hopf bifurcation and stability for a differential algebraic biological economic system abstract in this paper we analyze the stability and hopf bifurcation of the biological economic system based on the new normal form and the hopf bifurcation theorem the basic model we consider is owed to a ratio dependent predator prey system with harvesting compared with other researches on dynamics of predator prey population this system is described by differential algebraic equations due to economic factor here as bifurcation parameter it is found that periodic solutions arise from stable stationary states when the parameter increases close to a certain limit finally numerical simulations illustrate the effectiveness of our results

remarks on codes from hermitian curves corresp parameters and generator matrices are given for the codes obtained by applying goppa s algebraic geometric construction method to hermitian curves in pg 2 q where q 2 2s for some s in bf n automorphisms of these codes are also discussed and some results on self duality and weak self duality are given

spatiotemporal dynamics and spatial pattern in a diffusive intraguild predation model with delay effect abstract based on biological meaning a kind of diffusive intraguild predation igp resource ig prey and ig predator model with delay effect is investigated in this paper the model has holling type i functional response between resource ig prey and resource ig predator holling type ii functional response between ig prey and ig predator we first give sufficient conditions on the stability of possible nonnegative constant steady state solutions for the proposed model which give us a complete picture of the global dynamics then we investigate hopf bifurcation near the unique positive constant steady state solution by taking delay as bifurcation parameter and derive the hopf bifurcation threshold it is shown that the delay can induce three types of bistability node node bistability node cycle bistability and cycle cycle bistability periodic oscillations and irregular oscillations triggering spatiotemporal chaos in the diffusive igp model numerical simulations are performed to illustrate our theoretical results and suggest that delay can even trigger the emergence of self organised spatiotemporal patterns which evolve from spiral patterns to irregular spatial patterns via spatiotemporal hopf bifurcation in addition the impact of diffusion on the model s dynamics under certain time delay are also explored

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

dynamic analysis of a hybrid bioeconomic plankton system with double time delays and stochastic fluctuations abstract in this paper we establish a double delayed hybrid bioeconomic plankton system with stochastic fluctuations and commercial harvesting on zooplankton where maturation delay for toxin producing phytoplankton and gestation delay for zooplankton are considered stochastic fluctuations are incorporated into the proposed system in form of gaussian white noises to depict stochastic environmental factors in plankton system for deterministic system without double time delays existence of singularity induced bifurcation is studied due to variation of economic interest of commercial harvesting and state feedback controllers are designed to eliminate singularity induced bifurcation for deterministic system with double time delays positivity and uniform persistence of solutions are studied and some sufficient conditions associated with asymptotic stability of interior equilibrium are investigated for stochastic system without double time delays stochastic stability and existence of stochastic hopf bifurcation are discussed based on singular boundary theory of diffusion process and invariant measure theory for stochastic system with double time delays existence and uniqueness of global positive solution are investigated and asymptotic behaviors of the interior equilibrium are studied by constructing appropriate lyapunov functions numerical simulations are carried out to validate theoretical analysis

global stability for a class of predator prey systems this paper deals with the question of global stability of the positive locally asymptotically stable equilibrium in a class of predator prey systems the dulac s criterion is applied and liapunov functions are constructed to establish the global stability

optimal control of harvest and bifurcation of a prey predator model with stage structure abstract this paper describes a prey predator model with stage structure for prey the adult prey and predator populations are harvested in the proposed system the dynamic behavior of the model system is discussed it is observed that singularity induced bifurcation phenomenon is appeared when variation of the economic interest of harvesting is taken into account state feedback controller is incorporated to stabilize the model system in case of positive economic interest harvesting of prey and predator population are used as controls to develop a dynamic framework to investigate the optimal utilization of the resource sustainability properties of the stock and the resource rent earned from the resource the pontryagin s maximum principle is used to characterize the optimal controls the optimality system is derived and then solved numerically using an iterative method with runge kutta fourth order scheme simulation results show that the optimal control scheme can achieve sustainable ecosystem

pattern formation of a spatial predator prey system abstract there are random and directed movements of predator and prey populations in many natural systems which are strongly influenced and modified by spatial factors to investigate how these migration directed movement and diffusion random movement affect predator prey systems we have studied the spatiotemporal complexity in a predator prey system with holling tanner form a theoretical analysis of emerging spatial pattern is presented and wavelength and pattern speed are calculated at the same time we present the properties of pattern solutions the results of numerical simulations show that migration has prominent effect on the pattern formation of the population i e changing turing pattern into traveling pattern this study suggests that modelling by migration and diffusion in predator prey systems can account for the dynamical complexity of ecosystems

