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.     

Mathematical analysis of a two-patch model of tuberculosis disease with staged progression

The spread of tuberculosis is studied through a two-patch epidemiological system SE₁ ? E_nI which incorporates migrations from one patch to another just by susceptible individuals. Our model is consider with bilinear incidence and migration between two patches, where infected and infectious individuals cannot migrate from one patch to another, due to medical reasons. The existence and uniqueness of the associated endemic equilibria are discussed. Quadratic forms and Lyapunov functions are used to show that when the basic reproduction ratio is less than one, the disease-free equilibrium (DFE) is globally asymptotically stable, and when it is greater than one there exists in each case a unique endemic equilibrium (boundary equilibria and endemic equilibrium) which is globally asymptotically stable. Numerical simulation results are provided to illustrate the theoretical results.     

A CLASS OF VOLTERRA INTEGRAL EQUATIONS ARISING IN DELAYED-RECRUITMENT POPULATION MODELS

We formulate a Volterra integral equation which contains as special cases the differential-difference equation model of Blythe, Gurney and Nisbet for populations with delayed recruitment and a differential-difference equation with two delays related to the epidemic model of Wilson and Burke. We establish upper and lower bounds for positive solutions and give a classification of equilibria with conditions to determine whether an equilibrium is stable for all delays (absolutely stable), unstable for all delays, or switches from stable to unstable as the delay increases.     

DYNAMICAL ANALYSIS OF A SINGLE-SPECIES POPULATION MODEL WITH MIGRATIONS AND HARVEST BETWEEN PATCHES

A single-species population model with migrations and harvest between the protected patch and the unprotected patch is formulated and investigated in this paper. We study the local stability and the global stability of the equilibria. The research points out, under some suitable conditions, the singlespecies population model admits a unique positive equilibrium, which is globally asymptotically stable. We also derive that the trivial solution is globally asymptotically stable when the harvesting rate exceeds the threshold. Further,we discuss the practical effects of the protection zones and the harvest. The main results indicate that the protective zones indeed eliminate the extinction of the species under some cases, and the theoretical threshold of harvest to the practical management of the endangered species is provided as well. To end this contribution and to check the validity of the main results, numerical simulations are separately carried out to illustrate these results.     

Spread of disease in a patchy environment

For a two patches SIR model, it is shown that its dynamic behavior is determined by several quantities. We have shown that if R₀ < 1, then the disease-free equilibrium is globally asymptotically stable, otherwise it is unstable. Some sufficient conditions for the local stability of boundary equilibria are obtained. Numerical simulations indicate that travel between patches can reduces oscillations in both patches; may enhances oscillations in both patches; or travel switches oscillations from one patch to another.     

The stability of a diffusion model of plankton allelopathy with spatio–temporal delays

A reaction–diffusion system with non-local delay is proposed to describe two competitive planktonic growths in aquatic ecology. The local and global stability of the axial equilibria as well as the positive equilibrium are discussed. Our results show that the delay has no effect on the stability of the axial equilibria; on the other hand, the positive equilibrium can be induced to be locally unstable by the delay. Finally, the corresponding numerical simulations are also demonstrated.     

两种群相互竞争的高维SEIR传染病模型全局渐近稳定性

研究了一类两种群相互竞争的非线性高维SEIR传染病数学模型动力学性质,综合利用Lasalle不变集原理,Lyapunov函数,Routh-Hurwitz判据和Krasnoselskii等多种方法,得到了边界平衡点的全局渐近稳定和正平衡点局部渐近稳定的阈值条件.     

Dynamics analysis of a delayed viral infection model with logistic growth and immune impairment

In this paper, the dynamics behavior of a delayed viral infection model with logistic growth and immune impairment is studied. It is shown that there exist three equilibria. By analyzing the characteristic equations, the local stability of the infection-free equilibrium and the immune-exhausted equilibrium of the model are established. By using suitable Lyapunov functional and LaSalle invariant principle, it is proved that the two equilibria are globally asymptotically stable. In the following, the stability of the positive equilibrium is investigated. Furthermore, we investigate the existence of Hopf bifurcation by using a delay as a bifurcation parameter. Finally, numerical simulations are carried out to explain the mathematical conclusions.     

