuss the potential integrability of the system the nilpotent point of multiplicity 3 occurs with an invariant line and can have a codimension up to 4 but because it occurs with an invariant line the effective highest codimension is 3 we develop normal forms in which the invariant line is preserved for studying of the nilpotent saddle bifurcation for b 0 the reversibility of the nilpotent saddle is discussed we study the type of the heteroclinic loop and its cyclicity the phase portraits of the bifurcations diagram partially conjectured via the results obtained allow us to give a biological interpretation of the behavior of the two species

dynamical behavior of lotka volterra competition systems non autonomous bistable case and the effect of telegraph noise this article is concerned with the study of trajectory behavior of lotka volterra competition bistable systems and systems with telegraph noises we proved that for bistable systems there exists a unique solution bounded above and below by positive constants the oscillatory situation of systems with telegraph noises is pointed out

profile of the unique limit cycle in a class of general predator prey systems many predator prey systems with oscillatory behavior possess a unique limit cycle which is globally asymptotically stable for a class of general predator prey system we show that the solution orbit of the limit cycle exhibits the temporal pattern of a relaxation oscillator when a certain parameter is small

taylor s power law and fluctuation scaling explained by a central limit like convergence a power function relationship observed between the variance and the mean of many types of biological and physical systems has generated much debate as to its origins this taylor s law or fluctuation scaling has been recently hypothesized to result from the second law of thermodynamics and the behavior of the density of states this hypothesis is predicated on physical quantities like free energy and an external field the correspondence of these quantities with biological systems though remains unproven questions can be posed as to the applicability of this hypothesis to the diversity of observed phenomena as well as the range of spatial and temporal scales observed with taylor s law we note that the cumulant generating functions derived from this thermodynamic model correspond to those derived over a quarter century earlier for a class of probabilistic models known as the tweedie exponential dispersion models these latter models are characterized by variance to mean power functions their phenomenological basis rests with a central limit theorem like property that causes many statistical systems to converge mathematically toward a tweedie form we review evaluations of the tweedie poisson gamma model for taylor s law and provide three further cases to test the clustering of single nucleotide polymorphisms snps within the horse chromosome 1 the clustering of genes within human chromosome 8 and the mertens function this latter case is a number theoretic function for which a thermodynamic model cannot explain taylor s law but where tweedie convergence remains applicable the tweedie models are applicable to diverse biological physical and mathematical phenomena that express power variance functions over a wide range of measurement scales they provide a probabilistic description for taylor s law that allows mechanistic insight into complex systems without the assumption of a thermodynamic mechanism

scaling in currency exchange we study the scaling behavior in currency exchange rates our results suggest that they satisfy scaling with an exponent close to 0 5 but that it differs qualitatively from that of a simple random walk indeed price variations cannot be considered as independent variables and subtle correlations are present furthermore we introduce a novel statistical analysis for economic data which makes the physical properties of a signal more evident and eliminates the systematic effects of time periodicity

bifurcation analysis of modified leslie gower predator prey model with michaelis menten type prey harvesting abstract in the present paper we discuss bifurcation analysis of a modified leslie gower prey predator model in the presence of nonlinear harvesting in prey we give a detailed mathematical analysis of the model to describe some significant results that may arise from the interaction of biological resources the model displays a complex dynamics in the prey predator plane the permanence stability and bifurcation saddle node bifurcation transcritical hopf andronov and bogdanov takens of this model are discussed we have analyzed the effect of prey harvesting and growth rate of predator on the proposed model by considering them as bifurcation parameters as they are important from the ecological point of view the local existence and stability of the limit cycle emerging through hopf bifurcation is given the emergence of homoclinic loops has been shown through simulation when the limit cycle arising though hopf bifurcation collides with a saddle point this work reflects that the feasible upper bound of the rate of harvesting for the coexistence of the species can be guaranteed numerical simulations using matlab are carried out to demonstrate the results obtained

hopf bifurcation of a predator prey system with stage structure and harvesting abstract in this paper a two species predator prey system with stage structure and harvesting is investigated the existence of hopf bifurcations of the system is given and the stability and directions of hopf bifurcations are determined by applying the normal form theory and the center manifold theorem

