a stage structured prey predator model with discrete time delay abstract a stage structured predator prey model stage structure on prey with gestation delay has been presented in this paper linear stability analysis reveals that certain thresholds on the rate at which juveniles becomes adults in terms of the predation rate and other system parameters a stability switch in the system due to variation of delay parameter has been observed so is the phenomena of hopf bifurcation and stable limit cycle the estimation of the length of delay to preserve stability has also been calculated numerical simulation with a hypothetical set of data has been done to support the analytical findings

global stability of gompertz model of three competing populations the model of three competitive populations with gompertz growth is studied the periodic solutions are ruled out by generalized dulac criteria on the basis of the analysis we obtain conditions that ensure the asymptotic behavior of the model is simple

periodic solutions of a predator prey model with stage structure for predator a delayed periodic predator prey model with stage structure for predator is proposed it is assumed that immature individuals and mature individuals of the predator are divided by a fixed age and that immature predators do not have the ability to attack prey sufficient conditions are derived for the existence uniqueness and global asymptotic stability of positive periodic solutions of the model numerical simulations are presented to illustrate the feasibility of our main results

modeling the dynamics of stage structure predator prey system with monod haldane type response function a stage structure predator prey model is proposed and analyzed in this paper in which predators are divided into juvenile and mature predators using monod haldane type response function the dynamical behavior of this system both analytically and numerically is investigated from the view point of stability and bifurcation we investigate global stability around the interior equilibrium point e by constructing suitable lyapunov function our model simulation indicates that the conversion of prey population to juvenile predators can destabilize the model system which lead to limit cycle oscillations we also investigate that the rate of juvenile predators becoming mature predators play an important role to destabilize the model system for the stable coexistence of both the populations we carried out extensive numerical simulations of the model to confirm the analytical findings

global attractivity of a positive periodic solution for a nonautonomous stage structured population dynamics with time delay and diffusion by employing the continuation theorem of coincidence degree theory the existence of a positive periodic solution for a nonautonomous stage structured population dynamics with time delay and diffusion is established further by constructing a lyapunov functional and using the result of the existence of positive periodic solution the attractivity of a positive periodic solution for above system is obtained

harmless delays for uniform persistence abstract this paper studies a two dimensional predator prey system with a finite number of discrete delays our purpose is to demonstrate that the time delays are harmless for uniform persistence of the solutions of the system the results are obtained by constructing suitable persistence functionals

deconstructing an integrable lattice equation we show that an integrable lattice equation obtained by j hietarinta using the consistency around a cube method without the tetrahedron assumption is indeed solvable by linearization we also present its nonautonomous extension

periodic solutions of nonautonomous stage structured cooperative system abstract by using the continuation theorem of coincidence degree theory the existence of a positive periodic solution for the nonautonomous stage structured cooperative system is established

persistence and stability of a stage structured predator prey model with time delays a delayed predator prey model with stage structure for the predator is proposed and investigated sufficient conditions are derived for persistence the local and global asymptotic stability of a positive equilibrium of the model numerical simulations are presented to illustrate the feasibility of our main results

global convergence of a reaction diffusion predator prey model with stage structure and nonlocal delays in this paper a lotka volterra type reaction diffusion predator prey model with stage structure for the prey and nonlocal delays due to gestation of the predator is investigated in the case of a general domain sufficient conditions are obtained for the global convergence of positive solutions of the proposed problem by using the energy function method numerical simulations are carried out to illustrate the main results

a reaction diffusion predator prey model with stage structure and nonlocal delay abstract in this paper a reaction diffusion predator prey model with nonlocal delay due to gestation of the predator and stage structure for the prey is investigated in the case of a general domain sufficient conditions are derived for the global convergence of positive solutions to the proposed problem by using the energy function method numerical simulations are carried out to illustrate the main results

a new stage structured predator prey gomportz model with time delay and impulsive perturbations on the prey abstract in this paper we formulate a robust prey dependent consumption predator prey gomportz model with periodic harvesting catching or poisoning for the prey and stage structure for the predator with constant maturation time delay through stage time delay and perform a systematic mathematical and ecological study by use of the discrete dynamical system determined by the stroboscopic map we obtain a predator extinction periodic solution further we show it is globally attractive when some parameters of the system are under appropriate conditions by using the theory on delay functional and impulsive differential equation we obtain sufficient condition with time delay for the permanence of the system in this paper the main feature is that we introduce time delay and pulse into the predator prey natural enemy pest model with stage structure exhibit a new modeling method which is applied to investigate delay impulsive differential equations and give some reasonable suggestions for pest management

