DCDS-B

We study the existence of traveling wave solutions
for the two-species Lotka-Volterra competition model in the form
of integro-differential equations. The model incorporates
asymmetric dispersal kernels that describe long distance dispersal
processes of competing species in space. Using lower and upper
traveling wave solutions, we show that the model has traveling
wave solutions that connect the origin and the coexistence
equilibrium with speeds greater than the spreading speed of each
species in the absence of its rival.

DCDS-B

A time-delayed reaction-diffusion epidemic model with stage structure and spatial heterogeneity is investigated, which describes the dynamics of disease spread only proceeding in the adult population. We establish the basic reproduction number $\mathcal{R}_0$ for the model system, which gives the threshold dynamics in the sense that the disease will die out if $\mathcal{R}_0＜1$ and the disease will be uniformly persistent if $\mathcal{R}_0>1.$ Furthermore, it is shown that there is at least one positive steady state when $\mathcal{R}_0>1.$ Finally, in terms of general birth function for adult individuals, through introducing two numbers $\check{\mathcal{R}}_0$ and $\hat{\mathcal{R}}_0$, we establish sufficient conditions for the persistence and global extinction of the disease, respectively.

CPAA

In this paper, we consider the following perturbed nonlocal elliptic equation

where $\Omega$ is a smooth bounded domain in $\mathbb{R}{^N}$, $\lambda$ is a real parameter and $g$ is a non-odd perturbation term. If $f$ is odd in $u$ and satisfies various superlinear growth conditions at infinity in $u$, infinitely many solutions are obtained in spite of the lack of the symmetry of this problem for any $\lambda\in \mathbb{R}$. The results obtained in this paper may be seen as natural extensions of some classical theorems to the case of nonlocal operators. Moreover, the methods used in this paper can be also applied to obtain some new results for the classical Laplace equation with Dirichlet boundary conditions.

MBE

In this paper, a time periodic and two–group reaction–diffusion epidemic model with distributed delay is proposed and investigated. We firstly introduce the basic reproduction number $R_0$ for the model via the next generation operator method. We then establish the threshold dynamics of the model in terms of $R_0$, that is, the disease is uniformly persistent if $R_0 > 1$, while the disease goes to extinction if $R_0 < 1$. Finally, we study the global dynamics for the model in a special case when all the coefficients are independent of spatio–temporal variables.

MCRF

In this paper, we study controllability for a parabolic system of
chemotaxis. With one control only, the local exact controllability to
positive trajectory of the system is obtained by applying Kakutani's fixed
point theorem and the null controllability of associated linearized
parabolic system. The positivity of the state is shown to be remained in the
state space. The control function is shown to be in $L^\infty(Q)$, which is
estimated by using the methods of maximal regularity and $L^p$-$L^q$
estimate for parabolic equations.

NACO

Recently, some methods for solving optimization problems without derivatives have been proposed.
The main part of these methods is to form a suitable model function that can be minimized for obtaining a new iterative point. An important strategy is geometry-improving iteration for a good model, which needs a lot of calculations. Besides, Marazzi and Nocedal (2002) proposed a wedge trust region method for derivative free optimization. In this paper, we propose a new self-correcting geometry procedure with less computational efforts, and combine it with the wedge trust region method. The global convergence of new algorithm is established. The limited numerical experiments show that the new algorithm is efficient and competitive.

DCDS-B

In this paper, we propose a nonlocal and time-delayed reaction-diffusion predator-prey model with stage structure. It is assumed that prey individuals undergo two stages, immature and mature, and the conversion of consumed prey biomass into predator biomass has a retardation. In terms of the principal eigenvalue of nonlocal elliptic eigenvalue problems, we establish the uniform persistence and global extinction for the model. In particular, the uniform persistence implies the existence of positive steady states. Finally, we investigate a specially spatially homogeneous predator-prey system and show the complicated dynamics of the system due to the non-local delay in the prey equation.

DCDS-B

An almost periodic epidemic model with age structure in a patchy
environment is considered. The existence of the almost periodic
disease-free solution and the definition of the basic reproduction
ratio $R_{0}$ are given. Based on those, it is shown that a
disease dies out if the basic reproduction number $R_{0}$ is less
than unity and persists in the population if it is greater than
unity.