On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex
Mats Gyllenberg Jifa Jiang Lei Niu Ping Yan
Discrete & Continuous Dynamical Systems - A 2018, 38(2): 615-650 doi: 10.3934/dcds.2018027
We propose the generalized competitive Atkinson-Allen map
$T_i(x)=\frac{(1+r_i)(1-c_i)x_i}{1+\sum_{j=1}^nb_{ij}x_j}+c_ix_i, 0 <c_i <1, b_{ij}, r_i>0, i, j=1, ···, n, $
which is the classical Atkson-Allen map when
for all
$i=1, ..., n$
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.
keywords: Discrete-time competitive model carrying simplex generalized competitive Atkinson-Allen model classification Neimark-Sacker bifurcation Chenciner bifurcation invariant closed curve heteroclinic cycle
Dynamics of a reaction-diffusion system of autocatalytic chemical reaction
Jifa Jiang Junping Shi
Discrete & Continuous Dynamical Systems - A 2008, 21(1): 245-258 doi: 10.3934/dcds.2008.21.245
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.
keywords: autocatalytic chemical reaction asymptotic autonomous system convergence to equilibrium.
The complete classification on a model of two species competition with an inhibitor
Jifa Jiang Fensidi Tang
Discrete & Continuous Dynamical Systems - A 2008, 20(3): 659-672 doi: 10.3934/dcds.2008.20.659
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.
keywords: strong monotonicity inhibitor convergence Lotka-Volterra two species competition nonexistence for periodic solutions
Epidemic dynamics on complex networks with general infection rate and immune strategies
Shouying Huang Jifa Jiang
Discrete & Continuous Dynamical Systems - B 2018, 23(6): 2071-2090 doi: 10.3934/dcdsb.2018226

This paper mainly aims to study the influence of individuals' different heterogeneous contact patterns on the spread of the disease. For this purpose, an SIS epidemic model with a general form of heterogeneous infection rate is investigated on complex heterogeneous networks. A qualitative analysis of this model reveals that, depending on the epidemic threshold $R_0$, either the disease-free equilibrium or the endemic equilibrium is globally asymptotically stable. Interestingly, no matter what functional form the heterogeneous infection rate is, whether the disease will disappear or not is completely determined by the value of $R_0$, but the heterogeneous infection rate has close relation with the epidemic threshold $R_0$. Especially, the heterogeneous infection rate can directly affect the final number of infected nodes when the disease is endemic. The obtained results improve and generalize some known results. Finally, based on the heterogeneity of contact patterns, the effects of different immunization schemes are discussed and compared. Meanwhile, we explore the relation between the immunization rate and the recovery rate, which are the two important parameters that can be improved. To illustrate our theoretical results, the corresponding numerical simulations are also included.

keywords: Heterogeneous network epidemic spreading infection rate global stability immunization strategy
Daozhou Gao Shigui Ruan Jifa Jiang
Mathematical Biosciences & Engineering 2017, 14(5&6): i-ii doi: 10.3934/mbe.201705i
Theoretical investigation on models of circadian rhythms based on dimerization and proteolysis of PER and TIM
Jifa Jiang Qiang Liu Lei Niu
Mathematical Biosciences & Engineering 2017, 14(5&6): 1247-1259 doi: 10.3934/mbe.2017064

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.

keywords: Circadian rhythm positive feedback steady state limit cycle period

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