Dynamics in a diffusive predator–prey system with double Allee effect and modified Leslie–Gower scheme

In this paper, we investigate a diffusive modified Leslie–Gower predator–prey system with double Allee effect on prey. The global existence, uniqueness and a priori bound of positive solutions are determined. The existence and local stability of constant steady–state solutions are analyzed. Next, we induce the nonexistence of nonconstant positive steady–state solutions, which indicates the effect of large diffusivity. Furthermore, we discuss the steady–state bifurcation and the existence of nonconstant positive steady–state solutions by the bifurcation theory. In addition, Hopf bifurcations of the spatially homogeneous and inhomogeneous periodic orbits are studied. Finally, we make some numerical simulations to validate and complement the theoretical analysis. Our results demonstrate that the dynamics of the system with double Allee effect and modified Leslie–Gower scheme are richer and more complex.

New Criteria for Sharp Oscillation of Second-OrderNeutral Delay Differential Equations

In this paper, new oscillation criteria for second-order half-linear neutral delay differential equations are established, using a recently developed method of iteratively improved monotonicity properties of a nonoscillatory solution. Our approach allows removing several disadvantages which were commonly associated with the method based on a priori bound for the nonoscillatory solution, and deriving new results which are optimal in a nonneutral case. It is shown that the newly obtained results significantly improve a large number of existing ones.

New results and applications on the existence results for nonlinear coupled systems

In this manuscript, we study a certain classical second-order fully nonlinear coupled system with generalized nonlinear coupled boundary conditions satisfying the monotone assumptions. Our new results unify the existence criteria of certain linear and nonlinear boundary value problems (BVPs) that have been previously studied on a case-by-case basis; for example, Dirichlet and Neumann are special cases. The common feature is that the solution of each BVPs lies in a sector defined by well-ordered coupled lower and upper solutions. The tools we use are the coupled lower and upper solutions approach along with some results of fixed point theory. By means of the coupled lower and upper solutions approach, the considered BVPs are logically modified to new problems, known as modified BVPs. The solution of the modified BVPs leads to the solution of the original BVPs. In our case, we only require the Nagumo condition to get a priori bound on the derivatives of the solution function. Further, we extend the results presented in (Franco et al. in Extr. Math. 18(2):153–160, 2003; Franco et al. in Appl. Math. Comput. 153:793–802, 2004; Franco and O’Regan in Arch. Inequal. Appl. 1:423–430, 2003; Asif et al. in Bound. Value Probl. 2015:134, 2015). Finally, as an application, we consider the fully nonlinear coupled mass-spring model.

Singularity and Decay Estimates for a Degenerate Parabolic Equation

In this paper, a degenerate parabolic equation u t − div x θ ∇ u = x a u p with p > 1 and θ < 2 , a ∈ ℝ , is considered. Based on rescaling arguments combined with a doubling property, the space-time singularity and decay estimates are established. Moreover, a universal and a priori bound of global nonnegative solutions for the corresponding initial boundary value problem is derived.

Item Aggregation and Column Generation for Online-Retail Inventory Placement

Problem definition: This paper studies an online retailer’s problem of choosing fulfillment centers in which to place items. We formulate the problem as a mixed-integer program that models thousands or millions of items to be placed in dozens of fulfillment centers and shipped to dozens of customer regions. The objective is to minimize the sum of shipping and fixed costs over one planning period. Academic/practical relevance: A good placement plan can significantly reduce the operational cost, which is crucial for online-retail businesses because they often have a low profit margin. The placement problem can be difficult to solve with existing techniques or off-the-shelf software because of the large number of items and the fulfillment center fixed costs and capacity constraints. Methodology: We propose a large-scale optimization framework that aggregates items into clusters, solves the cluster-level problem with column generation, and disaggregates the solution into item-level placement plans. We develop an a priori bound on the optimality gap, and we also apply the framework to a numerical example that consists of 1,000,000 items. Results: The a priori bound provides insights on how to select the appropriate aggregation criteria. For the numerical example, our framework produces a near-optimal solution in a few hours, significantly outperforming a sequential placement heuristic that approximates the status quo. Managerial implications: Our study provides a computationally efficient approach for solving online-retail inventory placement as well as similar large-scale optimization problems in practice.

A-priori bounds for quasilinear problems in critical dimension

We establish uniform a-priori bounds for solutions of the quasilinear problems $$\begin{array}{} \displaystyle \begin{cases} -{\it\Delta}_Nu=f(u)\quad&\mbox{in }{\it\Omega},\\ u=0\quad&\mbox{on }{\partial{\it\Omega}}, \end{cases} \end{array}$$ where Ω ⊂ ℝN is a bounded smooth convex domain and f is positive, superlinear and subcritical in the sense of the Trudinger-Moser inequality. The typical growth of f is thus exponential. Finally, a generalisation of the result for nonhomogeneous nonlinearities is given. Using a blow-up approach, this paper completes the results in [1, 2], extending the class of nonlinearities for which the uniform a-priori bound applies.

A Numerical Approximation Method for the Inverse Problem of the Three-Dimensional Laplace Equation

In this article, an inverse problem with regards to the Laplace equation with non-homogeneous Neumann boundary conditions in a three-dimensional case is investigated. To deal with this problem, a regularization method (mollification method) with the bivariate de la Vallée Poussin kernel is proposed. Stable estimates are obtained under a priori bound assumptions and an appropriate choice of the regularization parameter. The error estimates indicate that the solution of the approximation continuously depends on the noisy data. Two experiments are presented, in order to validate the proposed method in terms of accuracy, convergence, stability, and efficiency.

Optimal Algorithms for Computing Average Temperatures

A numerical algorithm is presented for computing average global temperature (or other quantities of interest such as average precipitation) from measurements taken at speci_ed locations and times. The algorithm is proven to be in a certain sense optimal. The analysis of the optimal algorithm provides a sharp a priori bound on the error between the computed value and the true average global temperature. This a priori bound involves a computable compatibility constant which assesses the quality of the measurements for the chosen model. The optimal algorithm is constructed by solving a convex minimization problem and involves a set of functions selected a priori in relation to the model. It is shown that the solution promotes sparsity and hence utilizes a smaller number of well-chosen data sites than those provided. The algorithm is then applied to canonical data sets and mathematically generic models for the computation of average temperature and average precipitation over given regions and given time intervals. A comparison is provided between the proposed algorithms and existing methods.