stability and bifurcation analysis of a stage structured predator prey model with time delay abstract in this paper we proposed and analyzed a prey predator system with stage structured for the predator population a time delay is incorporated due to the gestation for the matured predator all the possible non negative equilibria are obtained and their local as well as global behavior are studied choosing delay as a bifurcation parameter the existence of the hopf bifurcation of the system has been investigated moreover we use the normal form method and the center manifold theorem to examine the direction of the hopf bifurcation and the nature of the bifurcating periodic solution some numerical simulations are given to support the analytical results some interesting conclusions are obtained from our analysis and it is given at the end of the paper

a lotka volterra type food chain model with stage structure and time delays abstract a three species lotka volterra type food chain model with stage structure and time delays is investigated it is assumed in the model that the individuals in each species may belong to one of two classes the immatures and the matures the age to maturity is presented by a time delay and that the immature predators immature top predators do not have the ability to feed on prey predator by using some comparison arguments we first discuss the permanence of the model by means of an iterative technique a set of easily verifiable sufficient conditions are established for the global attractivity of the nonnegative equilibria of the model

permanence and stability of a stage structured predator prey model abstract a predator prey model with a stage structure for the predator which improves the assumption that each individual predator has the same ability to capture prey is proposed it is assumed that immature individuals and mature individuals of the predator are divided by a fixed age and that immature predators do not have the ability to attack prey we obtain conditions that determine the permanence of the populations and the extinction of the populations furthermore we establish necessary and sufficient conditions for the local stability of the positive equilibrium of the model by exploiting the monotonicity of one equation of the model we obtain conditions for the global attractivity of the positive equilibrium which allow for long delay as long as the predator birth rate is large or the death rate of immature predators is small by constructing liapunov functionals we also obtain conditions under which the positive equilibrium is globally stable when the delay is small

optimal harvesting policy for single population with stage structure it is the purpose of this paper to analyze the exploitation of the stage structured singleautonomous population model we analyze the harvesting population equation and obtain existence local and global geometric properties of nonnegative equilibria explicit expressions are obtained for the optimal harvesting effort the maximum sustainable yield and the corresponding population level

global stability of a lotka volterra type predator prey model with stage structure and time delay a delayed lotka volterra type predator prey model with stage structure for predator is investigated it is assumed in the model that the individuals in the predator population may belong to one of two classes the immatures and the matures the age to maturity is presented by a time delay and that the immature predators do not have the ability to prey by analyzing characteristic equations and using an iterative technique a set of easily verifiable sufficient conditions are derived for the local and global stability of the nonnegative equilibria of the model numerical simulations are carried out to illustrate the validity of our results

asymptotic behaviors of competitive lotka volterra system with stage structure in this paper a two species competitive model with stage structure is presented and studied results on the global extinction and permanence are given which generalize the well known three theorems for the two species competitive system and moreover they confirm the negative effect of stage structure on the permanence of populations as well as estimate the degree of such effect conclusions in this paper suggest that for a competitive community stage structure is also one of the important reasons that cause permanence and extinction

the role of space in stage structured cannibalism with harvesting of an adult predator the dynamics of a predator prey reaction diffusion system with holling type iii functional response where the predator has two stages a juvenile stage and an adult stage is proposed and analyzed to study the effect of predation with non constant harvesting of an adult predator our analysis leads to different thresholds in terms of the model parameters acting as conditions under which the organisms associated with the system cannot thrive even in the absence of predation local stability of the system is obtained in the absence of one or more organisms and in the presence of all the organisms moreover it is shown that the system undergoes hopf bifurcation when the intrinsic growth rate of herbivorous prey crosses certain critical value computer simulations have been carried out to illustrate various analytical results