stochastic persistence and stationary distribution in a holling tanner type prey predator model abstract in this paper we study a stochastic predator prey model with beddington deangelis type functional response and logistic growth for predators the deterministic model is already well studied and we recall some important results here we construct the stochastic model from the deterministic model by introducing multiplicative noise terms into the growth equations of prey and predator populations for the stochastic model we show that the system admits unique positive global solution starting from the positive initial value then we prove that the system is strongly persistent in mean when the intensity of environmental forcing is less than some threshold magnitudes finally we show that the system has a stationary distribution under certain parametric restrictions numerical simulations are carried out to substantiate the analytical results

chaotic dynamics of a three species prey predator competition model with noise in ecology in this paper we investigate a three species ecosystem consisting of a prey a predator and a top predator the predator survives on the prey and the top predator survives on both the prey and the predator the mathematical model consists of non linear simultaneous differential equations all the eight equilibrium points of the model are identified and by using routh hurwitz criteria as well as lyapunov function we derive the criteria for local and global stabilities also the population intensities of fluctuations variances around the positive equilibrium due to noise are computed their stability is also analyzed with computer simulation using matlab some numerical simulations for justifying the theoretical analysis are also provided the main conclusions are supplied in conclusion

mapping multiplicative to additive noise the langevin formulation of a number of well known stochastic processes involves multiplicative noise in this work we present a systematic mapping of a process with multiplicative noise to a related process with additive noise which may often be easier to analyse the mapping is easily understood in the example of the branching process in a second example we study the random neighbour or infinite range contact process which is mapped to an ornstein uhlenbeck process with absorbing wall the present work might shed some light on absorbing state phase transitions in general such as the role of conditional expectation values and finite size scaling and elucidate the meaning of the noise amplitude while we focus on the physical interpretation of the mapping we also provide a mathematical derivation

global analysis in a predator prey system with nonmonotonic functional response a predator prey system with nonmonotonic functional response is considered global qualitative and bifurcation analyses are combined to determine the global dynamics of the model the bifurcation a

bounded solutions and wavefronts for discrete dynamics this paper deals with the second order nonlinear difference equation dd r k dd u k q kg u k 1 0 where r k and q k are positive real sequences defined on n cup 0 and the nonlinearity g r to r is nonnegative and nontrivial sufficient and necessary conditions are given for the existence of bounded solutions starting from a fixed initial condition u 0 the same dynamic with f instead of g such that uf u 0 for u not 0 was recently extensively investigated on the contrary our nonlinearity g is of a small appearance in the discrete case its introduction is motivated by the analysis of wavefront profiles in biological and chemical models the paper emphasizes the many different dynamical behaviors caused by such a g with respect to the equation involving function f some applications in the study of wavefronts complete this work

on the stability of non autonomous perturbed lotka volterra models the paper is devoted to an extended lotka volterra system of differential equations of predator prey model the extension is proposed with perturbation terms which are null for the positive equilibrium state in the original lotka volterra system the equilibrium state is not asymptotically stable due to the fact that perturbations are periodic in time the aim of the paper is to characterize a form of perturbation terms guaranteeing the asymptotic stability or instability of equilibrium state the reason of the proposed model is that for large time scale the lotka volterra model is too simple to be realistic in the paper the non autonomous perturbations do not change the equilibrium state but introduce functions of time as well as for additional perturbed terms as for the main part of the equations modified from lotka volterra model theorems are proposed in a renormalized form of the differential equations for time and the two variables the key point of the paper comes from the use of a liapunov function introduced in section 2 which allows to obtain conditions for the asymptotic stability section 3 and instability section 4 by using a cetaiev instability theorem following conditions on the renormalized coefficients in time of system 6 an appendix recalls the main results of the liapunov direct method for non autonomous binary systems of ordinary differential equations

topological degree in the generalized gause prey predator model abstract we consider a generalized gause prey predator model with t periodic continuous coefficients in the case where the poincare map p over time t is well defined the result of the paper can be explained as follows we locate a subset u of r 2 such that the topological degree d i p u equals to 1 the novelty of the paper is that the later is done under only continuity and some monotonicity assumptions for the coefficients of the model a suitable integral operator is used in place of the poincare map to cope with possible nonuniqueness of solutions the paper therefore provides a new framework for studying the generalized gause model with functional differential perturbations and multi valued ingredients

competing subcommunities of mutualist and a generalized kamke theorem a model of a biological community of species is considered the community consists of two competing subcommunities each of which has the property that each pair of species of the subcommunity interact in a mutually beneficial or noninterfering manner sufficient conditions are given that the two subcommunities can coexist the coexistence state being a stable equilibrium the domains of attraction of stable equilibria are considered the sharpest results are for the special case of lotka volterra systems mathematically the fundamental tool is a generalized kamke theorem

