DCDS-B
Global stability and backward bifurcation of a general viral infection model with virus-driven proliferation of target cells
Hongying Shu Lin Wang
In this paper, a general viral model with virus-driven proliferation of target cells is studied. Global stability results are established by employing the Lyapunov method and a geometric approach developed by Li and Muldowney. It is shown that under certain conditions, the model exhibits a global threshold dynamics, while if these conditions are not met, then backward bifurcation and bistability are possible. An example is presented to provide some insights on how the virus-driven proliferation of target cells influences the virus dynamics and the drug therapy strategies.
keywords: global stability Virus dynamics backward bifurcation. in-host model
MBE
Modeling the spread of bed bug infestation and optimal resource allocation for disinfestation
Ali Gharouni Lin Wang
A patch-structured multigroup-like $SIS$ epidemiological model is proposed to study the spread of the common bed bug infestation. It is shown that the model exhibits global threshold dynamics with the basic reproduction number as the threshold parameter. Costs associated with the disinfestation process are incorporated into setting up the optimization problems. Procedures are proposed and simulated for finding optimal resource allocation strategies to achieve the infestation free state. Our analysis and simulations provide useful insights on how to efficiently distribute the available exterminators among the infested patches for optimal disinfestation management.
keywords: Bed bug infestation threshold dynamics optimal resource allocation disinfestation. $SIS$ model
MBE
Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy
Shengqiang Liu Lin Wang
Global stability is analyzed for a general mathematical model of HIV-1 pathogenesis proposed by Nelson and Perelson [11]. The general model include two distributed intracellular delays and a combination therapy with a reverse transcriptase inhibitor and a protease inhibitor. It is shown that the model exhibits a threshold dynamics: if the basic reproduction number is less than or equal to one, then the HIV-1 infection is cleared from the T-cell population; whereas if the basic reproduction number is larger than one, then the HIV-1 infection persists and the viral concentration maintains at a constant level.
keywords: HIV-1; Global stability; delay; steady state; Lyapunov functional.
MBE
Analyzing the causes of alpine meadow degradation and the efficiency of restoration strategies through a mathematical modelling exercise
Hanwu Liu Lin Wang Fengqin Zhang Qiuying Li Huakun Zhou

As an important ecosystem, alpine meadow in China has been degraded severely over the past few decades. In order to restore degraded alpine meadows efficiently, the underlying causes of alpine meadow degradation should be identified and the efficiency of restoration strategies should be evaluated. For this purpose, a mathematical modeling exercise is carried out in this paper. Our mathematical analysis shows that the increasing of raptor mortality and the decreasing of livestock mortality (or the increasing of the rate at which livestock increases by consuming forage grass) are the major causes of alpine meadow degradation. We find that controlling the amount of livestock according to the grass yield or ecological migration, together with protecting raptor, is an effective strategy to restore degraded alpine meadows; while meliorating vegetation and controlling rodent population with rodenticide are conducive to restoring degraded alpine meadows. Our analysis also suggests that providing supplementary food to livestock and building greenhouse shelters to protect livestock in winter may contribute to alpine meadow degradation.

