hopf bifurcation and turing instability in spatial homogeneous and inhomogeneous predator prey models abstract this paper is concerned with two species spatial homogeneous and inhomogeneous predator prey models with beddington deangelis functional response for the spatial homogeneous model the asymptotic behavior of the interior equilibrium and the existence of hopf bifurcation of nonconstant periodic solutions surrounding the interior equilibrium are considered furthermore the direction of hopf bifurcation and the stability of bifurcated periodic solutions are investigated for the model with no flux boundary conditions turing instability of the interior equilibrium solution is studied in particular turing instability region regarding the parameters is established finally to verify our theoretical results some numerical simulations are also included

global well posedness and stability analysis of prey predator model with indirect prey taxis abstract this paper deals with a prey predator model with indirect prey taxis which means chemical of prey causes the directional movement of the predator we prove the global existence and uniform boundedness of solutions to the model for general functional responses in any spatial dimensions moreover through linear stability analysis it turns out that prey taxis is an essential factor in generating pattern formations this result differs in that the destabilizing effect of taxis does not occur in the direct prey taxis case in addition we show the global stability of the semi trivial steady state and coexistence steady state for some specific functional responses we give numerical examples to support the analytic results

turing patterns in the lengyel epstein system for the cima reaction the first experimental evidence of the turing pattern was observed by de kepper and her associates 1990 on the cima reaction in an open unstirred gel reactor almost 40 years after turing s prediction lengyel and epstein characterized this famous experiment using a system of reaction diffusion equations in this paper we report some fundamental analytic properties of the lengyel epstein system our result also indicates that if either of the initial concentrations of the reactants the size of the reactor or the effective diffusion rate are not large enough then the system does not admit nonconstant steady states a priori estimates are fundamental to our approach for this nonexistence result the degree theory was combined with the a priori estimates to derive existence of nonconstant steady states

turing instability for a ratio dependent predator prey model with diffusion abstract ratio dependent predator prey models have been increasingly favored by field ecologists where predator prey interactions have to be taken into account the process of predation search in this paper we study the conditions of the existence and stability properties of the equilibrium solutions in a reaction diffusion model in which predator mortality is neither a constant nor an unbounded function but it is increasing with the predator abundance we show that analytically at a certain critical value a diffusion driven turing type instability occurs i e the stationary solution stays stable with respect to the kinetic system the system without diffusion we also show that the stationary solution becomes unstable with respect to the system with diffusion and that turing bifurcation takes place a spatially non homogenous non constant solution structure or pattern arises a numerical scheme that preserve the positivity of the numerical solutions and the boundedness of prey solution will be presented numerical examples are also included

hopf bifurcation and turing instability of 2 d lengyel epstein system with reaction diffusion terms in this paper we study 2 d lengyel epstein l e reaction diffusion system with homogeneous neumann boundary condition by employing stability theory hopf bifurcation theorem and turing s theory we present some sufficient conditions ensuring the equilibrium point of system to be stable and derive conditions on the parameters so that spatial homogenous hopf bifurcation and turing instability occur these conditions obtained have important leading significance in applications of l e system finally we show some numerical examples to verity the theoretical analysis

steady states of a predator prey model with prey taxis abstract we study the steady states of a predator prey model with prey taxis incorporating holling type ii functional response under the homogeneous neumann boundary condition the stability of equilibrium points and the existence of non constant steady states are investigated we obtain that the prey tactic sensitivity coefficient delays the stability of the unique positive constant solution but for other equilibrium points stability the prey tactic sensitivity coefficient does not influence on it furthermore we derive some sufficient conditions relative to the prey tactic sensitivity coefficient which confines the existence of steady states and find that even if the interaction coefficient is sufficiently large there also exist non constant positive steady states under some conditions

the accuracy and stability of an implicit solution method for the fractional diffusion equation we have investigated the accuracy and stability of an implicit numerical scheme for solving the fractional diffusion equation this model equation governs the evolution for the probability density function that describes anomalously diffusing particles anomalous diffusion is ubiquitous in physical and biological systems where trapping and binding of particles can occur the implicit numerical scheme that we have investigated is based on finite difference approximations and is straightforward to implement the accuracy of the scheme is o dx 2 in the spatial grid size and o dt 1 c in the fractional time step where 0 email protected 1 is the order of the fractional derivative and c 1 is standard diffusion we have provided algebraic and numerical evidence that the scheme is unconditionally stable for 0 c 1