dynamics of a class of nonautonomous semi ratio dependent predator prey systems with functional responses abstract in this paper we investigate the dynamics of a class of the so called semi ratio dependent predator prey interaction models with functional responses based on systems of nonautonomous differential equations with time dependent parameters the functional responses are classified into five types and typical examples of each type are provided then we establish sufficient criteria for the boundedness of solutions the permanence of system and the existence uniqueness and globally asymptotic stability of positive periodic solution and positive almost periodic solution some conclusive discussion is presented at the end of this paper

stability analysis on a predator prey system with distributed delays abstract this paper concerns the local and global dynamical properties of the nonnegative and positive equilibria of a lotka volterra predator prey system with distributed delays it is shown that while the positive equilibrium does not exist the nonnegative equilibrium is globally asymptotically stable or globally attractive as long as the delays are small enough if the positive equilibrium exists it is shown that it is locally asymptotically stable when the delays are suitably small furthermore an explicit asymptotic stability region for the positive equilibrium is also obtained based on a liapunov functional

the concept of force in population dynamics abstract the area of population dynamics has a rich history of the development and analysis of models of biological and social phenomena using ordinary differential equations this paper describes a method for understanding the influence one variable exerts on another in such models as a force with the relative effects of these forces providing a narrative explanation of the curvature in variable behaviour using the stock flow form of a model a symbolic notation is developed that identifies the forces with the causal pathways of the model s feedback loops a force is measured by its impact defined as the ratio of acceleration to rate of change computed by differentiation along its associated pathway between variables different phases of force dominance are determined to enhance the standard stability analysis of the models providing an explanation of model behaviour in newtonian mechanical terms the concepts developed are applied to well known models from mathematical biology the spruce budworm model where force dominance identifies scenarios that give clarity to intervention points and the lotka volterra predator prey model where the analysis highlights the importance of dissipative forces in achieving stability conclusions are drawn on the explanatory power of this approach with suggestions made for future work

an efficient solution of the firing mob problem abstract an efficient solution of the firing mob problem which is the generalization of the well known firing squad synchronization problem to finite bounded degree networks is presented first a method of synchronizing tree connected networks is given this method is extended to general networks the total synchronization time is 3 5r where r is the radius of the network no solution can work in time less than 3r on all networks moreover it is shown why our solution will approach this value in the limit case when the number of states used becomes arbitrarily large

sample and population exponents of generalized taylor s law has been verified extensively in ecology where it applies to population abundance physics and other natural sciences its ubiquitous empirical verification suggests a context independent mechanism sample exponents b measured empirically via the scaling of sample mean and variance typically cluster around the value b 2 some theoretical models of population growth however predict a broad range of values for the population exponent b pertaining to the mean and variance of population density depending on details of the growth process is the widely reported sample exponent b 2 the result of ecological processes or could it be a statistical artifact here we apply large deviations theory and finite sample arguments to show exactly that in a broad class of growth models the sample exponent is b 2 regardless of the underlying population exponent we derive a generalized tl in terms of sample and population exponents bjk for the scaling of the kth vs the jth cumulants the sample exponent bjk depends predictably on the number of samples and for finite samples we obtain bjk k j asymptotically in time a prediction that we verify in two empirical examples thus the sample exponent b 2 may indeed be a statistical artifact and not dependent on population dynamics under conditions that we specify exactly given the broad class of models investigated our results apply to many fields where tl is used although inadequately understood

periodic solutions for a delayed predator prey system with stage structured predator on time scales in this paper a modified delayed predator prey system with stage structure for predator is proposed and studied which generalizes and incorporates as special cases some known models with the help of continuation theorem based on the gaines and mawhin coincidence degree theory we investigate the existence of periodic solutions for the proposed delayed predator prey system with stage structured predator and two kinds of functional responses called holling iii functional response and beddington deangelis functional response on time scales in particular when the time scale is chosen as the set of the real numbers or the integers the existence of the periodic solutions of the corresponding continuous time and discrete time models follows

persistence of periodic patterns for perturbed biological oscillators abstract we apply the perturbation method and the implicit function theorem to study the persistence of periodic patterns for two perturbed general biological oscillators one weakly coupled ordinary differential equation and one reaction diffusion system the proof relies also on the formal adjoint equation theory and the lyapunov schmidt method for the perturbed reaction diffusion model spatial temporal patterns such as rotating waves are shown to exist numerical simulations are presented to illustrate our analytical results

