Nonlinear dynamics of a mathematical model on action potential duration and calcium transient in paced cardiac cells
Jiying Ma Dongmei Xiao
In the paper we focus on the dynamics of a two-dimensional discrete-time mathematical model, which describes the interaction between the action potential duration (APD) and calcium transient in paced cardiac cells. By qualitative and bifurcation analysis, we prove that this model can undergo period-doubling bifurcation and Neimark-Sacker bifurcation as parameters vary, respectively. These results provide theoretical support on some experimental observations, such as the alternans of APD and calcium transient, and quasi-periodic oscillations between APD and calcium transient in paced cardiac cells. The rich and complicated bifurcation phenomena indicate that the dynamics of this model are very sensitive to some parameters, which might have important implications for the control of cardiovascular disease.
keywords: period-doubling bifurcation Discrete-time mathematical model APD restitution Neimark-Sacker bifurcation. calcium transient
Qualitative analysis of a model for co-culture of bacteria and amoebae
Laura Fumanelli Pierre Magal Dongmei Xiao Xiao Yu
In this article we analyze a mathematical model presented in [11]. The model consists of two scalar ordinary differential equations, which describe the interaction between bacteria and amoebae. We first give the sufficient conditions for the uniform persistence of the model, then we prove that the model can undergo Hopf bifurcation and Bogdanov-Takens bifurcation for some parameter values, respectively.
keywords: Bogdanov-Takens bifurcation uniform persistence. Hopf bifurcation Population dynamics
Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models
Yu Yang Dongmei Xiao
A disease transmission model of SIRS type with latent period and nonlinear incidence rate is considered. Latent period is assumed to be a constant $\tau$, and the incidence rate is assumed to be of a specific nonlinear form, namely, $\frac{kI(t-\tau)S(t)}{1+\alpha I^{h}(t-\tau)}$, where $h\ge 1$. Stability of the disease-free equilibrium, and existence, uniqueness and stability of an endemic equilibrium for the model, are investigated. It is shown that, there exists the basic reproduction number $R_0$ which is independent of the form of the nonlinear incidence rate, if $R_0\le 1$, then the disease-free equilibrium is globally asymptotically stable, whereas if $R_0>1$, then the unique endemic equilibrium is globally asymptotically stable in the interior of the feasible region for the model in which there is no latency, and periodic solutions can arise by Hopf bifurcation from the endemic equilibrium for the model at a critical latency. Some numerical simulations are provided to support our analytical conclusions.
keywords: SIRS model; latent period; Nonlinear incidence rate; global stable; Hopf bifurcation.
Bifurcations in a discrete time Lotka-Volterra predator-prey system
Xiaoli Liu Dongmei Xiao
In this paper, a discrete-time system, derived from a predator-prey system by Euler's method with step one, is investigated in the closed first quadrant $R_+^2$. It is shown that the discrete-time system undergoes fold bifurcation, flip bifurcation and Neimark-Sacker bifurcation, and the discrete-time system has a stable invariant cycle in the interior of $R_+^2$ for some parameter values. Numerical simulations are provided to verify the theoretical analysis and show the complicated dynamical behavior. These results reveal far richer dynamics of the discrete model compared with the same type continuous model.
keywords: fixed point numerical simulations. Neimark-Sacker bifurcations Discrete time fold and flip bifurcation predator prey system
Multiple bifurcations of a predator-prey system
Dongmei Xiao Kate Fang Zhang
The bifurcation analysis of a generalized predator-prey model depending on all parameters is carried out in this paper. The model, which was first proposed by Hanski et al. [6], has a degenerate saddle of codimension 2 for some parameter values, and a Bogdanov-Takens singularity (focus case) of codimension 3 for some other parameter values. By using normal form theory, we also show that saddle bifurcation of codimension 2 and Bogdanov-Takens bifurcation of codimension 3 (focus case) occur as the parameter values change in a small neighborhood of the appropriate parameter values, respectively. Moreover, we provide some numerical simulations using XPPAUT to show that the model has two limit cycles for some parameter values, has one limit cycle which contains three positive equilibria inside for some other parameter values, and has three positive equilibria but no limit cycles for other parameter values.
keywords: Predator-prey system Bogdanov-Takens bifurcation of codimension 3. limit cycles saddle bifurcation of codimension 2
Identification of focus and center in a 3-dimensional system
Lingling Liu Bo Gao Dongmei Xiao Weinian Zhang
In this paper we identify focus and center for a generalized Lorenz system, a 3-dimensional quadratic polynomial differential system with four parameters $a$, $b$, $c$, $\sigma$. The known work computes the first order Lyapunov quantity on a center manifold and shows the appearance of a limit cycle for $a\neq b$, but the order of weak foci was not determined yet. Moreover, the case that $a=b$ was not discussed. In this paper, for $a\neq b$ we use resultants to decompose the algebraic varieties of Lyapunov quantities so as to prove that the order is at most 3. Further, we apply Sturm's theorem to determine real zeros of the first order Lyapunov quantity over an extension field so that we give branches of parameter curves for each order of weak foci. For $a=b$ we prove its Darboux integrability by finding an invariant surface, showing that the equilibrium of center-focus type is actually a rough center on a center manifold.
keywords: center-focus Darboux integrability. algebraic variety Generalized Lorenz system extension field
Dynamics of epidemic models with asymptomatic infection and seasonal succession
Yilei Tang Dongmei Xiao Weinian Zhang Di Zhu