spatiotemporal patterns in a reaction diffusion model with the degn harrison reaction scheme abstract spatial and temporal patterns generated in ecological and chemical systems have become a central object of research in recent decades in this work we are concerned with a reaction diffusion model with the degn harrison reaction scheme which accounts for the qualitative feature of the respiratory process in a klebsiella aerogenes bacterial culture we study the global stability of the constant steady state existence and nonexistence of nonconstant steady states as well as the hopf and steady state bifurcations in particular our results show the existence of turing patterns and inhomogeneous periodic oscillatory patterns while the system parameters are all spatially homogeneous these results also exhibit the critical role of the system parameters in leading to the formation of spatiotemporal patterns

turing pattern of the oregonator model abstract the oregonator model is the mathematical dynamics which describes the field koros noyes mechanics of the famous belousov zhabotinskii reaction in this work we establish some fundamental analytic properties of this dynamics and its corresponding steady state under various conditions on the parameters and the size of the reactor we examine the existence and non existence of non constant steady states in particular for some properly chosen parameter ranges we prove the occurrence of the turing pattern generated by this oregonator model our results exhibit interesting and very different roles of the diffusion rates and the reactor in the formation of the turing pattern our mathematical analysis mainly relies on a priori estimates and the topological degree argument

stability analysis of a fractional order diffused prey predator model with prey refuges abstract in this paper a fractional order diffused prey predator model with prey refuges is considered the existence uniqueness non negativity and boundedness of the solutions for the model are proved moreover some sufficient conditions are given to ensure the existence and uniform asymptotic stability of the equilibrium point of the studied system finally numerical simulations are presented in order to support the theoretical results

multiple stability switches and hopf bifurcations induced by the delay in a lengyel epstein chemical reaction system abstract this paper examines the dynamical analysis of the lengyel epstein system with a discrete delay in detail under the assumption that the unique positive equilibrium of the model is locally asymptotically stable in the absence of the delay the effect of the increase of delay on the stability of the unique positive equilibrium is analyzed in detail it is found that under suitable conditions on the other parameters the delay doesn t affect the stability of the equilibrium namely the equilibrium is absolutely stable while under the other conditions on the other parameters the equilibrium will become ultimately unstable after passing through multiple stability switches and hopf bifurcations at some certain critical values of delay particularly by means of the normal form method and the center manifold reduction for retarded functional differential equations the explicit formulae determining the direction of hopf bifurcations and the stability of the bifurcating periodic solutions are obtained to verify our theoretical conclusions some numerical simulations for specific examples are also included at the end of this article

periodic solution of a prey predator model with nonlinear state feedback control assume that when the number of pests reaches the certain threshold pest management strategy will be taken to control pests based on this assumption in this paper we propose a pest management model with nonlinear state feedback control we then analyze the dynamic behavior of the model more precisely we first investigate the singularity of the model by using method of qualitative analysis secondly the existence of periodic solution of the model is studied by using successor functions and poincare bendixson theorem and then it is followed by the study of the stability of periodic solution finally an example with numerical simulations is given to illustrate our conclusions

factorisation properties of group scheme actions let h be an algebraic group scheme over a field k acting on a commutative k algebra a which is a unique factorisation domain we show that under certain mild assumptions the monoid of nonzero h stable principal ideals in a is free commutative from this we deduce in certain special cases results about the monoid of nonzero semi invariants and the algebra of invariants we use an infinitesimal method which allows us to work over an arbitrary base field

spatiotemporal pattern formation of a diffusive bimolecular model with autocatalysis and saturation law in this paper a reaction diffusion bimolecular model with autocatalysis and saturation law is investigated firstly we provide some conditions for the stability turing instability of the constant positive solution then we mainly consider hopf bifurcations and steady state bifurcations which bifurcate from the unique constant positive solution of the system our results suggest the existence of spatially non homogeneous periodic orbit and non constant positive steady state solutions which implies the possibility of rich spatiotemporal patterns in this diffusive bimolecular model numerical examples are presented to support our theoretical analysis furthermore non existence of non constant steady state is investigated in terms of parameters

spatial patterns of a fractional type cross diffusion holling tanner model abstract in this paper a fractional type cross diffusion holling tanner model under homogeneous neumann boundary condition is considered using the maximum principle and the harnack inequality we obtain a priori estimates of the positive stationary solution nonexistence and existence of nonconstant positive stationary solution are obtained which reveal that fractional type cross diffusion can create spatial patterns but the random diffusion fails

