asymptotic pattern for a partial neutral functional differential equation abstract in the present article the asymptotic speed of spread and traveling wavefronts of a partial neutral functional differential equation pnfde are investigated a variable transform yields to a partial retarded functional differential equation prfde with infinite number of constant delays the relation between the above pnfde and prfde is studied and an equivalence in a certain meaning of these two equations is verified using this equivalence we in fact obtain the existence and uniqueness of solution for the initial value problem of the pnfde we then consider the truncated system of the above prfde which is a partial retarded functional differential equation with finite number of constant delays we establish the existence of the traveling wave solution and the spreading speed for this truncated system a limit argument and the method of finite delay approximation are used to obtain the existence of the traveling wave solutions and the spreading speed for the prfde with infinite number of constant delays this leads to the existence of the traveling wave solution and the spreading speed for the considered pnfde by the above equivalence at last we confirm that the spreading speed is coincidental with the minimal wave speed for all equations an example is given to illustrate the application of our results

abstract functional differential equations and reaction diffusion systems functional differential equations and reaction diffusion systems r h martin jr and h l smith abstract several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a banach space setting the ideas are applied to reaction diffusion systems that have time delays in the nonlinear reaction terms the techniques presented here include differential inequalities invariant sets and lyapunov functions and therefore they provide for a wide range of applicability the results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a banach space setting the ideas are applied to reaction diffusion systems that have time delays in the nonlinear reaction terms the techniques presented here include differential inequalities invariant sets and lyapunov functions and therefore they provide for a wide range of applicability the results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays suppose q is a bounded domain in rn with aq smooth and a is the laplacian operator on q also let m be a positive integer z a positive number and f fi m a continuous bounded function from 0 xc x q x c z 0 m into rm where c t 0 is the space of continuous functions from z 0 into r the purpose of this paper is to apply abstract results for semilinear functional differential equations in banach spaces to reactiondiffusion systems with time delays having the form a tu x t diau x t i t x ut x t a xeq i 1 m rdd ai x ui x t au i x t 81 x t t a x eaq u x a 0 x x 0 5 t 0 di 0 and ca q 0 o is c and i q x 0 oo r is c2 here an is the outward normal derivative on aq and if di 0 it is assumed that no boundary conditions are specified for this i also tu x t denotes the partial with respect to t whereas ut x denotes the member of c z 0 defined by 0 u x t 0 u x t 0 m our techniques provide basic existence criteria but the main point is that they can also be effectively applied to obtain estimates for solutions especially received by the editors october 7 1987 and in revised form october 7 1988 1980 mathematics subject classification 1985 revision primary 35r10 34k30

fuzzy group decision making for evaluating the rate if aggregative risk in software development in this paper we present a new algorithm to evaluate the rate of aggregative risk in software development using fuzzy set theory under the fuzzy group decision making environment the proposed algorithm has the following advantages 1 it does not need to form the fuzzy assessment matrices for attributes 2 it does not need to perform the complicated defuzzification operations of fuzzy numbers using the centroid method the proposed algorithm is more efficient and faster in evaluating the rate of aggregative risk in software development under the fuzzy group decision making environment

on existence of semi wavefronts for a non local reaction diffusion equation with distributed delay abstract we study the problem of existence of semi wavefront solutions for a non local delayed reaction diffusion equation with monostable nonlinearity in difference with previous works we consider non local interaction which can be asymmetric in space as a consequence of this asymmetry we must analyze the existence of expansion waves for both positive and negative speeds in the paper we use a framework of the general theory recently developed for a certain nonlinear convolution equation this approach allows us to prove the wave existence for the range of admissible speeds c r c c where the critical speeds c and c can be calculated explicitly from some associated equations the main result is then applied to several non local reaction diffusion epidemic and population models

a fuzzy approach to select the location of the distribution center abstract the location selection of distribution center dc is one of the most important decision issues for logistics managers owing to vague concept frequently represented in decision data a new multiple criteria decision making method is proposed to solve the distribution center location selection problem under fuzzy environment in the proposed method the ratings of each alternative and the weight of each criterion are described by linguistic variables which can be expressed in triangular fuzzy numbers the final evaluation value of each dc location is also expressed in a triangular fuzzy number by calculating the difference of final evaluation value between each pair of dc locations a fuzzy preference relation matrix is constructed to represent the intensity of the preferences of one plant location over another and then a stepwise ranking procedure is proposed to determine the ranking order of all candidate locations finally a numerical example is solved to illustrate the procedure of the proposed method at the end of this paper

