DCDS
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.
DCDS-B
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
DCDS-B
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
DCDS
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
DCDS-B
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.
DCDS
Propagation phenomena for CNNs with asymmetric templates and distributed delays
Zhixian Yu Xiao-Qiang Zhao
Discrete & Continuous Dynamical Systems - A 2018, 38(2): 905-939 doi: 10.3934/dcds.2018039

The aim of this work is to study propagation phenomena for monotone and nonmonotone cellular neural networks with the asymmetric templates and distributed delays. More precisely, for the monotone case, we establish the existence of the leftward ($c_{-}^*$) and rightward ($c_{+}^*$) spreading speeds for CNNs by appealing to the theory developed in [26,27], and $c_{-}^*+c_{+}^*>0$. Especially, if cells possess the symmetric templates and the same delayed interactions, then $c_{-}^*=c_{+}^*>0$. Moreover, if the effect of the self-feedback interaction $α f'(0)$ is not less than 1, then both $c_{-}^*>0$ and $c_{+}^*>0$. For the non-monotone case, the leftward and rightward spreading speeds are investigated by using the results of the spreading speed for the monotone case and squeezing the given output function between two appropriate nondecreasing functions. It turns out that the leftward and rightward spreading speeds are linearly determinate in these two cases. We further obtain the existence and nonexistence of travelling wave solutions under the weaker conditions than those in [46, 47] and show that the spreading speed coincides with the minimal wave speed.

keywords: Spreading speeds travelling waves monotone semiflows CNNs Non-monotone
MBE
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
DCDS
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
DCDS-B
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
DCDS-B
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

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