global attractivity in a periodic chemostat with general uptake functions a system of ode s is considered as a model of two populations competing for a nutrient where nutrient inputs and chemostat washouts are periodic as well general nutrient uptake functions which may also be time dependent and periodic are considered criteria are derived for there to exist a globally attracting positive periodic solution and a theorem on global monotonicity of attracting solutions between different systems is proved

exponential dispersion models on etudie les proprietes generales de la classe des modeles de dispersion exponentiels qui est la generalisation multivariable des modeles de distribution d erreur de nelder et wedderburn 1972

dynamics of a general prey predator model with prey stage structure and diffusive effects in this paper we propose and study the dynamics of a diffusive prey predator model with general functional response and stage structure for the prey firstly we consider the asymptotical stability of equilibrium points and hopf bifurcation for the reduced ode system secondly the existence and uniform boundedness of global solutions and stability of equilibrium points for the corresponding reaction diffusion system are discussed finally we establish the existence and the nonexistence of nonconstant positive steady states of this reaction diffusion system which indicates the effect of large diffusivity our results shows the importance of the diffusion rate of the predator species i e d 3 the large diffusion rate of the predator alone will help the generation of patterns however a large diffusion rate of the immature prey species or a large diffusion rate of the mature prey species can lead to the nonexistence of spatial patterns

zipf s and taylor s laws zipf s law states that the frequency of an observation with a given value is inversely proportional to the square of that value taylor s law instead describes the scaling between fluctuations in the size of a population and its mean empirical evidence of the validity of these laws has been found in many and diverse domains despite the numerous models proposed to explain the presence of zipf s law there is no consensus on how it originates from a microscopic process of individual dynamics without fine tuning here we show that zipf s law and taylor s law can emerge from a general class of stochastic processes at the individual level which incorporate one of two features environmental variability i e fluctuations of parameters or correlations i e dependence between individuals under these assumptions we show numerically and with theoretical arguments that the conditional variance of the population increments scales as the square of the population and that the corresponding stationary distribution of the processes follows zipf s law

qualitative analysis of a stochastic ratio dependent predator prey system a stochastic ratio dependent predator prey model is investigated in this paper by the comparison theorem of stochastic equations and ito s formula we obtain the global existence of a positive unique solution of the ratio dependent model besides a condition for species to be extinct is given and a persistent condition is established we also conclude that both the prey population and the ratio dependent function are stable in time average in the end numerical simulations are carried out to confirm our findings

analysis of autonomous lotka volterra competition systems with random perturbation abstract this paper discusses a randomized n species lotka volterra competition system we show that this system is stable in time average under certain conditions furthermore there is a stationary distribution of this system if extra conditions are satisfied also we give the extinction condition of this system finally numerical simulations are carried out to support our results

dynamics of age structured predator prey population models a predator prey model is investigated in which the prey population is assumed to have age structure and is governed by the mckendrick von foerster partial differential equation and the predator population is described by the classical volterra lotka ordinary differential equation quite general hypotheses are assumed for the mortality function the fertility function and the functional responses of predation existence and stability of three biologically meaningful equilibria corresponding to extinction of both species persistence of one species prey and coexistence of two species are studied a particular example and some numerical results are given

neimark sacker bifurcations in a non standard numerical scheme for a class of positivity preserving odes we discuss the nature of neimark sacker bifurcations occurring in a non standard numerical scheme for a class of positivity preserving systems of ordinary differential equations odes which undergoes a corresponding hopf bifurcation extending previous work alexander moghadas 2005a electron j diff eqn conf 12 9 19 it is shown that the type of neimark sacker bifurcation supercritical or subcritical may be affected by the integration time step the general form of the scheme in the vicinity of a fixed point is given from which the expression for the first lyapunov coefficient for two dimensional systems valid for arbitrary time step is explicitly derived the analysis shows that this coefficient undergoes an o shift with respect to the corresponding coefficient of the original ode this could lead to a type of bifurcation which differs from the corresponding hopf bifurcation in the ode due to changes in the sign of the first lyapunov coefficient as varies this is especially problematic in the vicinity of certain types of degenerate hopf bifurcation at which this coefficient may vanish we also present a general method to eliminate the possible o shift in the bifurcation parameter of the scheme however the first lyapunov coefficient may still be subjected to an o shift leading to a possibly erroneous type of bifurcation examples are given to illustrate the theoretical results of the paper with applications to mathematical biology