a delayed stage structured predator prey model with impulsive stocking on prey and continuous harvesting on predator abstract in this work we investigate a delayed stage structured holling ii predator prey model with impulsive stocking on prey and continuous harvesting on predator sufficient conditions of the global attractivity of predator extinction periodic solution and the permanence of the system are obtained these results show that the behavior of impulsive stocking on prey plays an important role for the permanence of the system we also prove that all solutions of the system are uniformly ultimately bounded our results provide reliable tactical basis for the biological resource management

effect of predator cannibalism and prey growth on the dynamic behavior for a predator stage structured population model with diffusion abstract in this paper a predator prey model with predator stage structured and diffusion is concerned we deal with the system by endowing it with the homogeneous neumann boundary conditions we first give the a priori estimates of positive solutions for the reduced reaction diffusion system secondly we discuss the effects of predator cannibalism and prey growth on the stability of nonnegative constant steady states of the model in detail thirdly we investigate the nonexistence and existence of nonconstant positive solutions finally we discuss the hopf bifurcation created by diffusion

stability analysis of a sis model with stage structured and distributed maturation delay abstract we consider a sis model with stage structure in which the maturation delay is modeled as a distribution to allow for the possibility that individuals may take different amounts of time to mature general birth function and incidence rate are used the basic reproductive number r 0 to determine whether the disease is extinct or not is found furthermore we obtain sufficient conditions of globally asymptotical stability for the disease free equilibrium and the endemic equilibrium

the effect of stage structure on the permanence of a predator prey system with time delay abstract a delayed predator prey system with stage structure for the prey and time delay due to the gestation of the predator is investigated by using an iteration technique and comparison arguments respectively sufficient conditions are derived for the global stability of the positive equilibrium and the two boundary equilibria of the proposed model numerical simulations are presented to illustrate the main results

hopf bifurcation and global stability for a delayed predator prey system with stage structure for predator abstract in this paper a delayed predator prey system with stage structure for predator is proposed and studied it is found that the time delay is harmless for permanence of the stage structured system if 1 we show that the unique positive equilibrium is locally asymptotically stable when time delay is sufficiently small while loss of stability by a hopf bifurcation can occur as the delay increases

periodic solutions of a nonautonomous predator prey system with stage structure and time delays a nonautonomous lotka volterra type predator prey model with stage structure and time delays is investigated it is assumed in the model that the individuals in each species may belong to one of two classes the immatures and the matures the age to maturity is presented by a time delay and that the immature predators do not feed on prey and do not have the ability to reproduce by some comparison arguments we first discuss the permanence of the model by using the continuation theorem of coincidence degree theory sufficient conditions are derived for the existence of positive periodic solutions to the model by means of a suitable lyapunov functional sufficient conditions are obtained for the uniqueness and global stability of the positive periodic solutions to the model

staged structured lotka volterra predator prey models for pest management abstract in this paper two predator prey models with stage structure are constructed and investigated in the first model continuous biological control is taken the existence and local stability of two equilibriums are studied by the liapunov stability theorem we obtain the condition for the global asymptotical stability of the trivial equilibrium i e pest eradication equilibrium in the second model impulsive biological control is taken by use of the floquet s theorem small amplitude perturbation method and comparison techniques we get the condition which guarantees the global asymptotical stability of the pest eradication periodic solution the sufficient condition for the permanence of the impulsive system is also obtained

persistence and periodicity of a delayed ratio dependent predator prey model with stage structure and prey dispersal a delayed periodic ratio dependent predator prey model with prey dispersal and stage structure for predator is investigated it is assumed that immature individuals and mature individuals of the predator species are divided by a fixed age and that immature predators don t have the ability to attack prey and that predator species is confined to one of the patches while the prey species can disperse between two patches we first discuss the uniform persistence and impermanence of the model by using gaines and mawhin s continuation theorem of coincidence degree theory a set of easily verifiable sufficient conditions are derived for the existence of positive periodic solutions of the proposed model numerical simulations are presented to illustrate the feasibility of our main results