traveling wavefronts for time delayed reaction diffusion equation ii nonlocal nonlinearity abstract in this paper we study a class of time delayed reaction diffusion equation with local nonlinearity for the birth rate for all wavefronts with the speed c c where c 0 is the critical wave speed we prove that these wavefronts are asymptotically stable when the initial perturbation around the traveling waves decays exponentially as x but the initial perturbation can be arbitrarily large in other locations this essentially improves the stability results obtained by mei so li and shen m mei j w h so m y li s s p shen asymptotic stability of traveling waves for the nicholson s blowflies equation with diffusion proc roy soc edinburgh sect a 134 2004 579 594 for the speed c 2 d m e p d m with small initial perturbation and by lin and mei c k lin m mei on travelling wavefronts of the nicholson s blowflies equations with diffusion submitted for publication for c c with sufficiently small delay time r 0 the approach adopted in this paper is the technical weighted energy method used in m mei j w h so m y li s s p shen asymptotic stability of traveling waves for the nicholson s blowflies equation with diffusion proc roy soc edinburgh sect a 134 2004 579 594 but inspired by gourley s a gourley linear stability of travelling fronts in an age structured reaction diffusion population model quart j mech appl math 58 2005 257 268 and based on the property of the critical wavefronts the weight function is carefully selected and it plays a key role in proving the stability for any c c and for an arbitrary time delay r 0

optimal allocation for the comparison of proportions optimal allocation strategies for the comparison of proportions p1 and p2 in two groups are discussed allocations which i maximize the precision in the estimation of the difference p1 p2 ii maximize the precision in the estimation of the ratio p1 p2 iii maximize the power to detect a group difference and iv minimize the total cost of a study are compared the recommended allocation varies markedly with the desired objective furthermore there is a wide range of allocation schemes which are effectively optimal with regard to each criterion considered a study of congenital malformations occurring in pregnancies of diabetic women is used to illustrate how these considerations apply to the planning of a study

spatial dynamics of a nonlocal and time delayed reaction diffusion system in this paper we investigate the spatial dynamics of a nonlocal and time delayed reaction diffusion system which is motivated by an age structured population model with distributed maturation delay the spreading speed c the existence of traveling waves with the wave speed c c and the nonexistence of traveling waves with c c are obtained it turns out that the spreading speed coincides with the minimal wave speed for monotone traveling waves

asymptotic speeds of spread and traveling waves for integral equations and delayed reaction diffusion models abstract the theory of asymptotic speeds of spread and monotone traveling waves is generalized to a large class of scalar nonlinear integral equations and is applied to some time delayed reaction and diffusion population models

spreading speeds and travelling waves for non monotone time delayed lattice equations a non monotone time delayed lattice system with global interaction is considered the spreading speed including the upward convergence is established by comparison arguments and a fluctuation met

allocation of subjects to test null relative risks smaller than one allocating a proportion k 1 1 radicalr 0 of subjects to an intervention is a practical approach to approximately maximize power for testing whether an intervention reduces relative risk of disease below a null ratio r 0 or r 0 0 33 2 3 for 0 33 or r 0 0 17 and 3 4 for 0 17 or r 0 or 0 10 both k and k s are easily calculated and invariant to alterations in disease rate estimates under null and alternative hypotheses when r 0 remains constant in examples that we studied allocating k or k s subjects to intervention achieved close to the minimum possible sample size given test size and power equivalently maximum power given test size and sample size for likelihood score tests compared to equal allocation k and k s reduced sample sizes by amounts ranging from approximately 5 5 per cent for r 0 0 50 to approximately 24 per cent for r 0 0 10 these sample size savings may be particularly important for large studies of prophylactic interventions such as vaccines while k was derived from variance minimization for an arcsine transformation we do not recommend the arcsine test since its true size exceeded the nominal value in contrast the true size for the uncorrected score test was less than the nominal size a skewness correction made the size of the score test very close to the nominal level and slightly increased power we recommend using the score test or the skewness corrected score test for planing studies designed to show a ratio of proportions is less than a prespecified null ratio r 0 1

