Dongmei Xiao

GLOBAL ANALYSIS IN A PREDATOR-PREY SYSTEM WITH NONMONOTONIC FUNCTIONAL RESPONSE ∗

A predator-prey system with nonmonotonic functional response is considered. Global qualitative and bifurcation analyses are combined to determine the global dynamics of the model. The bifurcation a...

Global analysis of an epidemic model with nonmonotone incidence rate

n Abstractn n In this paper we study an epidemic model with nonmonotonic incidence rate, which describes the psychological effect of certain serious diseases on the community when the number of infectives is getting larger. By carrying out a global analysis of the model and studying the stability of the disease-free equilibrium and the endemic equilibrium, we show that either the number of infective individuals tends to zero as time evolves or the disease persists.n n

BIFURCATIONS OF A RATIO-DEPENDENT PREDATOR-PREY SYSTEM WITH CONSTANT RATE HARVESTING ∗

The ratio-dependent predator-prey model exhibits rich interesting dynamics due to the singularity of the origin. The objective of this paper is to study the dynamical properties of the ratio-depend...

On the Delayed Ross–Macdonald Model for Malaria Transmission

AbstractnThe feedback dynamics from mosquito to human and back to mosquito involve considerable time delays due to the incubation periods of the parasites. In this paper, taking explicit account of the incubation periods of parasites within the human and the mosquito, we first propose a delayed Ross–Macdonald model. Then we calculate the basic reproduction number R0 and carry out some sensitivity analysis of R0 on the incubation periods, that is, to study the effect of time delays on the basic reproduction number. It is shown that the basic reproduction number is a decreasing function of both time delays. Thus, prolonging the incubation periods in either humans or mosquitos (via medicine or control measures) could reduce the prevalence of infection.n

On the uniqueness and nonexistence of limit cycles for predator?prey systems

In this paper, a general predator–prey system is considered. By utilizing simple but crucial changes of variables, the system is reduced to a generalized Lienard system where a wealth of existing methods and results are applicable. A new uniqueness theorem of limit cycle for generalized Lienard system is obtained. Some conditions of known uniqueness theorems and nonexistence theorems are modified so that they are more easily applied. As an application, criteria for the uniqueness of limit cycles and global stability of a unique positive equilibrium of the general predator–prey system are derived, which include some results of Kuang and Freedman (1988 Math. Biosci. 88 67–84). Several examples are given to illustrate our results.

Relaxation Oscillations in a Class of Predator-Prey Systems

Abstract We consider a class of three-dimensional, singularly perturbed predator–prey systems having two predators competing exploitatively for the same prey in a constant environment. By using dynamical systems techniques and the geometric singular perturbation theory, we give precise conditions which guarantee the existence of stable relaxation oscillations for systems within the class. Such result shows the coexistence of the predators and the prey with quite diversified time response which typically happens when the prey population grows much faster than those of predators. As an application, a well-known model will be discussed in detail by showing the existence of stable relaxation oscillations for a wide range of parameters values of the model.

Multiple Focus and Hopf Bifurcations in a Predator-Prey System with Nonmonotonic Functional Response

In this paper, we develop a criterion to calculate the multiplicity of a multiple focus for general predator-prey systems. As applications of this criterion, we calculate the largest multiplicity of a multiple focus in a predator-prey system with nonmonotonic functional response

STABILITY AND BIFURCATION IN A DELAYED RATIO-DEPENDENT PREDATOR-PREY SYSTEM

p(x)=frac{x}{ax^2+bx+1}

CODIMENSION TWO BIFURCATIONS IN A PREDATOR PREY SYSTEM WITH GROUP DEFENSE

studied by Zhu, Campbell, and Wolkowicz [SIAM J. Appl. Math., 63 (2002), pp. 636-682] and prove that the degenerate Hopf bifurcation is of codimension two. Furthermore, we show that there exist parameter values for which this system has a unique positive hyperbolic stable equilibrium and exactly two limit cycles, the inner one unstable and outer one stable. Numerical simulations for the existence of the two limit cycles bifurcated from the multiple focus are also given in support of the criterion.

Multiparametric bifurcations of an epidemiological model with strong Allee effect

Recently, ratio-dependent predator–prey systems have been regarded by some researchers as being more appropriate for predator–prey interactions where predation involves serious searching processes. Due to the fact that every population goes through some distinct life stages in real-life, one often introduces time delays in the variables being modelled. The presence of time delay often greatly complicates the analytical study of such models. In this paper, the qualitative behaviour of a class of ratio-dependent predator–prey systems with delay at the equilibrium in the interior of the first quadrant is studied. It is shown that the interior equilibrium cannot be absolutely stable and there exist non-trivial periodic solutions for the model. Moreover, by choosing delay