stability and bifurcation for an seis epidemic model with the impact of media abstract a novel seis epidemic model with the impact of media is introduced by analyzing the characteristic equation of equilibrium the basic reproduction number is obtained and the stability of the steady states is proved the occurrence of a forward backward and hopf bifurcation is derived numerical simulations and sensitivity analysis are performed our results manifest that media can regard as a good indicator in controlling the emergence and spread of the epidemic disease

examination of the 1 d expansion method from exact enumeration for a self interacting self avoiding walk the 1 d expansion method for polymer chains is examined by comparing these expansions for several thermodynamic and structural quantities with the results of standard series analysis of exact enumeration data the comparisons cover a wide range of spatial dimensions d including non integer ones and are performed for particular values of interaction energy good agreement is generally found for d 4 whereas discrepancies become conspicuous as d decreases to d 2 reasonable values are obtained for the exponents v and gamma in d 2 4 by applying the coherent anomaly method of suzuki 1986 to our 1 d expansions through fifth order in d 1

statistics of collapsing lattice animals we consider several lattice models of branched polymers in dilute solution in which the polymer molecule is modelled as a tree or animal in general the thermodynamic properties of the polymer are determined by monomer monomer and monomer solvent interactions we examine a two variable model which incorporates both types of interaction and discuss its relationship to other models which have previously been investigated in particular we discuss the collapse transition in these models

exact enumeration study of free energies of interacting polygons and walks in two dimensions we present analyses of substantially extended series for both interacting self avoiding walks isaw and polygons isap on the square lattice we argue that these provide good evidence that the free energies of both linear and ring polymers are equal above the temperature thus extending the application of a theorem of tesi et al to two dimensions below the temperature the conditions of this theorem break down in contradistinction to three dimensions but an analysis of the ratio of the partition functions for isap and isaw indicates that the free energies are in fact equal at all temperatures within at least any perceived difference can be interpreted as the difference in the size of corrections to scaling in both problems this may be used to explain the vastly different values of the crossover exponent previously estimated for isap to that predicted theoretically and numerically confirmed for isaw an analysis of newly extended neighbour avoiding self avoiding walk series is also given

geometrical arguments against the alexander orbach conjecture for lattice animals and diffusion limited aggregates geometrical arguments are given which suggest that the alexander orbach conjecture 1982 does not hold for lattice animals in d 2 and for the diffusion limited aggregates for large dimensions

representations of o n spin models by self avoiding random walks we establish that correlation functions of classical lattice spin models can be represented by series expansions in terms of self avoiding random walks using this we get new upper bounds of critical temperatures of the o n symmetric classical heisenberg models

1 d expansions for critical amplitudes the author shows how to expand critical amplitudes in inverse powers of the dimensionality d the technique which is rather general is applied to the classical n vector model with particular emphasis on the self avoiding walk n 0 and ising n 1 cases the existence of these expansions and of similar ones for critical points and critical constants suggests that 1 d expansions for non universal quantities are analogous to the epsilon expansions for universal quantities

study of the θ point by enumeration of self avoiding walks on the triangular lattice the author reports series expansion analyses of self avoiding walks with nearest neighbour bond interactions the estimates 2 nu t 1 07 or 0 05 and phi 0 64 or 0 05 for the correlation and crossover exponents at the theta point were obtained by examining the number of walks and the end to end distance data up to 16 steps on the triangular lattice

on the ferromagnetic ising model in noninteger spatial dimension using the necessary and sufficient conditions in terms of the high field series coefficients that the yang lee theorem holds we prove rigorously by counterexample that it cannot be extended to general noninteger dimension when such models are defined by the natural analytic continuation

1 d expansions for the free energy of weakly embedded site animal models of branched polymers we give the 1 d expansion to order 1 d5 for the limiting reduced free energy of a weakly embedded two variable site animal model of branched polymers hence we are able to derive expansions for the free energy of related one variable models and the growth constant of weakly and strongly embedded trees again to order 1 d5 we argue that the free energy expansions are asymptotically correct for a small range of fugacities only exact results show that animals with a compact hypercubic structure make the overwhelming contribution to the total number of weakly embedded site animals especially for large d this result is not reflected in the free energy 1 d expansions where trees make the dominant contribution we also derive new exact enumeration data for lattice animals using intermediate calculations in the derivation of the 1 d expansions thus we give partition functions for the general d dimensional hypercubic lattice for one to 13 sites

potts model formulation of branched polymers in a solvent a potts model formulation of the statistics of branched polymers or lattice animals in a solvent is given the migdal kadanoff renormalisation group is employed to study the critical behaviour or fractal dimension of the branched polymer four different critical behaviours are found corresponding to random animal collapse or theta point percolation and compact cluster the theta point behaviour is described by a tricritical point while percolation corresponds to a higher order critical point where the effect of the solvent on the branched polymer is the same as the screening effect of the other clusters in percolation