stability of steady states and existence of travelling waves in a vector disease model in this paper a host vector model is considered for a disease without immunity in which the current density of infectious vectors is related to the number of infectious hosts at earlier times spatial spread in a region is modelled in the partial integro differential equation by a diffusion term for the general model we first study the stability of the steady states using the contracting convex sets technique when the spatial variable is one dimensional and the delay kernel assumes some special form we establish the existence of travelling wave solutions by using the linear chain trick and the geometric singular perturbation method

on the uniqueness of semi wavefronts for non local delayed reaction diffusion equations abstract we establish the uniqueness of semi wavefront solution for a non local delayed reaction diffusion equation this result is obtained by using a generalization of the diekmann kaper theory for a nonlinear convolution equation several applications to the systems of non local reaction diffusion equations with distributed time delay are also considered

subject allocation and study curtailment for fixed event comparative poisson trials comparative poisson trials of prophylactic interventions such as vaccines can be lengthy and costly we evaluate two easily implemented approaches to reduce numbers of disease cases and person years of follow up n u t for comparative poisson trials with fixed numbers of cases t i altering k the portion of n u t allocated to treatment and ii curtailed stopping before t cases if numbers of cases in the treatment or control group indicate that h 0 has already been rejected or will not be rejected at t cases normal and arcsine approximations as well as discrete exact tests are evaluated for studies not stopped early allocating about 1 1 sqrt r a of person years to treatment roughly minimizes t needed for given size and power where r a is the alternative hypothesized relative disease incidence in treated subjects used to power the study this reduces t moderately vs equal allocation k 0 5 by 2 3 cases in our examples with exact tests however the common practice of allocating k 0 5 of the person years to treatment may be the overall best strategy to minimize n u t for studies that are not stopped early for studies analysed by exact test and planned with a one sided alpha ranging from 0 005 to 0 025 beta from 0 1 to 0 2 and r a from 0 2 to 0 5 curtailed stopping reduces both the number of disease cases and n u t by 6 40 per cent depending on true treatment benefit with curtailed stopping allocating k 0 5 person years to treatment approximately minimizes the numbers of cases and person years under most conditions although k as large as 0 6 often performs comparably well if a specific localized k is selected to minimize disease cases for a study analysed by exact test the study may be underpowered should the final allocation deviate even slightly from that k when it is conducted

a new fuzzy weighted average fwa method based on left and right scores an application for determining a suitable location for a gas oil station the selection of a facility location from alternative locations is a multiple criteria decision making mcdm problem including both quantitative and qualitative criteria in many real life cases determining the exact values for mcdm problems and especially for facility location selection problems is difficult or impossible so the values of alternatives with respect to the criteria or and the values of criteria weights are considered as fuzzy values fuzzy numbers such that the conventional crisp approaches for solving facility location selection problems and other mcdm problems tend to be less effective for dealing with the imprecise or vagueness nature of the linguistic assessments in such conditions fuzzy mcdm methods are applied for facility location selection problem and other fuzzy mcdm problems in this paper we propose a new fuzzy weighted average fwa method based on left and right scores for fuzzy mcdm problems moreover we apply the proposed method to a real application as a result we found that the proposed method is practical for facility location selection problems besides it seems that the proposed fwa method is very accurate flexible simple and easy to use when compared to other versions of the fwa method

use of unequal allocation in survival trials in some two treatment clinical survival trials a large imbalance in the allocation of patients to treatments will result in approximately the same power for the logrank test as equal allocation this fact can be used in some trials to reduce significantly the number of patients allocated to a potentially inferior treatment

traveling waves in a delayed sir epidemic model with nonlinear incidence we study the traveling waves of a diffusive sir epidemic model with a general nonlinear incidence we obtain the threshold dynamics for spatial spread of the disease we show that c is the minimum wave speed for the existence of traveling wave solutions we establish the existence and non existence of traveling wave solutions for a diffusive sir model with a general nonlinear incidence the existence proof is shown by introducing an auxiliary system applying schauder s fixed point theorem and then a limiting argument the nonexistence proof is obtained by two sided laplace transform when the speed is less than the critical velocity numerical simulations support the theoretical results we also point out the effects of the delay and the diffusion rate of the infective individuals on the spreading speed

