Stage-structured models for interacting wild and sterile mosquitoes

Using the sterile insect technique,in which sterile mosquitoes are released to reduce or eradicate the wild mosquito population,is an effective weapon to prevent transmission of mosquito-borne diseases. To study the impact of the sterile insect technique on the disease transmissions,we formulate stage-structured discrete-time mathematical models,based on difference equations,for the interactive dynamics of the wild and sterile mosquitoes. We incorporate different strategies for releasing sterile mosquitoes,investigate the model dynamics,and compare the impact of the different release strategies.Numerical examples are also provided to demonstrate dynamical features of the models.

ON MODELING THE GROWTH DYNAMICS OF A STAGE STRUCTURED POPULATION

A Lagrangian modeling approach is applied to the numerical simulation of the temporal dynamics of a stage-structured population. The growth dynamics is determined only by the main biological processes： development of an individual, mortality, reproduction. Different approaches in modeling the development process of an individual are implemented： stochastic advection-diffusion models （backward-forward dispersion models）, and stochastic development models where regression effects, defined as negative development on the status of an individual, are forbidden （forward dispersion models）. Some properties of the residence times of an individual in a stage are investigated： in particular, their role in the calibration of the development models and in the estimation of some parameters introduced in the model equation. As a study case a multi-stage pelagic copepod population is considered. Trying to separate the effects of the main biological processes on the temporal dynamics, numerical simulations have been carried out in some idealized situations： first only the development of the individuals, neglecting mortality and reproduction, is considered; then the mortality process is introduced, and finally both the mortality and reproduction processes. The results of the numerical simulations, are compared and discussed.

PREDATOR-PREY SYSTEM WITH STAGE STRUCTURE AND DELAY

A system of retarded functional differential equations is proposed as a predator-prey model with stage structure in the prey.The invariance of non-negativity,nature of boundary equilibria and global stability are analyzed.It is shown that in this model the time delay can make a stable equilibrium become unstable and even a switching of stabilities.

Global Stability of a Predator-prey Model with Stage Structure

A Holling type III predator-prey model with stage structure for prey is investi-gated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria of the system is discussed. By using the uniformly persistence theory, the system is proven to be permanent if the coexistence equilibrium exists. By using Lyapunov functionals and LaSalle’s invariance principle, it is shown that the two boundary equilibria is globally asymptotically stable when the coexistence equilibrium is not feasible. By using compound matrix theory, the sucient conditions are obtained for the global stability of the coexistence equilibrium. At last, numerical simulations are carried out to illustrate the main results.

OPTIMAL HARVESTING AND STABILITY FOR FISHING MODELS WITH STAGE STRUCTURE IN INSHORE-OFFSHORE AREAS

A fishing model with stage structure in inshore and offshore areas is considered.Steady states of the dynamical system representing fishery are derived and their local,as well as global,stability is discussed.The optimal harvest policy is formulated and solved as an optimal control problem.A numerical example is given to illustrate the solution procedure.

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.

Localized-domains staging structure and evolution in lithiated graphite

Intercalation provides to the host materials a means for controlled variation of many physical/chemical properties and dominates the reactions in metal‐ion batteries.Of particular interest is the graphite intercalation compounds with intriguing staging structures,which however are still unclear,especially in their nanostructure and dynamic transition mechanism.Herein,the nature of the staging structure and evolution of the lithium(Li)‐intercalated graphite was revealed by cryogenic‐transmission electron microscopy and other methods at the nanoscale.The intercalated Li‐ions distribute unevenly,generating local stress and dislocations in the graphitic structure.Each staging compound is found macroscopically ordered but microscopically inhomogeneous,exhibiting a localized‐domains structural model.Our findings uncover the correlation between the long‐range ordered structure and short‐range domains,refresh the insights on the staging structure and transition of Li‐intercalated/deintercalated graphite,and provide effective ways to enhance the reaction kinetic in rechargeable batteries by defect engineering.

Global Attracting Behavior of Non-autonomous Stage-structured Population Dynamical System with Diffusion

A non-autonomous single species dispersal model is considered, in which individual member of the population has a life history that goes through two stages, immature and mature. By applying the theory of monotone and concave operators to functional differential equations, we establish conditions under which the system admits a positive periodic solution which attracts all other positive solutions.

GLOBAL STABILITY OF A STAGE-STRUCTURED PREDATOR-PREY MODEL WITH MODIFIED LESLIE-GOWER AND HOLLING-TYPE II SCHEMES

In this paper, we consider a stage-structured predator-prey model with modified Leslie- Gower and Holling-type II schemes. Using an iterative technique, we investigate the global stability of the positive equilibrium of the system. Finally, some examples are presented to verify our main result.

Global Stability of a Predator-Prey System with Stage Structure for the Predator

In this paper, some feasibly suffcient conditions are obtained for the global asymptotic stability of a positive steady state of a predator-prey system with stage structure for the predator by using the theory of competitive systems, compound matrices and stability of periodic orbits, and then the work of Wang [4] is improved.

ON AN SIS EPIDEMIC MODEL WITH STAGE STRUCTURE

A SIS infectious disease model with stage structure consisting of immature and mature stages is proposed using a discrete time delay. The aim of the paper is to investigate under which conditions the disease becomes endemic or not and to find the difference between the model with stage structure and the corresponding model without stage structure. It is shown that either there exists a unique endemic equilibrium point which is globally asymptotically stable or the disease dies out by using an iterative scheme. The effect of the time delay on the populations at equilibria is considered.