In this paper, we consider a compartmental SIRS epidemic model with asymptomatic infection and seasonal succession, which is a periodic discontinuous differential system. The basic reproduction number $\mathcal{R}_0$ is defined and evaluated directly for this model, and uniform persistence of the disease and threshold dynamics are obtained. Specially, global dynamics of the model without seasonal force are studied. It is shown that the model has only a disease-free equilibrium which is globally stable if $\mathcal{R}_0≤ 1$, and as $\mathcal{R}_0>1$ the disease-free equilibrium is unstable and there is an endemic equilibrium, which is globally stable if the recovering rates of asymptomatic infectives and symptomatic infectives are close. These theoretical results provide an intuitive basis for understanding that the asymptomatically infective individuals and the seasonal disease transmission promote the evolution of the epidemic, which allow us to predict the outcomes of control strategies during the course of the epidemic.

keywords: Epidemic model asymptomatic infection seasonal succession basic reproduction number threshold dynamics
Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment
Yuan Lou Dongmei Xiao Peng Zhou
We study a two-species Lotka-Volterra competition model in an advective homogeneous environment. It is assumed that two species have the same population dynamics and diffusion rates but different advection rates. We show that if one competitor disperses by random diffusion only and the other assumes both random and directed movements, then the one without advection prevails. If two competitors are drifting along the same direction but with different advection rates, then the one with the smaller advection rate wins. Finally we prove that if the two competitors are drifting along the opposite direction, then two species will coexist. These results imply that the movement without advection in homogeneous environment is evolutionarily stable, as advection tends to move more individuals to the boundary of the habitat and thus cause the distribution of species mismatch with the resources which are evenly distributed in space.
keywords: Reaction-diffusion-advection stability advective environment co-existence steady state. Lotka-Volterra competition
A nonlinear diffusion problem arising in population genetics
Peng Zhou Jiang Yu Dongmei Xiao
In this paper we investigate a nonlinear diffusion equation with the Neumann boundary condition, which was proposed by Nagylaki in [19] to describe the evolution of two types of genes in population genetics. For such a model, we obtain the existence of nontrivial solutions and the limiting profile of such solutions as the diffusion rate $d\rightarrow0$ or $d\rightarrow\infty$. Our results show that as $d\rightarrow0$, the location of nontrivial solutions relative to trivial solutions plays a very important role for the existence and shape of limiting profile. In particular, an example is given to illustrate that the limiting profile does not exist for some nontrivial solutions. Moreover, to better understand the dynamics of this model, we analyze the stability and bifurcation of solutions. These conclusions provide a different angle to understand that obtained in [17,21].
keywords: Nonlinear diffusion equation existence bifurcation. limiting profile stability
The Within-Host dynamics of malaria infection with immune response
Yilong Li Shigui Ruan Dongmei Xiao
Malaria infection is one of the most serious global health problems of our time. In this article the blood-stage dynamics of malaria in an infected host are studied by incorporating red blood cells, malaria parasitemia and immune effectors into a mathematical model with nonlinear bounded Michaelis-Menten-Monod functions describing how immune cells interact with infected red blood cells and merozoites. By a theoretical analysis of this model, we show that there exists a threshold value $R_0$, namely the basic reproduction number, for the malaria infection. The malaria-free equilibrium is global asymptotically stable if $R_0<1$. If $R_0>1$, there exist two kinds of infection equilibria: malaria infection equilibrium (without specific immune response) and positive equilibrium (with specific immune response). Conditions on the existence and stability of both infection equilibria are given. Moreover, it has been showed that the model can undergo Hopf bifurcation at the positive equilibrium and exhibit periodic oscillations. Numerical simulations are also provided to demonstrate these theoretical results.
keywords: periodic oscillations. Malaria infection within-host dynamics thresh- old mathematical model

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