wavefronts and global stability in a time delayed population model with stage structure we formulate and study a one dimensional single species diffusive delay population model the time delay is the time taken from birth to maturity without diffusion the delay differential model extends the well known logistic differential equation by allowing delayed constant birth processes and instantaneous quadratically regulated death processes this delayed model is known to have simple global dynamics similar to that of the logistic equation through the use of a sub supersolution pair method we show that the diffusive delay model continues to generate simple global dynamics this has the important biological implication that quadratically regulated death processes dramatically simplify the growth dynamics we also consider the possibility of travelling wavefront solutions of the scalar equation for the mature population connecting the zero solution of that equation with the positive steady state our main finding here is that our fronts appear to be all monotone regardless of the size of the delay this is in sharp contrast to the frequently reported findings that delay causes a loss of monotonicity with the front developing a prominent hump in some other delay models

a reaction diffusion model for a single species with age structure i travelling wavefronts on unbounded domains in this paper we derive the equation for a single species population with two age classes and a fixed maturation period living in a spatially unbounded environment we show that if the mature deat

optimal allocation for bayesian inference about an odds ratio summary the choice of ratio of sample sizes of two groups of binary observations for comparing two proportions is studied using a bayesian approach prior uncertainty about the unknown proportions thus enters the design calculations

sample size and optimal design for logistic regression with binary interaction there is no consensus on what test to use as the basis for sample size determination and power analysis some authors advocate the wald test and some the likelihood ratio test we argue that the wald test should be used because the z score is commonly applied for regression coefficient significance testing and therefore the same statistic should be used in the power function we correct a widespread mistake on sample size determination when the variance of the maximum likelihood estimate mle is estimated at null value in our previous paper we developed a correct sample size formula for logistic regression with single exposure statist med 2007 26 18 3385 3397 in the present paper closed form formulas are derived for interaction studies with binary exposure and covariate in logistic regression the formula for the optimal control case ratio is derived such that it maximizes the power function given other parameters our sample size and power calculations with interaction can be carried out online at www dartmouth edu approximately eugened

existence and uniqueness of traveling waves for non monotone integral equations with applications abstract a class of integral equations without monotonicity is investigated it is shown that there is a spreading speed c 0 for such an integral equation and that its limiting integral equation admits a unique traveling wave up to translation with speed c c and no traveling wave with c c these results are also applied to some nonlocal reaction diffusion population models

power and sample size calculations for exact conditional tests with ordered categorical data we develop an algorithm for computing sample sizes equal or unequal for categorical data we illustrate its use in the two sample setting using the wilcoxon rank sum statistic but the algorithm accommodates the entire class of linear rank statistics and can be extended to include nonlinear rank statistics as well the sample size determinations can be based on either exact power or on a very precise monte carlo estimate of it to reduce the computations further power can be computed as a function of asymptotic critical values when the number of categories is not too small for the wilcoxon statistic we show that this approximation works well if there are more than five response categories

fuzzy group decision making for facility location selection the selection of a facility location among alternative locations is a multicriteria decision making problem including both quantitative and qualitative criteria the conventional approaches to facility location problem tend to be less effective in dealing with the imprecise or vagueness nature of the linguistic assessment under many situations the values of the qualitative criteria are often imprecisely defined for the decision makers the aim of the paper is to solve facility location problems using different solution approaches of fuzzy multi attribute group decision making the paper includes four different fuzzy multi attribute group decision making approaches the first one is a fuzzy model of group decision proposed by blin the second is the fuzzy synthetic evaluation the third is yager s weighted goals method and the last one is fuzzy analytic hierarchy process although four approaches have the same objective of selecting the best facility location alternative they come from different theoretic backgrounds and relate differently to the discipline of multi attribute group decision making these approaches are extended to select the best facility location alternative by taking into account quantitative and qualitative criteria a short comparative analysis among the approaches and a numeric example to each approach are given