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

permanence of variable coefficients predator prey system with stage structure abstract we studied a finite delay predator prey model with stage structure for predator by analyzing right hand of function and the standard comparison theorem some new sufficient conditions are derived for the permanence of population and some biological explanations are made

permanence of a predator prey system with stage structure and time delay abstract a stage structured predator prey system stage structure for both predator and prey with discrete time delay has been presented in this paper by analyzing the model and using the standard comparison theorem the sufficient conditions are derived for permanence of the system and non permanence of predators

global stability for a delayed hiv 1 infection model with nonlinear incidence of infection abstract in this paper a delayed hiv 1 infection model with nonlinear incidence of infection is reinvestigated it is shown that if the reproduction number r 1 then the system is permanent and the infective equilibrium of the system is globally asymptotically stable thus the global dynamics of the system is completely determined by the reproduction number r the results obtained enrich and improve the corresponding results given by wang et al x wang y tao x song a delayed hiv 1 infection model with beddington deangelis functional response nonlinear dynamics 62 2010 67 72 the conclusions we established also verify the numerical simulation results on the global asymptotic stability of the infective equilibrium in the paper d li w ma asymptotic properties of an hiv 1 infection model with time delay j math anal appl 335 2007 683 691

permanence and extinction of a periodic predator prey delay system with functional response and stage structure for prey abstract in this paper we consider a periodic coefficients predator prey system with functional response and infinite delay in which the prey has a history that takes them through two stages immature and mature sufficient conditions which guarantee the permanence and extinction of the system are obtained finally we give an example

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

hopf bifurcation analysis in a diffusive food chain model with time delay in this paper we consider the direction and stability of time delay induced hopf bifurcation in a three species food chain model with diffusion by means of analyzing eigenvalue spectrum and lyapunov functional we show that the positive equilibrium is asymptotically stable in the absence of time delay but loses its stability via the hopf bifurcation when the time delay increases beyond a threshold using the norm form and the center manifold theory we investigate the stability and direction of the hopf bifurcation the instability of the hopf bifurcation leads to the emergence of spatial patterns numerical calculations are performed to illustrate our theoretical results

dynamic behavior of a beddington deangelis type stage structured predator prey model abstract this paper deals with a robust stage structured predator prey model with beddington deangelis type functional response the proposed mathematical model consists of a system of three nonlinear ordinary differential equations to stimulate the interactions between prey population juvenile predators and adult predator population the positivity boundedness and the conditions for uniform persistence have been derived the dynamical behavior of the system both analytically and numerically investigated from the point of view of local stability persistence and global stability the global stability of the system has been derived by using the theory of competitive systems stability of periodic orbits and compound matrices for the interior equilibrium point depending on the conversion rate of the prey population to juvenile predator the model exhibits hopf bifurcation the model admit periodic solutions which is produced from the stage structure of the predator populations numerical simulations have been accomplished to validate our analytical findings

global stability of a stage structured predator prey model with prey dispersal a delayed lotka volterra type predator prey model with stage structure for predator and prey dispersal in two patch environments is investigated it is assumed that immature individuals and mature individuals of predator species are divided by a fixed age and that immature predators do not feed on prey and do not have the ability to reproduce on the other hand it is assumed that the prey species can disperse between one patch with a low level of food and without predation and one patch with a higher level of food but with predation by means of two different kinds of lyapunov functionals sufficient conditions are derived respectively for the global asymptotic stability of a positive equilibrium of the model by analyzing the characteristic equation criterion is established for the local stability of the positive equilibrium numerical simulations are presented to illustrate our main results

dynamics behaviors of a delayed stage structured predator prey model with impulsive effect in this paper we analyzed a delayed stage structured predator prey model with impulsive stocking on prey and continuous harvesting on predator sufficient conditions for the global attractivity of predator eradication periodic solution and permanence of the system are obtained the results show that the behavior of impulsive stocking on prey plays an important role for the permanence of the system

analysis of a model representing stage structured population growth with state dependent time delay a stage structured model of population growth is proposed where the time to ma turity is itself state dependent it is shown that under appropriate assumptions all solutions are positive and bounded criteria for the existence of positive equilibria and further conditions for the uniqueness of the equilibria are given the stability of the equilibria are also discussed in addition an attracting region is determined for solutions such that this region collapses to the unique positive equilibrium in the state independent case

stable periodic orbits for a class of three dimensional competitive systems abstract it is shown that for a dissipative three dimensional competitive and irreducible system of ordinary differential equations having a unique equilibrium point at which point the jacobian matrix has negative determinant either the equilibrium point is stable or there exists an orbitally stable periodic orbit if in addition the system is analytic then there exists an orbitally asymptotically stable periodic orbit when the equilibrium is unstable the additional assumption of analyticity can be replaced by the assumption that the equilibrium point and every periodic orbit are hyperbolic in this case the morse smale conditions hold and the flow is structurally stable