on the asymptotic stability of the time fractional lengyel epstein system abstract this paper concerns a time fractional version of the conventional lengyel epstein cima reaction model we define the invariant regions of the system and establish sufficient conditions for the unique equilibrium s local and global asymptotic stability numerical results are presented to illustrate the effect of the fractional order on system dynamics

steady states of a leslie gower model with diffusion and advection abstract this paper focuses on a stationary leslie gower model with diffusion and advection firstly some existence conditions of nonconstant positive solutions are obtained by means of the leray schauder degree theory as diffusion and advection of one of the species both tend to infinity we obtain a limiting system which is a semi linear elliptic equation with nonlocal constraint in the simplified 1d case the global bifurcation structure of nonconstant solutions of the limiting system is classified

turing hopf bifurcations in a predator prey model with herd behavior quadratic mortality and prey taxis abstract in this paper we propose a predator prey model with herd behavior and prey taxis then we analyze the stability and bifurcation of the positive equilibrium of the model subject to the homogeneous neumann boundary condition by using an abstract bifurcation theory and taking prey tactic sensitivity coefficient as the bifurcation parameter we obtain a branch of stable nonconstant solutions bifurcating from the positive equilibrium our results show that prey taxis can yield the occurrence of spatial patterns

limit sets and the poincare bendixson theorem in impulsive semidynamical systems we consider semidynamical systems with impulse effects at variable times and we discuss some properties of the limit sets of orbits of these systems such as invariancy compactness and connectedness as a consequence we obtain a version of the poincare bendixson theorem for impulsive semidynamical systems

turing instability and pattern induced by cross diffusion in a predator prey system with allee effect to have spatial patterns form small allee effect admits relatively large range of predator mortality rate found cross diffusion is the key mechanism of spatial pattern formation obtained amplitude equations suggest both supercritical and subcritical bifurcation may occur weak allee effect admits supercritical bifurcation while strong one allows subcritical bifurcation in this paper we first propose a mathematical model for a spatial predator prey system with allee effect and then by using the proposed model we investigate the turing instability and the phenomena of pattern formation we show how cross diffusion destabilizes the spatially uniform steady state the method of multiple time scales is employed to derive the amplitude equations which is the cubic stuart landau equation in the supercritical case and the quintic in the subcritical case based on the amplitude equations we obtain the asymptotic solutions of the model close to the onset of instability

the zassenhaus variety of a reductive lie algebra in positive characteristic let g be the lie algebra of a connected reductive group g over an algebraically closed field k of characteristic p 0 let z be the centre of the universal enveloping algebra u u g of g its maximal spectrum is called the zassenhaus variety of g we show that under certain mild assumptions on g the field of fractions frac z of z is g equivariantly isomorphic to the function field of the dual space g with twisted g action in particular frac z is rational this confirms a conjecture of j alev furthermore we show that z is a unique factorisation domain confirming a conjecture of a braun and c hajarnavis recently a premet used the above result about frac z a result of colliot thelene kunyavskii popov and reichstein and reduction mod p arguments to show that the gelfand kirillov conjecture cannot hold for simple complex lie algebras that are not of type a c or g2

exponential stability and extended dissipativity criteria for generalized discrete time neural networks with additive time varying delays abstract this paper is concerned with exponential stability and extended dissipativity criteria for generalized discrete time neural networks gdnns with additive time varying delays the generalized dissipativity analysis combines a few previous results into a framework such as l 2 l performance h performance passivity performance strictly q s r dissipative and strictly q s r dissipative the definition of exponential stability for gdnns is given with a new and more appropriate expression a novel augmented lyapunov krasovskii functional lkf which involves more information about the additive time varying delays is constructed by introducing more zero equalities and using a new double summation inequality together with finsler s lemma an improved delay dependent exponential stability and extended dissipativity criterion are derived in terms of convex combination technique cct finally numerical examples are given to illustrate the usefulness and advantages of the proposed methods

simple and superlattice turing patterns in reaction diffusion systems bifurcation bistability and parameter collapse abstract we use equivariant bifurcation theory to investigate pattern selection at the onset of a turing instability in a general two component reaction diffusion system the analysis is restricted to patterns that periodically tile the plane in either a square or hexagonal fashion both simple periodic patterns stripes squares hexagons and rhombs and superlattice patterns are considered the latter correspond to patterns that have structure on two disparate length scales the short length scale is dictated by the critical wave number from linear theory while the periodicity of the pattern is on a larger scale analytic expressions for the coefficients of the leading nonlinear terms in the bifurcation equations are computed from the general reaction diffusion system using perturbation theory we show that no matter how complicated the reaction kinetics might be the nonlinear reaction terms enter the analysis through just four parameters moreover for hexagonal problems all patterns bifurcate unstably unless a particular degeneracy condition is satisfied and at this degeneracy we find that the number of effective system parameters drops to two allowing a complete characterization of the possible bifurcation results at this degeneracy for example we find that rhombs squares and superlattice patterns always bifurcate unstably we apply these general results to some specific model equations including the lengyel epstein cima model to investigate the relative stability of patterns as a function of system parameters and to numerically test the analytical predictions