Symmetrization in neumann problems

This paper is concerned with a model of a predator–prey system, where both populations disperse among n patches forming their habitat. Criteria are given tor both survival and extinction of the predator population. In case the predator survives, conditions are derived which guarantee a globally asymptotically stable positive equilibrium     

A bifurcation theorem for nonlinear matrix models of population dynamics

ABSTRACT

We prove a general theorem for nonlinear matrix models of the type used in structured population dynamics that describes the bifurcation that occurs when the extinction equilibrium destabilizes as a model parameter is varied. The existence of a bifurcating continuum of positive equilibria is established, and their local stability is related to the direction of bifurcation. Our theorem generalizes existing theorems found in the literature in two ways. First, it allows for a general appearance of the bifurcation parameter (existing theorems require the parameter to appear linearly). This significantly widens the applicability of the theorem to population models. Second, our theorem describes circumstances in which a backward bifurcation can produce stable positive equilibria (existing theorems allow for stability only when the bifurcation is forward). The signs of two diagnostic quantities determine the stability of the bifurcating equilibrium and the direction of bifurcation. We give examples that illustrate these features.     

A predator–prey–disease model with immune response in infected prey

In this paper, a predator–prey–disease model with immune response in the infected prey is formulated. The basic reproduction number of the within-host model is defined and it is found that there are three equilibria: extinction equilibrium, infection-free equilibrium and infection-persistent equilibrium. The stabilities of these equilibria are completely determined by the reproduction number of the within-host model. Furthermore, we define a basic reproduction number of the between-host model and two predator invasion numbers: predator invasion number in the absence of disease and predator invasion number in the presence of disease. We have predator and infection-free equilibrium, infection-free equilibrium, predator-free equilibrium and a co-existence equilibrium. We determine the local stabilities of these equilibria with conditions on the reproduction and invasion reproduction numbers. Finally, we show that the predator-free equilibrium is globally stable.     

Global Qualitative Analysis for a Ratio-Dependent Predator-Prey Model with Delay

A ratio-dependent predator-prey model with time lag for predator is proposed and analyzed. Mathematical analyses of the model equations with regard to boundedness of solutions, nature of equilibria, permanence, and stability are analyzed. We note that for a ratio-dependent system local asymptotic stability of the positive steady state does not even guarantee the so-called persistence of the system and, therefore, does not imply global asymptotic stability. It is found that an orbitally asymptotically stable periodic orbit exists in that model. Some sufficient conditions which guarantee the global stability of positive equilibrium are given.     

Time delayed parabolic system in a two-species competitive model with stage structure

A two-species competitive model with stage structure is discussed. The dynamics of coupled system of semilinear parabolic equations with time delays are investigated. Results on the local and global stabilities of the axial equilibria and positive equilibrium are given. Our results show that the introduction of diffusion does not affect the permanence and extinction of the species though the introduction of stage structure brings negative effect on it.     

Study on a HIV/AIDS model with application to Yunnan province,China

This paper presents an epidemic model aiming at the prevalence of HIV/AIDS in Yunnan, China. The total population in the model is restricted within high risk population. By the epidemic characteristics of HIV/AIDS in Yunnan province, the population is divided into two groups: injecting drug users (IDUs) and people engaged in commercial sex (PECS) which includes female sex workers (FSWs), and clients of female sex workers (C). For a better understanding of HIV/AIDS transmission dynamics, we do some necessary mathematical analysis. The conditions and thresholds for the existence of four equilibria are established. We compute the reproduction number for each group independently, and show that when both the reproduction numbers are less than unity, the disease-free equilibrium is globally stable. The local stabilities for other equilibria including two boundary equilibria and one positive equilibrium are figured out. When we omit the infectivity of AIDS patients, global stability of these equilibria are obtained. For the simulation, parameters are chosen to fit as much as possible prevalence data publicly available for Yunnan. Increasing strength of the control measure on high risk population is necessary to reduce the HIV/AIDS in Yunnan.     