dynamics of a stochastic holling tanner predator prey model abstract in the paper we study a stochastic holling tanner predator prey system first by constructing a suitable stochastic lyapunov function we establish sufficient conditions for the existence of a unique ergodic stationary distribution of the positive solutions to the system after that sufficient causes for extinction of the predator population is obtained meanwhile we also obtain the threshold between persistence and extinction of the predator population at last some examples and numerical simulations are provided to illustrate our theoretical results

limit cycles for two families of cubic systems abstract in this paper we study the number of limit cycles of two families of cubic systems introduced in previous papers to model real phenomena the first one is motivated by a model of star formation histories in giant spiral galaxies and the second one comes from a model of volterra type to prove our results we develop a new criterion on the non existence of periodic orbits and we extend a well known criterion on the uniqueness of limit cycles due to kuang and freedman both results allow to reduce the problem to the control of the sign of certain functions that are treated by algebraic tools moreover in both cases we prove that when the limit cycles exist they are non algebraic

periodic solutions and permanence for a delayed nonautonomous ratio dependent predator prey model with holling type functional response by using the continuation theorem of coincidence degree theory the existence of positive periodic solutions for a delayed ratio dependent predator prey model with holling type iii functional response x t x t a t b t t k t s x s ds c t x2 t y t m2y2 t x2 t y t y t e t x2 t m2y2 t x2 t d t is established where a t b t c t e t and d t are all positive periodic continuous functions with period 0 m 0 and k s is a measurable function with period is a nonnegative constant the permanence of the system is also considered in particular if k s 0 s where 0 s is the dirac delta function at s 0 our results show that the permanence of the above system is equivalent to the existence of positive periodic solution

homoclinic bifurcation of prey predator model with impulsive state feedback control abstract in this paper homoclinic bifurcations of a prey predator system with state impulse are investigated firstly the existence of order 1 homoclinic cycle of system 1 2 with 0 is investigated secondly choosing q as a control parameter the sufficient conditions of existence and stability of order 1 periodic solution of system 1 2 with 0 are obtained by means of the geometry theory of semi continuous dynamic systems thirdly on the basis of the theory of rotated vector fields homoclinic bifurcation of system 1 2 about parameter are also studied finally some simulations are provided to prove the main results the methods used in this paper are intuitive to prove the existence of order 1 homoclinic cycle and homoclinic bifurcations

existence of a polycycle in non lipschitz gause type predator prey models abstract we study gause type predator prey models when the interaction between predator and prey is not locally lipschitz continuous in the absence of one of them we shall show that in this case there appears a polycycle which affects the existence of limit cycles for the system we apply the results to study the existence of limit cycles for a classical gause system

dynamical and numerical analyses of a generalized food chain model this paper focuses on the qualitative and quantitative analysis of a generalized three species food chain model the conditions for the existence and stability of the boundary and positive equilibria of the model are established by constructing an appropriate bifurcation function it is shown that the model undergoes a hopf bifurcation from the positive equilibrium for certain parameter values the root of this bifurcation function gives the bifurcation value a robust non standard numerical method is constructed and used to obtain the solution of the model numerical simulations using this method show that the hopf bifurcation is supercritical with a stable limit cycle in the positive octant

stochastic effects in mean field population growth the quasi gaussian approximation to the case of a taylor s law distributed substrate abstract we analyze the effect of the stochastic growth rates which may originate from heterogeneity of resource distribution on the average unrestricted growth of an ensemble of non interacting sub populations of consumers with holling i and ii functional responses assuming normal gaussian and anomalous tweedie density distributions of resources we show that the variation in the resource availability between the sub population habitats leads to a non malthusian dynamics of the initial phase of the average biomass growth and that this phase of growth can be successfully approximated using quasi gaussian approach this conclusion is illustrated by numerical simulations

scaling in ecosystems and the linkage of macroecological laws scaling provides an elegant framework for understanding power law behavior and deducing relationships between critical exponents we demonstrate that scaling theory can be generalized to develop a framework for the analysis of diverse empirical macroecological relationships traditionally treated as independent our mathematical arguments predict links between the species area relationship the relative species abundance and community size spectra in excellent accord with empirical data