OPTIMAL HARVESTING POLICY AND STABILITY FOR SINGLE-SPECIES GROWTH MODEL WITH STAGE STRUCTURE

Abstract. In this paper, we consider a stage structure population model with two lifestages, immature and mature, with harvesting mature population and stocking immaturepopulation. It is shown that under suitable hypotheses there exists a globally asymptoti-cally stable positive equilibrium. The effect of the delay on the populations at equilibriumand the optimal harvesting policy for mature population are also considered.

A PREDATOR-PREY SYSTEM WITH STAGE STRUCTURE AND HARVESTING FOR PREDATOR

In this paper, we consider a predator-prey system with stage structure and harvesting (where the predator population has two stages, an immature stage and a mature stage with harvesting, and the growth of the prey population is of Lotka-Volterra nature). We obtain the conditions of the globally asymptotic stability for three nonne-gative equilibria of this system.

Dynamics of stochastic predator-prey models with distributed delay and stage structure for prey

In this paper,we analyze two stochastic predator-prey models with distributed delay and stage structure for prey.For the nonautonomous periodic case of the model,by using Khasminskii’s theory of periodic solution,we show that the system has at least one positive T-periodic solution.For the model which is disturbed by both white and telegraph noises,we obtain sufficient criteria for positive recurrence of the solutions to the model by constructing a suitable stochastic Lyapunov function with regime switching.The positive recurrence implies that both prey and predator populations will be persistent in the long term.

Global attractivity of the predator-prey system for cotton aphid and seven-spot ladybird beetle with stage structure

It is well known that the cotton aphid is the major pest in cotton fields of Northwest China, and seven-spot ladybird is an important natural enemy among the various possi- ble natural enemies of cotton aphid. In order to increase the applications of population dynamics in integrated pest management and control the cotton aphids biologically, we need to understand the population dynamics of cotton aphid and their natural enemies. A delay predator prey system on cotton aphid and seven-spot ladybird beetle are pro- posed in this paper. Based on the comparison theorem and an iterative method, we investigate the global attra^tivity of the equilibrium points which have important bio- logical meanings. Furthermore, some numerical simulations were carried out to illustrateand expand our theoretical results, in which a conjecture to generalize the well-known Theorem 16.4 in H. R. Thiemes book was put forward, which was taken as the open problem. The numerical simulations show coexistence of periodic solution, confirming the theoretical prediction.

PERMANENCE OF A PERIODIC PREDATOR-PREY SYSTEM WITH GENERAL HOLLING TYPE FUNCTIONAL RESPONSE AND STAGE STRUCTURE FOR PREY

In this paper, we give some sufficient and necessary condition of permanence for a periodic predator-prey system with general Holling type functional response and stage structure for preys. By doing this, we improve some known results.

RATIO-DEPENDENT PREDATOR-PREY MODEL WITH STAGE STRUCTURE AND TIME DELAY

Ratio-dependent predator prey models are favored by many animal ecologists recently as more suitable ones for predator-prey interactions where predation involves searching process. In this paper, a ratio-dependent predator prey model with stage structure and time delay for prey is proposed and analyzed. In this model, we only consider the stage structure of immature and mature prey species and not consider the stage structure of predator species. We assume that the predator only feed on the mature prey and the time for prey from birth to maturity represented by a constant time delay. At first, we investigate the permanence and existence of the proposed model and sufficient conditions are derived. Then the global stability of the nonnegative equilibria are derived. We also get the sufficient criteria for stability switch of the positive equilibrium. Finally, some numerical simulations are carried out for supporting the analytic results.

Globally asymptotic stability of a predator-prey model with stage structure incorporating prey refuge

This paper focuses on the stabilities of the equilibria to a predator-prey model with stage structure incorporating prey refuge. By analyzing the characteristic functions, we obtain that the equilibria of the model are locally stable when some suitable conditions are being satisfied. According to the comparison theorem and iteration technique, the globally asymptotic stability of the positive equilibrium is discussed. And, the sufficient conditions of the global stability to the trivial equilibrium and the boundary equilibrium are derived. The study shows that the prey refuge will enhance the density of the prey species, and it will decrease the density of predator species. Finally, some numerical simulations are carried out to show the efficiency of our main results.

Periodic Solutions of a Delayed Predator-Prey Model with Stage Structure for Prey

A periodic predator-prey model with stage structure for prey and time delays due to negative feed-back and gestation of predator is proposed.By using Gaines and Mawhin's continuation theorem of coincidencedegree theory,sufficient conditions are derived for the existence of positive periodic solutions to the proposedmodel.Numerical simulations are presented to illustrate the feasibility of our main result.

Bistability of a two-species competition model with stage structure

In this paper,we focus on the dynamic behaviors of the bistability for a two-species Lotka–Volterra competition model with stage structure.Using the theory of generalized saddle-point behaviors,it is shown that there exists a C1-separatrixΓon its state space such that Species 2 wins the competition whenever the initial distribution is aboveΓ,while Species 1 wins the competition whenever the initial distribution is belowΓ.Combining with the previous conclusions of the system,we give a complete classification for global competitive dynamics.Furthermore,we carefully analyze the changes of the competition outcome after introducing the stage structure.Finally,some numerical simulations are provided to illustrate the effectiveness of the theoretical results.