a fuzzy rule based approach for screening international distribution centres in practical situations a decision maker often has to deal with numerous alternatives in evaluating and selecting the most suitable international distribution centre idc location for enhancing the organizational competitiveness to improve the effectiveness of the decision making process it is critical to reduce the number of alternatives through screening so that the decision maker can focus on the most attractive alternatives for further consideration this paper presents a fuzzy rule based approach to effectively screen decision alternatives in adequately solving the problem of evaluating and selecting idc locations case based reasoning with the use of fuzzy production rules is adopted to screen out unattractive alternatives based on past experience in a specific decision making situation a decision support system based on the proposed approach is presented to improve the effectiveness of the screening process an example is given to demonstrate the applicability of the proposed approach in improving the effectiveness of the decision making process in multicriteria decision making

deviance estimates of sample size for equivalence tests in vaccine trials this paper proposes a sample size procedure for both equivalence and conventional tests for the comparison of two binomial proportions based on the signed square root of the deviance when the comparison is based on the odds ratio i describe an alternate close conditional exact method that gives results that support those given by the deviance method i summarize the advantages of the deviance based method and also show that in general equivalence situations the sample size estimate depends upon the measure of comparison selected odds ratio risk ratio or risk difference

planning and analysis of three arm non inferiority trials with binary endpoints three arm trials including an experimental treatment an active control and a placebo group are frequently preferred for the assessment of non inferiority in contrast to two arm non inferiority studies these designs allow a direct proof of efficacy of a new treatment by comparison with placebo as a further advantage the test problem for establishing non inferiority can be formulated in such a way that rejection of the null hypothesis assures that a pre defined portion of the unknown effect the reference shows versus placebo is preserved by the treatment under investigation we present statistical methods for this study design and the situation of a binary outcome variable asymptotic test procedures are given and their actual type i error rates are calculated approximate sample size formulae are derived and their accuracy is discussed furthermore the question of optimal allocation of the total sample size is considered power properties of the testing strategy including a pre test for assay sensitivity are presented the derived methods are illustrated by application to a clinical trial in depression

minimal wave speed and spread speed in a system modelling the geographic spread of black legged tick ixodes scapularis abstract in a recent work 1 a mathematical model is derived to explore the role that the white tail deer plays in the geographic spread of the black legged tick ixodes scapularis in the northeast of the united states of america the threshold dynamics is rigorously investigated in terms of the basic reproduction number for the cases of the 1 d whole space r and general bounded spatial domain with homogeneous neumann and direchlet boundary conditions however the minimal wave speed and spread speed of the model which are the motivation of this model and thus most important are only explored numerically in the present paper we offer a rigorous theoretical confirmation of what are numerically observed in 1 concluding that if the basic reproduction number is larger than one the model allows a spread speed which is also the minimal speed of traveling wave fronts and this speed is linearly deterministic

entire solutions for nonlocal dispersal equations with spatio temporal delay monostable case abstract this paper deals with entire solutions for a general nonlocal dispersal monostable equation with spatio temporal delay i e solutions that are defined in the whole space and for all time t r we first derive a particular model for a single species and show how such systems arise from population biology then we construct some new types of entire solutions other than traveling wave solutions and equilibrium solutions of the equation under consideration with quasi monotone and non quasi monotone nonlinearities various qualitative properties of the entire solutions are also investigated in particular the relationship between the entire solutions and the traveling wave fronts which they originated is considered our main arguments are based on the comparison principle the method of super and sub solutions and the construction of auxiliary control systems

test statistics and sample size formulae for comparative binomial trials with null hypothesis of non zero risk difference or non unity relative risk when it is required to establish a materially significant difference between two treatments or alternatively to show that two treatments are equivalent standard test statistics and sample size formulae based on a null hypothesis of no difference no longer apply this paper reviews some of the test statistics and sample size formulae proposed for comparative binomial trials when the null hypothesis is of a specified non zero difference or non unity relative risk methods based on restricted maximum likelihood estimation are recommended and applied to studies of pertussis vaccine

optimal allocation to maximize the power of two sample tests for binary response summary we study allocations that maximize the power of tests of equality of two treatments having binary outcomes when a normal approximation applies the asymptotic power is maximized by minimizing the variance leading to a neyman allocation that assigns observations in proportion to the standard deviations this allocation which in general requires knowledge of the parameters of the problem is recommended in a large body of literature under contiguous alternatives the normal approximation indeed applies and in this case the neyman allocation reduces to a balanced design however when studying the power under a noncontiguous alternative a large deviations approximation is needed and the neyman allocation is no longer asymptotically optimal in the latter case the optimal allocation depends on the parameters but is rather close to a balanced design thus a balanced design is a viable option for both contiguous and noncontiguous alternatives finite sample studies show that a balanced design is indeed generally quite close to being optimal for power maximization this is good news as implementation of a balanced design does not require knowledge of the parameters