singular perturbations of the holling i predator prey system with a focus abstract we study the occurrence of limit cyles in a model for two species predator prey interaction referred to as holling i this model is based on the phenomenological fitting of the so called functional response function to observed data in nature by a continuous piecewise differentiable function of the original models presented by holling it is the only model for which the possible asymptotic behaviour of the prey and predator densities has not been determined yet we extend the work of liu who showed that for certain values of parameters of the holling i system at least two nested limit cycles will occur surrounding a focus his case corresponds to a bifurcation problem where limit cycles are created from a system with a continuum of singularities i e a singular perturbation problem with slow fast solutions we prove that exactly two hyperbolic limit cycles occur after perturbation

positive periodic solutions in a non selective harvesting predator prey model with multiple delays abstract this paper is concerned with a predator prey model with hassell varley type functional response non selective harvesting and multiple delays some new sufficient conditions are obtained for the existence of positive periodic solutions by applying the coincidence degree theorem in addition the results are delay dependent which is different from the previous work finally simulations illustrate the effectiveness of our results

stability and bifurcation analysis on a ratio dependent predator prey model with time delay a ratio dependent predator prey model with time delay due to the gestation of the predator is investigated by analyzing the corresponding characteristic equations the local stability of a positive equilibrium and a semi trivial boundary equilibrium is discussed respectively further it is proved that the system undergoes a hopf bifurcation at the positive equilibrium using the normal form theory and the center manifold reduction explicit formulae are derived to determine the direction of bifurcations and the stability and other properties of bifurcating periodic solutions by means of an iteration technique sufficient conditions are obtained for the global attractiveness of the positive equilibrium by comparison arguments the global stability of the semi trivial equilibrium is also addressed numerical simulations are carried out to illustrate the main results

permanence and stability in non autonomous predator prey lotka volterra systems with feedback controls the main purpose of this article is considering whether or not the feedback controls have an influence on a non autonomous predator prey lotka volterra type system general criteria on permanence are established which is described by an integral form and independent of some feedback controls by constructing suitable lyapunov functionals a set of easily verifiable sufficient conditions are derived for the global stability of any positive solution to the model

the basins of attraction in a modified may holling tanner predator prey model with allee affect abstract i analyse a modified may holling tanner predator prey model considering an allee effect in the prey and alternative food sources for predator additionally the predation functional response or predation consumption rate is linear the extended model exhibits rich dynamics and we prove the existence of separatrices in the phase plane separating basins of attraction related to oscillation co existence and extinction of the predator prey population we also show the existence of a homoclinic curve that degenerates to form a limit cycle and discuss numerous potential bifurcations such as saddle node hopf and bogdanov takens bifurcations we use simulations to illustrate the behaviour of the model

dynamic analysis of fractional order singular holling type ii predator prey system abstract in this paper a fractional order singular fos predator prey model with holling type ii functional response has been introduced and the mathematical behavior of the model from the aspect of local stability is investigated through the fractional calculus and economic theory a new and more realistic predator prey model has been extended and the solvability condition is presented besides numerical simulations are considered to illustrate the effectiveness of the numerical method and confirm the theoretical results to explore the impacts of fractional order and economic interest on the presented system in biological context it is found that the presence of fractional order in the differential model can improve the stability of the solutions and enrich the dynamics of system in addition singular models exhibit more complicated dynamics rather than standard models especially the bifurcation phenomena which can reveal the instability mechanism of systems

finite element approximation of spatially extended predator prey interactions with the holling type ii functional response we study the numerical approximation of the solutions of a class of nonlinear reaction diffusion systems modelling predator prey interactions where the local growth of prey is logistic and the predator displays the holling type ii functional response the fully discrete scheme results from a finite element discretisation in space with lumped mass and a semi implicit discretisation in time we establish a priori estimates and error bounds for the semi discrete and fully discrete finite element approximations numerical results illustrating the theoretical results and spatiotemporal phenomena are presented in one and two space dimensions the class of problems studied in this paper are real experimental systems where the parameters are associated with real kinetics expressed in nondimensional form the theoretical techniques were adapted from a previous study of an idealised reaction diffusion system garvie and blowey in eur j appl math 16 5 621 646 2005

persistence in dynamical systems soit un espace metrique localement compact de metrique d et soit e un sous ensemble clos de a frontiere e et interieur e on suppose que l on a un flot continu 7b f defini sur e tel que e est invariant par 7b f on considere la notion de persistance et on donne un critere de persistance