global stability of prey taxis systems abstract in this paper we prove the global boundedness and stability of the predator prey system with prey taxis in a two dimensional bounded domain with neumann boundary conditions by deriving an entropy like equality and a boundedness criterion we show that the intrinsic interaction between predators and preys is sufficient to prevent the population overcrowding even the prey taxis is included and strong furthermore by constructing appropriate lyapunov functionals we show that prey only steady state is globally asymptotically stable if the predation is weak and the co existence steady state is globally asymptotically stable under some conditions like the prey taxis is weak or the prey diffuses fast if the predation is strong the convergence rates of solutions to the steady states are derived in the paper

diffusion self diffusion and cross diffusion u1 x 0 u 1 0 x u2 x 0 u 2 0 x i n 0 where 2 n 1 2 x 2 is the laplace operator 0 is a bounded smooth domain of r n with n1 0 and 0 are the boundary and the closure of 0 respectively 0t 0 0 t and 0t 0 0 t for some t 0 is the outward unit normal vector on 0 di ai bi ci i 1 2 are all positive constants while ij i j 1 2 denote non negative constants the initial values u1 0 and u2 0 are non negative smooth func tions which are not identically zero the mathematical model 1 1 was proposed by shigesada et al 27 in their study of spatial segregation of interacting species where u1 and u2 represent the densities of two competing species d1 and d2 are their diffu sion rates a1 and a2 denote the intrinsic growth rates b1 and c2 account for intra specific competitions b2 and c1 are the coefficients of inter specific competitions 11 and 22 are usually referred as self diffusion pressures and 12 and 21 are cross diffusion pressures an interesting feature of 1 1 is that the movements of these two competing species are affected by the population pressures due to the mutual interference between the individuals and it was hoped that the nonlinear dispersive force due to the

on global bifurcation for quasilinear elliptic systems on bounded domains general second order quasilinear elliptic systems with nonlinear boundary conditions on bounded domains are formulated into nonlinear mappings between sobolev spaces it is shown that the linearized mapping is a fredholm operator of index zero this and the abstract global bifurcation theorem of jacobo pejsachowicz patrick j rabier degree theory for c1 fredholm mappings of index 0 j anal math 76 1998 289 319 allow us to carry out bifurcation analysis directly on these elliptic systems at the abstract level we establish a unilateral global bifurcation result that is needed when studying positive solutions finally we supply two examples of cross diffusion population model and chemotaxis model to demonstrate how the theory can be applied

zassenhaus varieties of general linear lie algebras abstract let g be a lie algebra over an algebraically closed field of characteristic p 0 and let u g be the universal enveloping algebra of g we prove in this paper for g gl n and g sl n that the centre of u g is a unique factorisation domain and its field of fractions is rational for g sl n our argument requires the assumption that p n while for g gl n it works for any p it turned out that our two main results are closely related to each other the first one confirms in type a a recent conjecture of a braun and c hajarnavis while the second answers a question of j alev

turing patterns of a strongly coupled predator prey system with diffusion effects abstract the main purpose of this work is to investigate the effects of cross diffusion in a strongly coupled predator prey system by a linear stability analysis we find the conditions which allow a homogeneous steady state stable for the kinetics to become unstable through a turing mechanism in particular it is shown that turing instability of the reaction diffusion system can disappear due to the presence of the cross diffusion which implies that the cross diffusion induced stability can be regarded as the cross stability of the corresponding reaction diffusion system furthermore we consider the existence and non existence results concerning non constant positive steady states patterns of the system we demonstrate that cross diffusion can create non constant positive steady state solutions these results exhibit interesting and very different roles of the cross diffusion in the formation and the disappearance of the turing instability