implementing optimal allocation in sequential binary response experiments for sequential experiments with k treatments we establish two formal optimization criteria to find optimal allocation strategies both criteria involve the sample sizes on each treatment and a concave noncentrality parameter from a multivariate test we show that these two criteria are equivalent we apply this result to specific questions 1 how do we maximize power of a multivariate test of homogeneity with binary response and 2 for fixed power how do we minimize expected treatment failures because the solutions depend on unknown parameters we describe a response adaptive randomization procedure that targets the optimal allocation and provides increases in power along the lines of 2 4 over complete randomization for equal allocation the increase in power contradicts the conclusions of other authors who have explored other randomization procedures for k 2 and have found that the variability induced by randomization negates any benefit of targeting an optimal allocation

quick and simple approximation of sample sizes for comparing two independent binomial distributions different sample size case given the ratio k of two sample sizes sample sizes for comparing two independent proportions can be approximated from the sample size for the equal sample size design the proposed method is compared to the method of fleiss tytun and ury 1980 biometrics 36 343 346 methods for choosing the ratio are also presented

stability analysis of a stage structure model with spatiotemporal delay effect this paper is concerned with a stage structure model with spatiotemporal delay and homogeneous dirichlet boundary condition the existence of steady state solution bifurcating from the trivial equilibrium is obtained by using lyapunovschmidt reduction the stability analysis of the positive spatially nonhomogeneous steady state solution is investigated by a detailed analysis of the characteristic equation using the properties of the omega limit set we obtain the global convergence of the solution with finite delay

a reaction diffusion lyme disease model with seasonality this paper is devoted to the study of a reaction diffusion lyme disease model with seasonality in the case of a bounded habitat we obtain a threshold result on the global stability of either disease free or endemic periodic solution in the case of an unbounded habitat we establish the existence of the disease spreading speed and its coincidence with the minimal wave speed for time periodic traveling wave solutions we also estimate parameter values based on some published data and use them to study the lyme disease transmission in port dove ontario our numerical simulations are consistent with the obtained analytic results

ranking fuzzy numbers with maximizing set and minimizing set abstract up to now these are five methods of ranking n fuzzy numbers in order but these methods contain some confusions and occasionally conflict with intuition this paper introduces the concept of maximizing set and minimizing set to decide the ordering value of each fuzzy number and uses these values to determine the order of the n fuzzy numbers in addition we give a method for calculating the ordering value of each fuzzy number with triangular trapezoidal and two sided drum like shaped membership functions

minimization of sample size when comparing two small probabilities in a non inferiority safety trial in clinical trials success rates of two treatments to be compared often range from 10 to 90 per cent when the comparison probabilities are much smaller than 10 per cent standard methods for sample size and power calculations may provide invalid results this situation may occur when there is interest in safety rather than in efficacy in such trials no more patients should be included than strictly necessary we compared the results of maximum likelihood methods for the computation of sample sizes in a non inferiority trial including exact procedures and considered unequal sample sizes for experimental and reference treatment an exact unequal sample size maximum likelihood procedure is advocated when the specified non zero risk difference under the null hypothesis is not too large such a procedure is also indicated when the parameter of interest is the relative risk rather than the risk difference

traveling waves for time delayed reaction diffusion equations with degenerate diffusion abstract this paper is concerned with time delayed reaction diffusion equations with degenerate diffusion when the term for birth rate is a nonlocal integral with a heat kernel the family of minimum wave speeds corresponding to all the degenerate diffusion coefficients is proved to admit a uniform positive infimum however when the term for birth rate is local there is no positive infimum of all the minimum wave speeds this difference indicates that the nonlocal effect plays a role as laplacian such that a positive lower bound independent of the degenerate diffusion exists for the minimum wave speeds the approach adopted for the proof is the monotone technique with the viscosity vanishing method the degeneracy of diffusion for the equation causes us essential difficulty in the proof a number of numerical simulations are also carried out at the end of the paper which further numerically confirm our theoretical results