Spatial dynamics of a nonlocal and delayed population model in a periodic habitat
Peixuan Weng Xiao-Qiang Zhao
Discrete & Continuous Dynamical Systems - A 2011, 29(1): 343-366 doi: 10.3934/dcds.2011.29.343
We derived an age-structured population model with nonlocal effects and time delay in a periodic habitat. The spatial dynamics of the model including the comparison principle, the global attractivity of spatially periodic equilibrium, spreading speeds, and spatially periodic traveling wavefronts is investigated. It turns out that the spreading speed coincides with the minimal wave speed for spatially periodic travel waves.
keywords: nonlocal effect periodic habitat spreading speeds Age-structured population model spatially periodic traveling wavefronts.
Kernel sections for processes and nonautonomous lattice systems
Xiao-Qiang Zhao Shengfan Zhou
Discrete & Continuous Dynamical Systems - B 2008, 9(3&4, May): 763-785 doi: 10.3934/dcdsb.2008.9.763
In this paper, we first establish a set of sufficient and necessary conditions for the existence of globally attractive kernel sections for processes defined on a general Banach space and a weighted space ℓ$_\rho ^p$ of infinite sequences ($p\geq 1)$, respectively. Then we obtain an upper bound of the Kolmogorov $\varepsilon$-entropy of kernel sections for processes on the Hilbert space ℓ$_\rho ^2 $. As applications, we investigate compact kernel sections for first order, partly dissipative, and second order nonautonomous lattice systems on weighted spaces containing bounded sequences.
keywords: Kernel sections lattice systems pullback asymptotic null. pullback $\omega $-limit compactness Kolmogorov's $\varepsilon $-entropy
Monostable waves and spreading speed for a reaction-diffusion model with seasonal succession
Manjun Ma Xiao-Qiang Zhao
Discrete & Continuous Dynamical Systems - B 2016, 21(2): 591-606 doi: 10.3934/dcdsb.2016.21.591
This paper is devoted to the study of propagation phenomena for a two-species competitive reaction-diffusion model with seasonal succession in the monostable case. By appealing to theory of traveling waves and spreading speeds for monotone semiflows, we establish the existence of the minimal wave speed for rightward traveling waves and its coincidence with the rightward spreading speed. We also obtain a set of sufficient conditions for the spreading speed to be linearly determinate.
keywords: minimal wave speed Reaction and diffusion linear determinacy. periodic traveling waves spreading speed seasonal succession
Global asymptotic stability of minimal fronts in monostable lattice equations
Shiwang Ma Xiao-Qiang Zhao
Discrete & Continuous Dynamical Systems - A 2008, 21(1): 259-275 doi: 10.3934/dcds.2008.21.259
The global asymptotic stability with phase shift of traveling wave fronts of minimal speed, in short minimal fronts, is established for a large class of monostable lattice equations via the method of upper and lower solutions and a squeezing technique.
keywords: minimal speed lattice equations. Asymptotic stability traveling waves
A vector-bias malaria model with incubation period and diffusion
Zhiting Xu Xiao-Qiang Zhao
Discrete & Continuous Dynamical Systems - B 2012, 17(7): 2615-2634 doi: 10.3934/dcdsb.2012.17.2615
This paper is devoted to the study of the global dynamics of a vector-bias malaria model with incubation period and diffusion. The global attractivity of the disease-free or endemic equilibrium is first proved for the spatially homogeneous system. Then the threshold dynamics is established for the spatially heterogeneous system in terms of the basic reproduction ratio. A set of sufficient conditions is further obtained for the global attractivity of the positive steady state.
keywords: reaction-diffusion basic reproduction ratio Malaria transmission global dynamics.
An extension of the formula for spreading speeds
Hans F. Weinberger Xiao-Qiang Zhao
Mathematical Biosciences & Engineering 2010, 7(1): 187-194 doi: 10.3934/mbe.2010.7.187
A well-known formula for the spreading speed of a discrete-time recursion model is extended to a class of problems for which its validity was previously unknown. These include migration models with moderately fat tails or fat tails. Examples of such models are given.
keywords: integrodifference equations linear determinacy. spreading speeds discrete-time recursions
The diffusive logistic model with a free boundary and seasonal succession
Rui Peng Xiao-Qiang Zhao
Discrete & Continuous Dynamical Systems - A 2013, 33(5): 2007-2031 doi: 10.3934/dcds.2013.33.2007
This paper concerns a diffusive logistic equation with a free boundary and seasonal succession, which is formulated to investigate the spreading of a new or invasive species, where the free boundary represents the expanding front and the time periodicity accounts for the effect of the bad and good seasons. The condition to determine whether the species spatially spreads to infinity or vanishes at a finite space interval is derived, and when the spreading happens, the asymptotic spreading speed of the species is also given. The obtained results reveal the effect of seasonal succession on the dynamical behavior of the spreading of the single species.
keywords: free boundary vanishing. spreading seasonal succession Diffusive logistic equation
Fisher waves in an epidemic model
Xiao-Qiang Zhao Wendi Wang
Discrete & Continuous Dynamical Systems - B 2004, 4(4): 1117-1128 doi: 10.3934/dcdsb.2004.4.1117
The existence of Fisher type monotone traveling waves and the minimal wave speed are established for a reaction-diffusion system modeling man-environment-man epidemics via the method of upper and lower solutions as applied to a reduced second order ordinary differential equation with infinite time delay.
keywords: monotone iterations. traveling waves Epidemic model upper and lower solutions
Threshold dynamics in a time-delayed periodic SIS epidemic model
Yijun Lou Xiao-Qiang Zhao
Discrete & Continuous Dynamical Systems - B 2009, 12(1): 169-186 doi: 10.3934/dcdsb.2009.12.169
The global dynamics of a periodic SIS epidemic model with maturation delay is investigated. We first obtain sufficient conditions for the single population growth equation to admit a globally attractive positive periodic solution. Then we introduce the basic reproduction ratio $\mathcal{R}_0$ for the epidemic model, and show that the disease dies out when $\mathcal{R}_0<1$, and the disease remains endemic when $\mathcal{R}_0>1$. Numerical simulations are also provided to confirm our analytic results.
keywords: Periodic epidemic model Periodic solutions Uniform persistence. Basic reproduction ratio Maturation delay
Asymptotic speed of spread and traveling waves for a nonlocal epidemic model
Dashun Xu Xiao-Qiang Zhao
Discrete & Continuous Dynamical Systems - B 2005, 5(4): 1043-1056 doi: 10.3934/dcdsb.2005.5.1043
By applying the theory of asymptotic speeds of spread and traveling waves to a nonlocal epidemic model, we established the existence of minimal wave speed for monotone traveling waves, and show that it coincides with the spreading speed for solutions with initial functions having compact supports. The numerical simulations are also presented.
keywords: traveling waves Spreading speed nonlocal epidemic model.

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