domain theoretic models of polymorphism abstract we give an illustration of a construction useful in producing and describing models of girard and reynolds polymorphic calculus the key unifying ideas are that of a grothendieck fibration and the category of continuous sections associated with it constructions used in indexed category theory the universal types of the calculus are interpreted as the category of continuous sections of the fibration as a major example a new model for the polymorphic calculus is presented in it a type is interpreted as a scott domain in fact understanding universal types of the polymorphic calculus as categories of continuous sections appears to be useful generally for example the technique also applies to the finitary projection model of bruce and longo and a recent model of girard indeed the work here was inspired by girard s and arose through trying to extend the construction of his model to scott domains it is hoped that by pin pointing a key construction this paper will help towards a deeper understanding of models for the polymorphic calculus and the relations between them

a predator prey system with stage structure for predator abstract this paper studies the asymptotoc behavior of a predator prey model with stage structure it is found that an orbitally asymptotically stable periodic orbit exists in that model when time delay due to gestation of predator and time delay from crowding effect of prey are incorporated we establish the condition for the permanence of populations and sufficient conditions under which positive equilibrium of the model is globally stable

qualitative analysis of a prey predator model with stage structure for the predator in this paper we propose a diffusive prey predator model with stage structure for the predator we first analyze the stability of the nonnegative steady states for the reduced ode system and then study the same question for the corresponding reaction diffusion system with homogeneous neumann boundary conditions we find that a hopf bifurcation occurs in the ode system but no turing pattern happens in the reaction diffusion system however when a natural cross diffusion term is included in the model we can prove the emergence of stationary patterns i e nonconstant positive stationary solutions for this system moreover these stationary patterns do not exist in the considered parameter regime when there is no cross diffusion

stability and delays in a predator prey system abstract sufficient conditions for both local and global stability of the positive equilibrium in a predator prey system with time delays are obtained by constructing suitable lyapunov functionals a remark on the mistake by k n murty and m a s srinivas j math anal appl 158 1991 333 341 is also included

almost periodic solution of the non autonomous two species competitive model with stage structure abstract sufficient conditions are obtained for the existence of a unique globally attractive strictly positive componentwise almost periodic solution of a non autonomous almost periodic competitive two species model with a stage structure in one species an example together with its numeric simulations shows the feasibility of our main results our result generalizes the main results of zeng et al g z zeng l s chen l h sun y liu permanence and the existence of the periodic solution of the non autonomous two species competitive model with stage structure adv complex syst 7 3 4 2004 385 393

global dynamics of a predator prey model with stage structure for the predator the global properties of a predator prey model with nonlinear functional response and stage structure for the predator are studied using lyapunov functions and lasalle s invariance principle it is found that under hypotheses which ensure the uniform persistence of the system and the existence of a unique positive steady state a feasible a priori lower bound condition on the abundance of the prey population ensures the global asymptotic stability of the positive steady state a condition which leads to the extinction of the predators is indicated we also obtain results on the existence and stability of periodic solutions in particular when 4 2 fails to hold and the unique positive steady state e becomes unstable the coexistence of prey and predator populations is ensured for initial populations not on the one dimensional stable manifold of e albeit with fluctuating population sizes

stability and hopf bifurcation analysis on a delayed leslie gower predator prey system incorporating a prey refuge in this paper a modified leslie gower predator prey system with time delays is investigated where the time delays are regarded as bifurcation parameters by analyzing the corresponding characteristic equation the local stability of a positive equilibrium is considered moreover we show that hopf bifurcations occur when time delay crosses some critical values by deriving the equation describing the flow on the center manifold we give the formula for determining the direction of the hopf bifurcation and the stability of bifurcating periodic solutions in addition we also try on the global existence of periodic solutions by using the global hopf bifurcation result of wu j wu symmetric functional differential equations and neural networks with memory trans amer math soc 350 1998 4799 4838 for functional differential equations numerical simulations are carried out to illustrate the theoretical results and they show that the time delays in the system under consideration can destroy the stability of the system

