DCDS

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.

DCDS-B

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.

DCDS

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.

DCDS-B

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.

DCDS

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.

MBE

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.

DCDS

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.

DCDS-B

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.

DCDS-B

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.

DCDS-B

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.