keywords: Alpine meadow degradation restoration strategy mathematical modelling stability
MBE
Bifurcation analysis and transient spatio-temporal dynamics for a diffusive plant-herbivore system with Dirichlet boundary conditions
Lin Wang James Watmough Fang Yu
In this paper, we study a diffusive plant-herbivore system with homogeneous and nonhomogeneous Dirichlet boundary conditions. Stability of spatially homogeneous steady states is established. We also derive conditions ensuring the occurrence of Hopf bifurcation and steady state bifurcation. Interesting transient spatio-temporal behaviors including oscillations in one or both of space and time are observed through numerical simulations.
keywords: Plant-herbivore interaction transient dynamics. Hopf bifurcation diffusion steady state bifurcation
MBE
Modeling diseases with latency and relapse
P. van den Driessche Lin Wang Xingfu Zou
A general mathematical model for a disease with an exposed (latent) period and relapse is proposed. Such a model is appropriate for tuberculosis, including bovine tuberculosis in cattle and wildlife, and for herpes. For this model with a general probability of remaining in the exposed class, the basic reproduction number $\R_0$ is identified and its threshold property is discussed. In particular, the disease-free equilibrium is proved to be globally asymptotically stable if $\R_0<1$. If the probability of remaining in the exposed class is assumed to be negatively exponentially distributed, then $\R_0=1$ is a sharp threshold between disease extinction and endemic disease. A delay differential equation system is obtained if the probability function is assumed to be a step-function. For this system, the endemic equilibrium is locally asymptotically stable if $\R_0>1$, and the disease is shown to be uniformly persistent with the infective population size either approaching or oscillating about the endemic level. Numerical simulations (for parameters appropriate for bovine tuberculosis in cattle) with $\mathcal{R}_0>1$ indicate that solutions tend to this endemic state.
keywords: bovine tuberculosis delay differential equation uniform persistence. endemic equilibrium global asymptotic stability disease-free equilibrium tuberculosis
MBE
Delay induced spatiotemporal patterns in a diffusive intraguild predation model with Beddington-DeAngelis functional response
Renji Han Binxiang Dai Lin Wang

A diffusive intraguild predation model with delay and Beddington-DeAngelis functional response is considered. Dynamics including stability and Hopf bifurcation near the spatially homogeneous steady states are investigated in detail. Further, it is numerically demonstrated that delay can trigger the emergence of irregular spatial patterns including chaos. The impacts of diffusion and functional response on the model's dynamics are also numerically explored.

keywords: Intraguild predation delay diffusion Beddington-DeAngelis functional response spatiotemporal dynamics Hopf bifurcation chaos
DCDS
Formula of entropy along unstable foliations for $C^1$ diffeomorphisms with dominated splitting
Xinsheng Wang Lin Wang Yujun Zhu

Metric entropies along a hierarchy of unstable foliations are investigated for $C^1 $ diffeomorphisms with dominated splitting. The analogues of Ruelle's inequality and Pesin's formula, which relate the metric entropy and Lyapunov exponents in each hierarchy, are given.

keywords: Metric entropy along unstable foliations dominated splitting Lyapunov exponents Ruelle inequality Pesin's entropy formula
CPAA
The dynamics of small amplitude solutions of the Swift-Hohenberg equation on a large interval
Ling-Jun Wang
We study the dynamics of the one dimensional Swift-Hohenberg equation defined on a large interval $(-l,l)$ with Dirichlet-Neumann boundary conditions, where $l>0$ is large and lies outside of some small neighborhoods of the points $n\pi$ and $(n+1/2)\pi,n \in N$. The arguments are based on dynamical system formulation and bifurcation theory. We show that the system with Dirichlet-Neumann boundary conditions can be reduced to a two-dimensional center manifold for each bifurcation parameter $O(l^{-2})$-close to its critical values when $l$ is sufficiently large. On this invariant manifold, we find families of steady solutions and heteroclinic connections with each connecting two different steady solutions. Moreover, by comparing the above dynamics with that of the Swift-Hohenberg equation defined on $R$ and admitting $2\pi$-spatially periodic solutions in [4], we find that the dynamics in our case preserves the main features of the dynamics in the $2\pi$ spatially periodic case.
keywords: Dirichlet-Neumann boundary. center manifold theory Swift-Hohenberg equation dynamics
JIMO
Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications
Ling Yun Wang Wei Hua Gui Kok Lay Teo Ryan Loxton Chun Hua Yang
In this paper, we consider a class of optimal control problems involving time delayed dynamical systems and subject to continuous state inequality constraints. We show that this type of problem can be approximated by a sequence of time delayed optimal control problems subject to inequality constraints in canonical form and with multiple characteristic time points appearing in the cost and constraint functions. We derive formulae for the gradient of the cost and constraint functions of the approximate problems. On this basis, each approximate problem can be solved using a gradient-based optimization technique. The computational method obtained is then applied to an industrial problem arising in the study of purification process of zinc sulphate electrolyte. The results are highly satisfactory.
keywords: zinc sulphate electrolyte purification process. continuous state inequality constraints optimal control multiple characteristic time points Time delayed system

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