phase diagram of branched polymer collapse the phase diagram of the collapse of a two dimensional infinite branched polymer interacting with the solvent and with itself through contact interactions is studied from the q ensuremath rightarrow 1 limit of an extension of the q state potts model exact solution on the bethe lattice and migdal kadanoff renormalization group calculations shows that there is a line of ensuremath theta transitions from the extended to a single compact phase the ensuremath theta line governed by three different fixed points consists of two lines of extended compact transitions which are in different universality classes and meet in a multicritical point on the other hand directed branched polymers are shown to be completely determined by the strongly embedded case and there is a single ensuremath theta transition which is in the directed percolation universality class textcopyright 1996 the american physical society

from dilute to dense self avoiding walks on hypercubic lattices a field theoretic representation is presented to count the number of configurations of a single self avoiding walk on a hypercubic lattice ind dimensions with periodic boundary conditions we evaluate the connectivity constant as a function of the fractionf of sites occupied by the polymer chain the meanfield approximation is exact in the limit of infinite dimensions and corrections to it in powers ofd 1 can be systematically evaluated the connectivity constant and the site entropy calculated throughout second order compare well with known results in two and three dimensions we also find that the entropy per site develops a maximum atf 1 2d 1 ford 2 d 3 this maximum occurs atf 0 80 f 0 86 and its value is about 50 30 higher than the entropy per site of a hamiltonian walk f 1

permanence of a stage structured predator prey system with impulsive stocking prey and harvesting predator abstract in this paper we propose a stage structured predator prey system with impulsive stocking immature prey and harvesting predator at different moment we analyze the global attractivity of the mature prey extinction periodic solution and obtain sufficient conditions for the permanence of the system numerical simulations are also inserted to verify the feasibility of the theoretical results moreover the obtained results show that impulsive stocking immature prey or harvesting predator may play a key role on the permanence of the system and provide tactical basis for the biological resources management

examination of the theta point from exact enumeration of self avoiding walks self avoiding walks on the square lattice with nearest neighbour attractive interactions are investigated as a model for the two dimensional version of a polymer in dilute solution temperature dependences of the exponent nu and the free energy of the chain are estimated from the exact enumeration data for up to 20 steps the value of nu at the theta point disagrees with the mean field theory the end distance distribution function at the theta point is also examined

scaling of self avoiding walks in high dimensions we examine self avoiding walks in dimensions 4 to 8 using high precision monte carlo simulations up to length n 16 384 providing the first such results in dimensions d 4 on which we concentrate our analysis we analyse the scaling behaviour of the partition function and the statistics of nearest neighbour contacts as well as the average geometric size of the walks and compare our results to 1 d expansions and to excellent rigorous bounds that exist in particular we obtain precise values for the connective constants 5 8 838 544 3 6 10 878 094 4 7 12 902 817 3 8 14 919 257 2 and give a revised estimate of 4 6 774 043 5 all of these are by at least one order of magnitude more accurate than those previously given from other approaches in d 4 and all approaches in d 4 our results are consistent with most theoretical predictions though in d 5 we find clear evidence of anomalous n 1 2 corrections for the scaling of the geometric size of the walks which we understand as a non analytic correction to scaling of the general form n 4 d 2 not present in pure gaussian random walks

collapse transition of randomly branched polymers renormalized field theory we present a minimal dynamical model for randomly branched isotropic polymers and we study this model in the framework of renormalized field theory for the swollen phase we show that our model provides a route to understand the well established dimensional reduction results from a different angle for the collapse transition we uncover a hidden becchi rouet stora supersymmetry signaling the sole relevance of tree configurations we correct the long standing one loop results for the critical exponents and we push these results on to two loop order for the collapse transition we find a runaway of the renormalization group flow which lends credence to the possibility that this transition is a fluctuation induced first order transition our dynamical model allows us to calculate for the first time the fractal dimension of the shortest path on randomly branched polymers in the swollen phase as well as at the collapse transition and related fractal dimensions

branched polymers with a prescribed number of cycles the authors consider the number of lattice animals with exactly c cycles for weakly embeddable bond clusters they show rigorously that the growth constant is independent of c and they derive upper and lower bounds on the critical exponents for each value of c they use series analysis methods to estimate the critical exponents for c 1 and 2 in two and three dimensions and find that the critical exponent does depend on the number of cycles evidence from series analysis results and from expansions in inverse powers of the dimension suggests that in the case of strong embeddings i e site clusters the growth constant is independent of c and the corresponding values of the critical exponents are identical to the values for weak embeddings i e bond clusters they discuss the relationship of these results to the field theory prediction that the critical exponent is independent of cycle fugacity

