DCDS-B

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.

MBE

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.

DCDS-B

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.

DCDS-B

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.

DCDS-B

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.

DCDS-B

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.

MBE

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.

DCDS

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.

DCDS

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].

MBE

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.