dynamics of a density dependent stage structured predator prey system with beddington deangelis functional response abstract in this paper we study the dynamics of a stage structured predator prey system with beddington deangelis functional response where not only the prey density dependence but also the predator density dependence are considered such that the studied predator prey system conforms to the realistically biological environment first we derive a sufficient and necessary condition for the existence of a unique positive equilibrium by analyzing the corresponding locations of hyperbolic curves then we provide a sufficient condition to assure the local asymptotic stability of this positive equilibrium by constructing a lyapunov function afterward by iteratively making use of the comparison theorem we propose a sufficient condition to assure its global attractiveness finally by investigating types of the limit set instead of making use of the persistence theory we prove that the predator coexists with the prey permanently if and only if positive equilibria exist

regularization of a backward problem for a lotka volterra competition system abstract lotka volterra systems are used to describe the dynamics of biological systems we study the backward problem for the lotka volterra system to determine the population density of species at preceding times the problem is ill posed in the sense that if the solution exists it does not depend continuously on the given data we propose two stable regularization methods to regularize the system furthermore under some a priori assumptions on the sought solution we show that the corresponding regularized solutions converge to the exact solutions in l 2 norm numerical simulations are presented to demonstrate the theoretical results

qualitative analysis of an sir epidemic model with stage structure abstract an sir epidemic model with stage structure consisting immature and mature stages is investigated the global stability of epidemic equilibrium and disease free equilibrium is proved meanwhile we obtained that global dynamics is completely determined by the basic reproduction number if 1 the disease free equilibrium is globally stable and the disease dies out if 1 a unique endemic equilibrium is globally stable in the interior of a feasible region and the disease persists at endemic equilibrium

extinction and permanence in nonautonomous competitive system with stage structure in this paper the n species nonautonomous stage structured competitive system is constructed and considered sufficient conditions for its extinction and permanence are obtained results here generalize and unify some previous ones moreover it is concluded that stage structure in this system is one of the important factors that effect the extinction and permanence of species

monotone semiflows generated by functional differential equations abstract this obtains sufficient condition for an autonomous functional differential equation to generate a strongly monotone semiflow on a suitable state space this allows the application to functional differential equations of very powerful recent results on strongly monotone semiflows due to hirsch and matano in addition a very striking relationship is established between such functional differential equations and corresponding ordinary differential equations an example involving a biochemical feedback loop is considered keywords stability steady states

permanence and stability of a predator prey system with stage structure for predator a stage structured predator prey system with delays for prey and predator respectively is proposed and analyzed mathematical analysis of the model equations with regard to boundedness of solutions permanence and stability are analyzed some sufficient conditions which guarantee the permanence of the system and the global asymptotic stability of the boundary and positive equilibrium respectively are obtained

persistence and global stability of a ratio dependent predator prey model with stage structure a ratio dependent predator prey model with stage structure for prey is investigated first sufficient conditions are derived for the uniform persistence and impermanence of the model next by constructing appropriate lyapunov functions a set of easily verifiable sufficient conditions are obtained for the global asymptotic stability of nonnegative equilibria of the model numerical simulations are presented to illustrate the validity of our main 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

cross diffusion induced stationary patterns in a prey predator system with parental care for predators abstract this paper is concerned about a strongly coupled nonlinear reaction diffusion system which describes a prey predator model with parental care for predators the long time behaviors of the solution are discussed stability and instability of the positive constant equilibrium are studied our results show that even though the unique positive constant steady state is stable for the kinetic system and for the self diffusion reaction system cross diffusion can generate the stationary patterns nonconstant positive steady states the numerical simulation is also given to illustrate the theoretical result

global stability in delay differential systems without dominating instantaneous negative feedbacks abstract most of the global stability or convergence results appearing so far for delayed lotka volterra type systems or other delay differential systems require that the instantaneous negative feedbacks dominate both delayed feedback and interspecific interactions such a requirement is rarely met in real systems since feedbacks are generally delayed this leads to the long standing question under what conditions will the global stability of a nonnegative steady state of a differential delay system persist when time delays involved in some part of the negative feedbacks are small enough this paper presents a partial answer to this open question roughly our results show that if the delay differential system remains dissipative when delays are introduced then the above open question has an affirmative answer our results also improve some recent findings