numerical study of an influenza epidemic model with diffusion abstract a diffusive epidemic model is investigated with a view to describe the transmission of influenza as an epidemic the equations are solved numerically using the splitting method under different initial distribution of population density it is shown that the initial population distribution and diffusion play an important role for spread of disease it is also shown that interventions medical and nonmedical significantly slow down the spread of disease stability of equilibria of the numerical solutions are also established

operator norm attainment and birkhoff james orthogonality abstract recently in sain d paul k operator norm attainment and inner product spaces linear algebra appl 439 2013 2448 2452 it was proved that if t is a linear operator on a finite dimensional real normed linear space x such that t attains norm only on d where d is a connected closed subset of s x then t satisfies the bhatia semrl bs property bhatia r semrl p orthogonality of matrices and distance problem linear algebra appl 287 1999 77 85 i e for a l x t b a implies that there exists x d such that t x b a x here we explore the converse of the above result we prove that in a real normed linear space x of dimension 2 a linear operator t satisfies the bs property if and only if the set of unit vectors on which t attains norm is connected in the corresponding projective space r p 1 s x x x motivated by the result in 2 dimensions we conjecture that this characterization of bs property is true in general n dimensional real normed linear spaces we further prove that if the space x is strictly convex then the set of operators in l x which satisfy the bs property is dense in l x

hopf bifurcation and spatial patterns of a delayed biological economic system with diffusion in this paper a delayed biological economic system which considers a plankton system with harvest effort on phytoplankton is proposed by using the theory of partial functional differential equations hopf bifurcation of the proposed system with delay as the bifurcation parameter is investigated it reveals that the discrete time delay has a destabilizing effect in the plankton dynamics and a phenomenon of hopf bifurcation occurs as the delay increases through a certain threshold then by numerical simulations the impact of delay diffusion and economic interest on plankton system are explored it is found that delay can cause system into chaos and can trigger the emergence of irregular spatial patterns via a hopf bifurcation moreover diffusion and economic profit can also affect the dynamic behavior of the system

stationary patterns of the holling tanner prey predator model with diffusion and cross diffusion this paper is concerned with the holling tanner prey predator model subject to the homogeneous neumann boundary condition we will show that under certain hypotheses even though the unique positive constant steady state is uniformly asymptotically stable for the dynamics with diffusion non constant positive steady states can exist due to the emergence of cross diffusion the biological implication of cross diffusion means that the prey species exercise a self defense mechanism to protect themselves from the attack of the predator our result also demonstrates that cross diffusion can create stationary patterns

numerical ranges of the powers of an operator abstract the numerical range w a of a bounded linear operator a on a hilbert space is the collection of complex numbers of the form a v v with v ranging over the unit vectors in the hilbert space in terms of the location of w a inclusion regions are obtained for w a k for positive integers k and also for negative integers k if a 1 exists related inequalities on the numerical radius w a sup w a and the crawford number c a inf w a are deduced

orthogonality of matrices and some distance problems abstract if a and b are matrices such that a zb a for all complex numbers z then a is said to be orthogonal to b we find necessary and sufficient conditions for this to be the case some applications and generalisations are also discussed

cyclotomic goppa codes the general goppa code construction is applied to certain so called cyclotomic fields of algebraic functions the resulting codes are metabelian group codes with binary subfield subcodes of minimum distance d or approximately 1 r n log sub 2 log sub 2 n for a given rate r and length n approaching infinity

orthogonality of matrices abstract let x be a real finite dimensional normed space with unit sphere sx and let l x be the space of linear operators from x into itself it is proved that x is an inner product space if and only if for a c l x a c u s x a au au cu where denotes birkhoff orthogonality

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

modeling and dynamical analysis of a triple delayed prey predator scavenger system with levy jumps abstract in this paper a hybrid triple delayed prey predator scavenger system with commercial harvesting and levy jumps is established where both maturation delay for prey population and gestation delays for predator and scavenger population are considered for deterministic system positivity and uniform permanence of solution are discussed local stability analysis of deterministic system around interior equilibrium is investigated due to variations of triple time delays for stochastic system without time delay sufficient conditions for stochastically ultimate boundedness and stochastic permanence are discussed for stochastic system with triple time delays existence and uniqueness of global positive solution are studied by constructing appropriate lyapunov functions asymptotic behavior of the hybrid stochastic system with triple time delays and levy jumps is discussed numerical simulations are supported to illustrate theoretical results

optimal control of a nutrient phytoplankton zooplankton fish system we consider the mathematical formulation analysis and numerical solution of an optimal control problem for a nonlinear nutrient phytoplankton zooplankton fish reaction diffusion system we study the existence of optimal solutions derive an optimality system and determine optimal solutions in the original spatially homogeneous formulation m scheffer oikos 62 1991 pp 271 282 the dynamics of plankton were investigated as a function of parameters for nutrient levels and fish predation rate on zooplankton in our paper the model is spatially extended and the parameter for fish predation treated as a multiplicative control variable the model has implications for the biomanipulation of food webs in eutrophic lakes to help improve water quality in order to illustrate the control of irregular spatiotemporal dynamics of plankton in the model we implement a semi implicit in time finite element method with mass lumping and present the results of numerical experiments in two space dimensions

