DCDS

The precise dynamics of a reaction-diffusion model of autocatalytic
chemical reaction is described. It is shown that exactly either one, two,
or three steady states exists, and the solution of dynamical problem always approaches
to one of steady states in the long run. Moreover it is shown that a global codimension one
manifold separates the basins of attraction of the two stable steady states.
Analytic ingredients include exact multiplicity of semilinear elliptic equation,
the theory of monotone dynamical systems and the
theory of asymptotically autonomous dynamical systems.

DCDS

Hetzer and Shen [3] considered a system
of a two-species Lotka-Volterra competition model with an
inhibitor, investigated its long-term behavior and proposed two
open questions: one is whether the system has a nontrivial
periodic solution; the other is whether one of two positive
equilibria is non-hyperbolic in the case that the system has
exactly two positive equilibria. The goal of this paper is first
to give these questions clear answers, then to present a complete
classification for its dynamics in terms of coefficients. As a
result, all solutions are convergent as $t$ goes to infinity.

DCDS

We propose the generalized competitive Atkinson-Allen map

which is the classical Atkson-Allen map when

and

for all

and a discretized system of the competitive Lotka-Volterra equations. It is proved that every

-dimensional map

of this form admits a carrying simplex Σ which is a globally attracting invariant hypersurface of codimension one. We define an equivalence relation relative to local stability of fixed points on the boundary of Σ on the space of all such three-dimensional maps. In the three-dimensional case we list a total of 33 stable equivalence classes and draw the corresponding phase portraits on each Σ. The dynamics of the generalized competitive Atkinson-Allen map differs from the dynamics of the standard one in that Neimark-Sacker bifurcations occur in two classes for which no such bifurcations were possible for the standard competitive Atkinson-Allen map. We also found Chenciner bifurcations by numerical examples which implies that two invariant closed curves can coexist for this model, whereas those have not yet been found for all other three-dimensional competitive mappings via the carrying simplex. In one class every map admits a heteroclinic cycle; we provide a stability criterion for heteroclinic cycles. Besides, the generalized Atkinson-Allen model is not dynamically consistent with the Lotka-Volterra system.

MBE

Circadian rhythms of physiology and behavior are widespread\break mechanisms in many organisms. The internal biological rhythms are driven by molecular clocks, which oscillate with a period nearly but not exactly $24$ hours. Many classic models of circadian rhythms are based on a time-delayed negative feedback, suggested by the protein products inhibiting transcription of their own genes. In 1999, based on stabilization of PER upon dimerization, Tyson et al. [J. J. Tyson, C. I. Hong, C. D. Thron, B. Novak, Biophys. J. 77 (1999) 2411-2417] proposed a crucial positive feedback to the circadian oscillator. This idea was mathematically expressed in a three-dimensional model. By imposing assumptions that the dimerization reactions were fast and dimeric proteins were in rapid equilibrium, they reduced the model to a pair of nonlinear ordinary differential equations of mRNA and total protein concentrations. Then they used phase plane analysis tools to investigate circadian rhythms. In this paper, the original three-dimensional model is studied. We explore the existence of oscillations and their periods. Much attention is paid to investigate how the periods depend on model parameters. The numerical simulations are in good agreement with their reduced work.

DCDS-B

In this paper we propose the stochastic epidemic model that relates directly to
the deterministic counterpart and reveal close connections between these
two models. Under the classic assumptions, the sample path of process eventually
converges to the disease-free equilibrium, even though the corresponding deterministic
flow converges to an endemic equilibrium. From the fact that disease can occur
sporadically, we adjust the stochastic model slightly
by introducing a stochastic incidence and establish precise connections between
the long-run behavior of the discrete stochastic process
and its deterministic flow approximation for large populations.

DCDS

We study the fixed point index on the carrying simplex of the
competitive map. The sum of the indices of the fixed points on the
carrying simplex for the three-dimensional competitive map is
unit. Based on that, we analyze the asymptotic behavior of the three-dimensional competitive Atkinson/Allen model.
We present all the equivalence classes relative to the boundary of the carrying simplex of the low-dimensional (two or three) map, depending upon relationship among the model coefficients. For the two-dimensional case, there are only three dynamic
scenarios, and every orbit converges to some fixed point. For the three-dimensional case, there are total $33$ stable
equivalence classes, and in $18$ of them all the compact limit sets are fixed points. Further, we focus on the analysis
of the dynamics of the other $15$ cases. Hopf bifurcation is studied and a necessary condition for it occurring is given, which implies that the classes $19$-$25$, $28$, $30$ and $32$ do not have any Hopf bifurcation. However, the class $26$ and class $27$ do admit Hopf bifurcations, which means that these two classes may have isolated invariant closed curves in their carrying simplex, and such an invariant closed curve corresponds to either a subharmonic or a quasiperiodic solution in continuous time systems. Each system in class $27$ has a heteroclinic cycle and the numerical simulation also reveals that there exist systems having May-Leonard
phenomenon: the existence of nonquasiperiodic oscillation.

MBE

In this paper, we develop and analyze an SIS epidemic model with a general nonlinear incidence rate, as well as degree-dependent birth and natural death, on heterogeneous networks. We analytically derive the epidemic threshold $R_0$ which completely governs the disease dynamics: when $R_0<1$, the disease-free equilibrium is globally asymptotically stable, i.e., the disease will die out; when $R_0>1$, the disease is permanent. It is interesting that the threshold value $R_0$ bears no relation to the functional form of the nonlinear incidence rate and degree-dependent birth. Furthermore, by applying an iteration scheme and the theory of cooperative system respectively, we obtain sufficient conditions under which the endemic equilibrium is globally asymptotically stable. Our results improve and generalize some known results. To illustrate the theoretical results, the corresponding numerical simulations are also given.