Uniform Persistence, Periodicity and Extinction in a Delayed Biological System with Stage Structure and Density-Dependent Juvenile Birth Rate

A delayed biological system of predator-prey interaction with stage structure and density dependent juvenile birth rate is investigated. It is assumed that the prey population has two stages: immature and mature. The growth of the immature prey is density dependent and is a function of the density of adult prey. Such phenomenon has been reported for beetles, tribolium, copepods, scorpions, several fish species and even crows. The growth of the predator is affected by the time delay due to gestation. By some Lemmas and methods of delay differential equation, the conditions for the uniform persistence and extinction of the system are obtained. Numerical simulations illustrate the feasibility of the main results and demonstrate that the density dependent coefficient has influence on the system populations’ densities though it has no effect on uniform persistence and extinction of the system.

A Novel Analysis Approach of Uniform Persistence for an Epidemic Model with Quarantine and Standard Incidence Rate

Inspired by the transmission characteristics of the coronavirus disease 2019(COVID-19),an epidemic model with quarantine and standard incidence rate is first developed,then a novel analysis approach is proposed for finding the ultimate lower bound of the number of infected individuals,which means that the epidemic is uniformly persistent if the control reproduction number R_(c)>1.This approach can be applied to the related biomat hem at ical models,and some existing works can be improved by using that.In addition,the infection-free equilibrium V^(0)of the model is locally asymptotically stable(LAS)if R_(c)<1 and linearly stable if R_(c)=1;while V^(0)is unstable if R_(c)>1.

UNIFORM PERSISTENCE AND GLOBAL ATTRACTIVITY FOR NONAUTONOMOUS COMPETITIVE SYSTEMS WITH DISPERSION

In this paper, we consider a nonautonomous competitive model with dispersion and a finite number of discrete delays. The system, which consists of two Lotka-Volterra patches, has two competitors: one can disperse between the two patches, but the other is confined to one patch and cannot disperse. Our purpose is to demonstrate that the dispersion rates have no effect on the uniform persistence of the solutions of the system. Furthermore, we establish the conditions under which the system admits a positive periodic solution which attracts all solutions.

PERSISTENCE AND GLOBAL STABILITY FOR A THREE-SPECIES RATIO-DEPENDENT PREDATOR-PREY SYSTEM WITH TIME DELAYS IN TWO-PATCH ENVIRONMENTS

A three-species ratio-dependent predator-prey diffusion model with time delays is investigated. It is shown that the system is uniformly persistent under some appropriate conditions, and sufficient conditions axe obtained for the global stability of the positive equilibrium of the system.

PERSISTENCE AND STABILITY IN A RATIO-DEPENDENT FOOD-CHAIN SYSTEM WITH TIME DELAYS

A delayed three-species ratio-dependent predator-prey food-chain model without dominating instantaneous negative feedback is investigated. It is shown that the system is permanent under some appropriate conditions,and sufficient conditions are obtained for the local asymptotic stability of a positive equilibrium of the system.

Existence and global attractivity of unique positive almost periodic solution for a model of hematopoiesis

Sufficient conditions are obtained which guarantee the uniform persistence and global attractivity of solutions for the model of hematopoiesis. Then some criteria are established for the existence, uniqueness and global attractivity of almost periodic solutions of almost periodic system.

QUALITATIVE ANALYSIS OF A DICRETE POPULATION MODEL

In this paper, authors study the qualitative behavior of solutions of the discrete population model xn-xn-1=xn (a+bxn-k-cx2n-k),where a ∈ (0, 1), b ∈ (-∞, 0),c ∈ (0,∞ ), and k is a positive integer. They hot only obtain necessary as well as sufficient and necessary conditions for the oscillation of ail eventually positive solutions about the positive equilibrium, but also obtain some sufficient conditions for the convergence of eventually positive solutions. Furthermore, authors also show that such model is uniformly persistent, and that all its eventually positive solutions are bounded.

Dynamic Behavior of a Predator-prey System with Stage-structure for Prey

In this paper, we considered the predator-prey system with stage-structure for prey, where the predators predate immature preys only. The positivity and boundedness of the solutions and asymptotic stability of equilibrium were firstly discussed, and then uniformly persistent sufficient conditions of populations were found.

Interactional models for adults of two populations with maturation delays

This paper is concerned with interactional models for adults of two species delayed by their mature periods. The existence and local stability of equilibria are discussed thoroughly for competitive systems, cooperative systems and predator-prey systems, respectively. For systems with interaction of competition and cooperation, it is found that the two populations are uniformly persistent if the positive equilibrium is stable. For predator-prey interaction, however, some further conditions are needed to guarantee the persistence of the systems.

Global Stability for a Diffusive System with Time Delay and Functional Response

In this paper,by applying Comparison Theorem of differential equation,Continuation Theorem of coincidence degree theory,Barbalat Lemma and Lyapunov Function,a diffusion system with distributive time delay and ratio-dependence functional response is studied.It is proved that the system is uniformly persistent under appropriate conditions.Further,if the system is a periodic one,it can have a strictly positive periodic solution which is globally asymptotically stable under appropriations.Some new results are obtained.

ON THE PERSISTENCE AND POSITIVE PERIODIC SOLUTION FOR PLANAR COMPETING LOTKA-VOLTERRA SYSTEMS

We study the qualitative property of solutions of planar periodic competing otka-Volterra systems New criteria of uniform persistence of solutins and existence, uniqueness and globally asymptotical stability of positive periodic solution are established. The results results in [1 -6] are summarized and improved in this paper.