stationary patterns of a predator prey model with spatial effect abstract in this paper spatial patterns of predator prey model with cross diffusion are investigated the hopf and turing bifurcation critical line in a spatial domain are obtained by using mathematical theory moreover exact turing space is given in two parameters space our results reveal that cross diffusion can induce stationary patterns which may be useful to help us better understand the dynamics of the real ecosystems

the bifurcation structure of the holling tanner model for predator prey interactions using two timing the holling tanner model for predator prey systems has two hopf bifurcation points for certain parameters the dependence of the environmental parameters on the underlying bifurcation structure is uncovered using two timing emphasis is on how the bifurcation diagram changes as the hopf bifurcation points separate two degenerate cases require a modification of conventional two timing when the two hopf bifurcation points nearly coalesce the two stable periodic solution branches are shown to be connected as a ratio of linear growth rates varies the hopf bifurcation points separate further and one limit cycle becomes unstable this situation can correspond to an outbreak in populations the modified two timing analysis analytically captures the unstable and stable limit cycles of the new branch

ecological chaos and the choice of optimal harvesting policy abstract harvested populations fluctuate because of two primary reasons the inherent nonlinearity contained in the interactions among the constituent species and the forces of harvesting acting on the oscillatory dynamics resulting from species interactions during the course of these fluctuations population densities make excursions to low densities when the ecological system executes chaotic motion extinction sized densities are common thus it is imperative to design harvesting strategies which aim at maximizing economic gains giving due consideration to the ecological health of the concerned ecological system the present study was designed and performed to figure out how to set harvesting strategies which optimize the economic gain the choice of optimal harvesting policy can be made only if dynamical features of the concerned ecological system are well understood in this paper we have consider the temporal and spatiotemporal interactions among phytoplankton zooplankton and fish population with holling type ii and holling type iii functional responses we have calculated stability analysis of the model system and performed the numerical simulations for both non spatial and spatial models to figure out the parameters that are responsible for chaotic dynamics of the model system the temporal model system shows rich dynamics including limit cycles and chaos whereas spatial model shows different types of patterns for population distribution in this work we have taken the case study of sundarban wetland ecosystem we have carried out the analysis of maximum sustainable yield and identified the parameters that are responsible for good health of wetland ecosystem through numerical simulation results

control of chaos due to additional predator in the hastings powell food chain model abstract a three species hastings and powell hp food chain model involving another predator of top prey is proposed and studied the modified food web model is analyzed to obtain the different conditions for which system exhibits stability around the biological feasible equilibrium points the permanence is established and global stability of boundary equilibrium point e x is discussed it is observed through numerical simulations that four dimensional model may show stable dynamics in contrast to chaotic dynamics that occurred in three species food chain varieties of dynamical behaviors in the food web are possible depending upon the sharing of food between the two predators of the top prey the results demonstrate that the additional predator play the crucial role in reducing the complexity in the dynamical behavior of the system

controlling bifurcation in a delayed fractional predator prey system with incommensurate orders this paper investigates an issue of bifurcation control for a novel incommensurate fractional order predator prey system with time delay firstly the associated characteristic equation is analyzed by taking time delay as the bifurcation parameter and the conditions of creation for hopf bifurcation are established it is demonstrated that time delay can heavily effect the dynamics of the proposed system and each order has a major influence on the creation of bifurcation simultaneously then a linear delayed feedback controller is introduced to successfully control the hopf bifurcation for such system it is shown that the control effort is markedly influenced by feedback gain it is further found that the onset of the bifurcation can be delayed as feedback gain decreases finally two illustrative examples are exploited to verify the validity of the obtained newly results

on the parameters of goppa codes a proof of v d goppa s 1983 lower bound to the dimension of subfield subcodes of his geometric codes is given a result on the minimum distance of these subfield subcodes is also given that generalizes the well known bound minimum distance of gamma l g or 2 deg g x 1 for classical goppa codes gamma l g over the field f sub 2 with a square free goppa polynomial g g x

relative numerical ranges abstract relying on the ideas of stampfli 14 and magajna 12 we introduce for operators s and t on a separable complex hilbert space a new notion called the numerical range of s relative to t at r t some properties of these numerical ranges are proved in particular it is shown that the relative numerical ranges are non empty convex subsets of the closure of the ordinary numerical range of s we show that the position of zero with respect to the relative numerical range of s relative to t at t gives an information about the distance between the involved operators this result has many interesting corollaries for instance one can characterize those complex numbers which are in the closure of the numerical range of s but are not in the spectrum of s