collapse transition and crossover scaling for self avoiding walks on the diamond lattice a monte carlo study of self avoiding walks on the diamond lattice is presented this model incorporates some of the steric effects and short range stiffness of real alkanes and for a nearest neighbour attractive interaction epsilon is found to have a collapse transition at kb theta epsilon 2 25 or 0 05 the behaviour of the chains in the vicinity of this theta temperature is analysed with the help of recent crossover scaling theories it is shown that for finite chain length n there is a rather broad theta region where the chains behaviour is quasi ideal the width of this region wtheta behaves as omega 0 varies as n 12 consistent with the blob picture the peak of the specific heat occurs at the boundary between the theta region and the region of collapsed chains the authors also give rough estimates for the scaling functions describing the crossovers of the end to end distance and structure factor of the chains

self avoiding walk enumeration via the lace expansion we introduce a new method for the enumeration of self avoiding walks based on the lace expansion we also introduce an algorithmic improvement called the two step method for self avoiding walk enumeration problems we obtain significant extensions of existing series on the cubic and hypercubic lattices in all dimensions d 3 we enumerate 32 step self avoiding polygons in d 3 26 step self avoiding polygons in d 4 30 step self avoiding walks in d 3 and 24 step self avoiding walks and polygons in all dimensions d 4 we analyze these series to obtain estimates for the connective constant and various critical exponents and amplitudes in dimensions 3 d 8 we also provide major extensions of 1 d expansions for the connective constant and for two critical amplitudes

new lattice model for interacting avoiding polymers with controlled length distribution a new lattice spin model for many self avoiding polymers is introduced in which the chain length distribution is fully controllable with a single generating magnetic field the model utilises spins with additional internal symmetry degrees of freedom to impose a causal connectivity of the polymer bonds on the lattice use of the method of random fields then produces an equivalent n to 0 limit field theory the flory huggins theory for a polymer solution emerges simply from this field theory in the mean field approximation polymer polymer interactions between polymer segments on nearest neighbour lattice are introduced into the field theory and the low polymer volume fraction limit of the theory reduces to the edwards type field theory for dilute through semidilute polymer solutions a sketch is provided towards the treatment of branched polymers with fully controllable chain and branch length distributions and branching probabilities as well as a kinetic polymerisation system governed by specified propagation and termination probabilities

a study of uniform stars using 1 d expansions and numerical methods we study a lattice model of an interacting uniform self avoiding star polymer with f branches a 1 d expansion for the limiting reduced free energy is derived through order 1 d for general f and for f 3 to order 1 d 2 the terms in the expansion are independent of f and agree term by term with the corresponding expansion for interacting self avoiding walks we also present a miscellany of numerical results obtained by more conventional series and monte carlo techniques all our results both past and present support the conjecture that the limiting reduced free energies of f stars walks and polygons are identical for all values of the interaction parameter

stability and bifurcation analysis for a delayed lotka volterra predator prey system the present paper deals with a delayed lotka volterra predator prey system by linearizing the equations and by analyzing the locations on the complex plane of the roots of the characteristic equation we find the necessary conditions that the parameters should verify in order to have the oscillations in the system in addition the normal form of the hopf bifurcation arising in the system is determined to investigate the direction and the stability of periodic solutions bifurcating from these hopf bifurcations to verify the obtained conditions a special numerical example is also included

the random walk representation of classical spin systems and correlation inequalities ferromagnetic lattice spin systems can be expressed as gases of random walks interacting via a soft core repulsion by using a mixed spinrandom walk representation we present a unified approach to many recently established correlation inequalities as an application of these inequalities we obtain a simple proof of the mass gap for the 4 2 quantum field model we also establish new upper bounds on critical temperatures

on directed interacting animals and directed percolation we study the phase diagram of fully directed lattice animals with nearest neighbour interactions on the square lattice this model comprises several interesting ensembles directed site and bond trees bond animals strongly embeddable animals as special cases and its collapse transition is equivalent to a directed bond percolation threshold precise estimates for the animal size exponents in the different phases and for the critical fugacities of these special ensembles are obtained from a phenomenological renormalization group analysis of the correlation lengths for strips of width up to n 17 the crossover region in the vicinity of the collapse transition is analysed in detail and the crossover exponent is determined directly from the singular part of the free energy we show using scaling arguments and an exact relation due to dhar that is equal to the fisher exponent governing the size distribution of large directed percolation clusters