PERIODIC SOLUTION AND PERSISTENCE FOR A THREE-SPECIES RATIO-DEPENDENT PREDATOR-PREY MODEL WITH TIME DELAYS IN TWO-PATCH ENVIRONMENTS

The author considers a three-species ratio-dependent predator-prey model with time delay in a two-patch environments. This model is of periodic coefficients, which incorporates the periodicity of the environment. By means of the coincidence degree theory, sufficient conditions for the existence of at least one positive periodic solution of this model are established. Moreover, The author shows that the system is uniformly persistent under the conditions.

PERSISTENCE IN A THREE SPECIES LOTKA-VOLTERRA PERIODIC PREDATOR-PREY SYSTEM

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Dynamical analysis of a diffusive malaria model with fixed latent period in the human and vector populations

To investigate the impact of the fixed latent periods in the human and vector populations on the disease transmission in heterogenous environment,we formulate a nonlocal and time-delayed reaction-diffusion(NLTD-RD)system.By appealing to the next generation operator(NGO),we define the basic reproduction number(BRN)Ro,and prove it as a threshold parameter for indicating whether disease persists or not.Specifically,if o<1,the disease-free equilibrium is globally asymptotically stable,while if Ro>1,the disease is shown to be uniformly persistent.In the homogeneous case that all parameters are assumed to be constants,the explicit expression of o is obtained.We further achieved the global attractivity of the constant equilibria by utilizing Lyapunov functionals.Numerical simulations are performed to verify the theoretical results and the effects of the diffusion rate on disease transmission.

Global dynamics in a model for anthrax transmission in animal populations

In this paper,we propose a deterministic model to study the transmission dynamics of anthrax disease,which includes live animals,carcasses,spores in the environment and vectors.We derive three biologically plausible and insightful quantities(reproduction numbers)that determine the stability of the equilibria.We carry out rigorous mathematical analysis on the model dynamics,the global stability of the disease-free and vector-free equilibrium,the disease-free equilibrium and the vector-free disease equilibrium is proved.The global stability of the endemic equilibrium as the basic reproduction number is greater than one is derived in the special case in which the disease-related death rate is zero.The possibility of backward bifurcation is briefly discussed.Numerical analyses are carried out to understand the transmission dynamics of anthrax and investigate effective control strategies for the outbreaks of the disease.Our studies suggest that the larval vector control measure should be taken as early as possible to control the vector population size,a vaccination policy and an animal carcass removal policy are useful methods to control the prevalence of the diseases in infected animal populations,the adult vector control measure is also necessary to prevent the transmission of anthrax.

Modeling the impact of awareness programs on the transmission dynamics of dengue and optimal control

In this work,we first propose a mathematical model to study the impact of awareness programs on dengue transmission.The basic reproduction number Ro is derived.The existence and stability of equilibria are investigated.The uniform persistence is established when Ro is larger than one.Our results suggest that awareness programs have significant impacts on dengue transmission dynamics although they cannot affect Ro.When o is less than one,awareness programs can shorten the prevailing time effectively.When Ro is larger than one,awareness programs may destabilize the unique interior equilibrium and a stable periodic solution appears due to Hopf bifurcation.In particular,we find that the occurrence of Hopf bifurcation depends not only on the intensity of awareness programs but also on the level of Ro.Besides,large fuctuations in the number of infected individuals caused by the stable periodic solution may bring pressure on limited medical resources.Therefore,different from intuitive ideas,blindly increasing the intensity of awareness programs is not necessarily conducive to control the transmission of dengue.The decision-making department should decide to adopt different publicity strategies according to the current level of Ro.Finally,we consider the optimal control problem of the model and find the optimal control strategy corresponding to awareness programs by Pontryagin's Maximum Principle.The results manifest that the optimal control strategy can effectively mitigate the transmission of dengue.

Conditions for Global Stability of n-Patches Ecological System with Dispersion

In this paper, a nonautonomous predator-prey dispersion model is studied, where all parameters are time-dependent. The system, which is consisted of n-patches, the prey specics can disperse among n-patches, but the predator species is confined to one patch and cannot disperse. It is proved the system is uniformly persistent under any dispersion rates effect. Furthermore, sufficient conditions are established for global stability of the system.

GLOBAL ASYMPTOTIC STABILITY OF PERIODIC SOLUTION IN N-SPECIES COMPETITION SYSTEM WITH TIME DELAYS

This paper devotes to study the N species competition system with time delays in a periodic environment. some verifiable sufficient conditions which are easy to be verified for dissipation, the existence of periodic solution and global asymptotic stability of periodic solution are obtained.

Attractivity in a Delayed Three-species Ratio-dependent Predator-prey System without Dominating Instantaneous Negative Feedback

Abstract A delayed three-species ratio-dependent predator-prey food-chain model without dominating instantaneous negative feedback is investigated. It is shown that the system is permanent under some appropriate conditions, and sufficient conditions are derived for the global attractivity of the positive equilibrium of the system.

THE POSITIVE PERIODIC SOLUTIONS OF PERIODIC RFDEs WITH INFINITE DELAY

This paper is concerned with the periodic retarded functional differential equati-ons(RFDEs) with infinite delay. The sufficient conditions for the existence of noncon-stant positive periodic solutions are established by combining the theory of monotone semiflows generated by RFDEs with infinite delay and the fixed point theorems of solution operators. A nontrivial application of the results obtained here to a well-known nonautonomous Lotka-Volterra system with infinite delay is also presented.