global stability of the steady states of an sis epidemic reaction diffusion model abstract in this work we investigate the sis epidemic reaction diffusion model under heterogeneous environment studied by allen et al in l j s allen b m bolker y lou a l nevai asymptotic profiles of the steady states for an sis epidemic reaction diffusion model discrete contin dyn syst a 21 1 2008 1 20 in the two cases 1 the diffusion rate d s of the susceptible individuals is equal to the diffusion rate d i of the infected individuals 2 x r x for any fixed constant r 0 where x and x respectively represent the rates of disease transmission and disease recovery we completely determine the global stability of the disease free equilibrium and the unique endemic equilibrium if it exists our results partially answer the conjecture proposed by allen et al

hopf bifurcation analysis of a reaction diffusion sel kov system abstract a reaction diffusion system known as the sel kov model subject to the homogeneous neumann boundary condition is investigated where detailed hopf bifurcation analysis is performed we not only show the existence of the spatially homogeneous non homogeneous periodic solutions of the system but also derive conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution

spatio temporal dynamics near the steady state of a planktonic system abstract the study of spatio temporal behaviour of ecological systems is fundamentally important as it can provide deep understanding of species interaction and predict the effects of environmental changes in this paper we first propose a spatial model with prey taxis for planktonic systems in which we also consider the herb behaviour in prey and effect of the hyperbolic mortality rate applying the homogeneous neumann boundary condition to the model and using prey tactic sensitivity coefficient as bifurcation parameter we then detailedly analyse the stability and bifurcation of the steady state of the system firstly we carry out a study of the equilibrium bifurcation showing the occurrence of fold bifurcation hopf bifurcation and the bt bifurcation then by using an abstract bifurcation theory and taking prey tactic sensitivity coefficient as the bifurcation parameter we investigate the turing hopf bifurcation obtaining a branch of stable non constant solutions bifurcating from the positive equilibrium and our results show that prey taxis can yield the occurrence of spatio temporal patterns finally numerical simulations are carried out to illustrate our theoretical results showing the existence of a periodic solution when the prey tactic sensitivity coefficient is away from the critical value

pattern formation in the three dimensional reaction diffusion systems existing group theoretic analysis of pattern formation in three dimensions t k callahan e knobloch symmetry breaking bifurcations on cubic lattices nonlinearity 10 1997 1179 1216 is used to make specific predictions about the formation of three dimensional patterns in two models of the turing instability the brusselator model and the lengyel epstein model spatially periodic patterns having the periodicity of the simple cubic sc face centered cubic fcc or body centered cubic bcc lattices are considered an efficient center manifold reduction is described and used to identify parameter regimes permitting stable lamellae sc fcc double diamond hexagonal prism bcc and bcci states both models possess a special wavenumber k at which the normal form coefficients take on fixed model independent ratios and both are described by identical bifurcation diagrams this property is generic for two species chemical reaction diffusion models with a single activator and inhibitor 1999 elsevier science b v all rights reserved

turing patterns in a reaction diffusion model with the degn harrison reaction scheme abstract in this paper we consider a reaction diffusion model with degn harrison reaction scheme some fundamental analytic properties of nonconstant positive solutions are first investigated we next study the stability of constant steady state solution to both ode and pde models our result also indicates that if either the size of the reactor or the effective diffusion rate is large enough then the system does not admit nonconstant positive solutions finally we establish the global structure of steady state bifurcations from simple eigenvalues by bifurcation theory and the local structure of the steady state bifurcations from double eigenvalues by the techniques of space decomposition and implicit function theorem

time local dissipative formulation and stable numerical schemes for a class of integrodifferential wave equations we consider integrodifferential equations of the abstract form mathbf h partial t phi mathbf g nabla phi f where mathbf h partial t is a diagonal convolution operator and mathbf g nabla is a linear anti selfadjoint differential operator on the basis of an original approach devoted to integral causal operators we propose and study a time local augmented formulation under the form of a cauchy problem partial t psi mathcal a psi mathcal b f such that phi mathcal c psi we show that under a suitable hypothesis on the symbol mathbf h p this new formulation is dissipative in the sense of a natural energy functional we then establish the stability of numerical schemes built from this time local formulation thanks to the dissipation of appropriate discrete energies finally the efficiency of these schemes is highlighted by concrete numerical results relating to a model recently proposed for 1d acoustic waves in porous media

cross diffusion induced turing instability for a competition model with saturation effect abstract competitive behavior like predation widely exists in the world which has influences on the dynamics of ecosystems though a good knowledge for the patten formation and selection of predator prey models with diffusion is known in the literature little is known for that of a diffused competition model as a result a competition model with saturation effect and self and cross diffusion is proposed and analyzed in this paper firstly by making use of nullcline analysis we derive the condition of existence and stability of positive equilibrium and prove global stability then using the linear stability analysis the turing bifurcation critical value and the condition of the occurrence of turing pattern are obtained when control parameter is selected finally we deduce the amplitude equations around the turing bifurcation point by using the standard multiple scale analysis meanwhile a series of numerical simulations are given to expand our theoretical analysis this work shows that cross diffusion plays a key role in the formation of spatial patterns for competitive model with saturation effect