effects of toxin delay on the dynamics of a phytoplankton zooplankton model abstract this paper is concerned with a phytoplankton zooplankton model with toxin delay it is shown that the positive equilibrium of the system is stable when the toxin delay is not included however we find that the incorporation of a discrete toxin delay not only can destabilize the positive equilibrium of the system but also can cause a hopf bifurcation at the positive equilibrium as it crosses some critical values in particular the normal form of the hopf bifurcation arising in the system is determined to investigate the direction and the stability of periodic solutions bifurcating from these hopf bifurcations to verify the obtained conditions a special numerical example is also included these results in this work demonstrate that the toxin delay plays an important role on deriving the complex dynamics

the stationary distribution and ergodicity of a stochastic phytoplankton zooplankton model with toxin producing phytoplankton under regime switching abstract this paper addresses a stochastic phytoplankton zooplankton model with toxin producing phytoplankton under regime switching the stochastic model in this paper includes white and colored noises for the model we first prove the existence of the global positive solution of the model and then by using khasminskii s method and lyapunov function we derive sufficient conditions for the existence of a unique ergodic stationary distribution finally numerical simulations are carried out to support our results

non universality in the collapse of two dimensional branched polymers in this paper we study the complete phase diagram of a model of interacting branched polymers the model we consider is a lattice animal one where the collapse transition can be driven both by a contact fugacity between two occupied nearest neighbours and by a fugacity related to each occupied edge using a potts model formulation of the problem we conjecture the existence of two different universality classes for the theta transitions with thermal exponents nu and phi equal to 1 2 2 3 and 8 15 8 15 separated by a higher order percolation point we also present convincing numerical evidence for these exponent values using a transfer matrix approach we discuss the possibility of a collapse collapse transition and we predict the behaviour of our model when an adsorbing surface is included

global asymptotic stability of a diffusive predator prey model with ratio dependent functional response abstract this paper is concerned with a diffusive leslie gower predator prey system with ratio dependent holling type iii functional response under homogeneous neumann boundary conditions the uniform persistence of the solutions semiflows the existence of global attractors local and global asymptotic stability of the positive constant steady state of the reaction diffusion model are discussed by using comparison principle the linearization method and the lyapunov functional method respectively the global asymptotic stability of the positive constant steady state shows that the prey and predator will be spatially homogeneously distributed as time converges to infinities

lattice trees with specified topologies the authors study the total numbers of lattice trees with specified topologies for strongly embeddable or site clusters with ni branching points of degree i they show how to prove rigorously that the growth constants exist and are all equal to the neighbour avoiding walk limit nu they derive some exact upper bounds for the critical exponents associated with the star comb and brush topologies exact enumeration data are derived and analysed for both weak and strong embeddings of some stars combs and brushes on the square triangular simple cubic and d 4 simple hypercubic lattices using the exact enumeration data for the general d dimensional simple hypercubic lattice and the exact results for the interior of a bethe lattice they derive expansions for the growth constants in inverse powers of the dimensionality these results are consistent with the growth constants being equal to the appropriate walk limits mu or nu

new lower bounds on the self avoiding walk connective constant we give an elementary new method for obtaining rigorous lower bounds on the connective constant for self avoiding walks on the hypercubic lattice d the method is based on loop erasure and restoration and does not require exact enumeration data our bounds are best for highd and in fact agree with the first four terms of the 1 d expansion for the connective constant the bounds are the best to date for dimensionsd 3 but do not produce good results in two dimensions ford 3 4 5 and 6 respectively our lower bound is within 2 4 0 43 0 12 and 0 044 of the value estimated by series extrapolation

marriage of exact enumeration and 1 d expansion methods lattice model of dilute polymers we consider the properties of a self avoiding polymer chain with nearestneighbor contact energy on ad dimensional hypercubic lattice general theoretical arguments enable us to prescribe the exact analytic form of then segment chain partition functionc n and unknown coefficients for chains of up to 11 segments are determined using exact enumeration data ind 2 6 this exact form provides the main ingredient to produce a large n expansion ind 1of the chain free energy through fifth order with the full dependence on the contact energy retained the dependent chain connectivity constant and free energy amplitude are evaluated within thed 1expansion too d 5 our general formulation includes for the first time self avoiding walks neighboravoiding walks theta and collapsed chains as particular limiting cases

free energy of self interacting uniform stars we consider uniform star polymers on a variety of lattices with an additional contact interaction between pairs of vertices which are unit distance apart and are not joined by an edge of the star we present some rigorous results and other evidence which indicate that these systems have the same limiting free energy as self interacting self avoiding walks we discuss the extension of these results to other homeomorphism types and to systems with an additional surface interaction term