A global bifurcation theorem for Darwinian matrix models

Motivated by models from evolutionary population dynamics, we study a general class of nonlinear difference equations called matrix models. Under the assumption that the projection matrix is non-negative and irreducible, we prove a theorem that establishes the global existence of a continuum with positive equilibria that bifurcates from an extinction equilibrium at a value of a model parameter at which the extinction equilibrium destabilizes. We give criteria for the global shape of the continuum, including local direction of bifurcation and its relationship to the local stability of the bifurcating positive equilibria. We discuss a relationship between backward bifurcations and Allee effects. Illustrative examples are given.     

Global dynamics of two phytoplankton-zooplankton models with toxic substances effect

In this paper, we investigate phytoplankton-zooplankton models with toxic substances effect and two different kinds of predator functional responses. For Holling type II predator functional response, it is shown that the local stability of the positive equilibrium implies global stability if there exists a unique positive equilibrium. When there exist multiple positive equilibria, the local stability of the positive equilibrium with small phytoplankton population density implies that the model occurs bistable phenomenon. These results also hold for Holling type III predator functional response under certain conditions.     

Global Analysis of Predator–Prey System with Hawk and Dove Tactics

Functional response of the Holling type II is incorporated into a predator–prey model with predators using hawk‐dove tactics to consider combination effects of nonlinear functional response and individual tactics. By mathematical analysis, it is shown that the model undergoes a sequence of bifurcations including saddle‐node bifurcation, supercritical Hopf bifurcation and homoclinic bifurcation. New phenomena are found that include the bistable coexistence of prey and predators in the form of a stable limit cycle and a stable positive equilibrium, the bistable coexistence of prey and predators in a large stable limit cycle that encloses three positive equilibria and a stable positive equilibrium within the cycle, and the bistable coexistence of two stable limit cycles.     

Analysis of a Lotka–Volterra food chain chemostat with converting time delays

A model of the food chain chemostat involving predator, prey and growth-limiting nutrients is considered. The model incorporates two discrete time delays in order to describe the time involved in converting processes. The Lotka–Volterra type increasing functions are used to describe the species uptakes. In addition to showing that solutions with positive initial conditions are positive and bounded, we establish sufficient conditions for the (i) local stability and instability of the positive equilibrium and (ii) global stability of the non-negative equilibria. Numerical simulation suggests that the delays have both destabilizing and stabilizing effects, and the system can produce stable periodic solutions, quasi-periodic solutions and strange attractors.     

Competitive equilibrium and singleton cores in generalized matching problems

We study competitive equilibria in generalized matching problems. We show that, if there is a competitive matching, then it is unique and the core is a singleton consisting of the competitive matching. That is, a singleton core is necessary for the existence of competitive equilibria. We also show that a competitive matching exists if and only if the matching produced by the top trading cycles algorithm is feasible, in which case it is the unique competitive matching. Hence, we can use the top trading cycles algorithm to test whether a competitive equilibrium exists and to construct a competitive equilibrium if one exists. Lastly, in the context of bilateral matching problems, we compare the condition for the existence of competitive matchings with existing sufficient conditions for the existence or uniqueness of stable matchings and show that it is weaker than most existing conditions for uniqueness.     

How to share the cost of a public good?

The paper deals with a simple model with only one private and one public good. The core of such an economy is shown to have strong properties, in particular, it is stable in thevon Neumann-Morgenstern sense and complete. A natural candidate for a selection within the core is the concept ofLindahl equilibrium which constitutes a generalization of the concept of competitive equilibrium in an economy with private goods only. Although theLindahl equilibria belong in general to the core, they do not have the same symmetry properties. It is shown that it is possible to select in the core allocations having stronger symmetry properties than theLindahl equilibria.