global bifurcation analysis and pattern formation in homogeneous diffusive predator prey systems abstract the dynamics of a general diffusive predator prey system is considered existence and nonexistence of non constant positive steady state solutions are shown to identify the ranges of parameters of spatial pattern formation bifurcations of spatially homogeneous and nonhomogeneous periodic solutions as well as non constant steady state solutions are studied

investigating the interactions between allee effect and harvesting behaviour of a single species model an evolutionary dynamics approach abstract in this study we propose an ecological model which is based on the smith growth model and subject to multiplicative allee effect we then investigate the equilibrium bifurcation induced by the harvesting behaviour and the allee effect and pitch fork bifurcation and saddle node bifurcation may occur assuming that the carrying capacity and harvesting rate obey a gaussian distribution we then consider the evolutionary dynamics by using continuous strategy evolutionary game theory our study shows that strong allee effect and low harvesting rate can induce bistability it also suggests that evolution prevents extinction even if population has strong allee effect numerical analysis shows that the evolutionary model can admit up to nine equilibrium points including up to four evolutionary stable strategy equilibria

swarming patterns in a two dimensional kinematic model for biological groups we construct a continuum model for the motion of biological organisms experiencing social interactions and study its pattern forming behavior the model takes the form of a conservation law in two spatial dimensions the social interactions are modeled in the velocity term which is nonlocal in the population density and includes a parameter that controls the interaction length scale the dynamics of the resulting partial integrodifferential equation may be uniquely decomposed into incompressible motion and potential motion for the purely incompressible case the model resembles one for fluid dynamical vortex patches there exist solutions which have constant population density and compact support for all time numerical simulations produce rotating structures which have circular cores and spiral arms and are reminiscent of naturally observed phenomena such as ant mills the sign of the social interaction term determines the direction of the rotation and the interaction length scale affects the degree o

global analysis and numerical simulations of a novel stochastic eco epidemiological model with time delay abstract in this paper a new stochastic eco epidemiological model is proposed and investigated and the high dimensional model includes time delay and a general incidence rate which make the study more complex first the existence and uniqueness of the global positive solution with any positive initial value are verified second using stochastic analysis methods and some differential inequality techniques we explore the long time asymptotic properties of the stochastic delayed system furthermore we prove that the solution of the stochastic delayed system has a unique stationary distribution and it has ergodic property third the sufficient conditions of permanence in mean or extinction for the ecological populations are obtained finally we introduce a series of computer simulations to show the performance of the theoretical results

permanence and periodicity of a delayed ratio dependent predator prey model with holling type functional response and stage structure a periodic and delayed ratio dependent predator prey system with holling type iii functional response and stage structure for both prey and predator is investigated it is assumed that immature predator and mature individuals of each species are divided by a fixed age and immature predator do not have the ability to attack prey sufficient conditions are derived for the permanence and existence of positive periodic solution of the model numerical simulations are presented to illustrate the feasibility of our main results

evolutionary dynamics of single species model with allee effect abstract by considering a continuous strategy evolutionary game theory egt a single species population subject to additive allee effect is investigated we assume that the carrying capacity and predator evasion ability following a gaussian distribution this assumption shows that evolution prevents extinction even if population has strong allee effect the ratio of competition and anti predation phenotypes has an important effect on determining global dynamics of the egt model the numerical simulations present the model can have nine equilibria and the model has multiple evolutionarily stable strategy equilibria ess

delay induced turing instability in reaction diffusion equations time delays have been commonly used in modeling biological systems and can significantly change the dynamics of these systems quite a few works have been focused on analyzing the effect of small delays on the pattern formation of biological systems in this paper we investigate the effect of any delay on the formation of turing patterns of reaction diffusion equations first for a delay system in a general form we propose a technique calculating the critical value of the time delay above which a turing instability occurs then we apply the technique to a predator prey model and study the pattern formation of the model due to the delay for the model in question we find that when the time delay is small it has a uniform steady state or irregular patterns which are not of turing type however in the presence of a large delay we find spiral patterns of turing type for such a model we also find that the critical delay is a decreasing function of the ratio of carrying capacity to half saturation of the prey density