We characterize global solvability versus blow-up in finite time for a nonlinear diffusion model in one space dimension with opposing fluxes at the boundary. Memory driven absorption at one side is introduced through the time integral of a power law, while dissipation at the other side is contained in a localized power law. Previously, a complete analysis appeared in the literature on global solvability for such a model incorporating a balance of local power laws according to orderings of the powers. The present study is part of our continuing investigation of impacts on solvability due to introducing memory within corresponding purely local diffusion models. Such memory boundary conditions have been introduced in models for the process of new capillary growth, as initiated by a pre-vascularized tumor, known as angiogenesis. Global solvability established for the model with memory flux emerges out of integral estimates, providing sharp conditions. We find either all solutions of such models are global or all nontrivial solutions blow up in finite time. This is notably in contrast to cases of the local model with both global and blow-up solutions, as well as equilibrium states available for establishing global boundedness.

In this paper, we consider a spatial SIS reaction-diffusion model with Dirichlet boundary conditions. We study the asymptotic profiles of the endemic equilibrium for small and large diffusion rates of the susceptible individuals and the infected individuals. Compared to a similar model with Neumann boundary conditions, our results indicate that the hostile exterior environment makes a distinct impact on the spread of infectious diseases.

In this paper, we study an SIS epidemic reaction-diffusion model with mass action infection mechanism and subject to homogeneous Dirichlet boundary conditions. We define a basic reproduction number R-0 for the model and prove that there exists a unique endemic equilibrium if R-0 > 1. We then establish the global attractivity of the disease-free equilibrium and the endemic equilibrium for a special case. Furthermore, we analyze the asymptotic profiles of the endemic equilibrium. We show that limited movement of the individuals cannot eradicate the disease because of a varying total population, while large population mobility can cause the disease extinction due to the hostile exterior environment.

This corrigendum corrects a result in [1].

In this paper, we consider a reaction-diffusion equation with continuous delay and spatial variable coefficients which models the evolution of a single species. We establish a sharp threshold dynamic result: there exists a critical value λ(a) such that if λ(a)<0 the positive steady state solution of the equation is globally attractive, while if λ(a)≥0 the trivial steady state is globally attractive. To this end, we analyze the ω-limit set of the equation and prove that it is a singleton. Moreover, we apply our method to obtain global attractivity of the positive steady state of a spatially nonlocal diffusive logistic model.

We study the characterization of global solvability versus blow up in finite time for a porous medium model including a balance of internal absorption with memory driven flux through the boundary. Such a boundary condition was previously investigated as part of a model for the transmission of tumour-released growth factor from the site of a pre-vascularized tumour up to and across the wall of a nearby capillary, initiating the process of new capillary growth known as angiogenesis. In previous studies of the model without absorption, we have established the characterization of global solvability in a manner that exactly parallels known results for the corresponding model with localized boundary flux conditions. To include models accounting for internal uptake of growth factor, this analysis has recently been extended to a heat equation with absorption, and herein, we consider the case of a porous medium equation with absorption. Conditions for global solvability emerge naturally out of integral estimates and again provide close parallels with results for localized boundary flux models. It is noted that the results provide a complete characterization in a wide range of models considered.

The life cycles of many species include separate dispersal and sedentary stages. To understand the population dynamics of such species, we study a hybrid model consisting of a reaction–diffusion equation that governs the random movement and settlement of dispersal individuals and an age-structured hyperbolic equation that describes the growth of sedentary individuals. We establish the existence and uniqueness of the solution of the model using the monotone method based on a comparison principle. We study the population persistence and extirpation by means of three measures. We also verify the theoretical results through numerical simulations.

A general result on global solvability is established for a diffusion–absorption model with memory-driven flux at the boundary. Such a boundary condition has been studied previously for application to the problem of new capillary growth as induced by a pre-metastatic tumor. In earlier results for the model without absorption, we provided a complete characterization of power law memory boundary conditions regarding either global solvability or blow-up in finite time. It turns out such results are identical to those for the corresponding model with localized power law flux conditions at the boundary. Now in the case of nonlinear absorption added to the model, which in fact better incorporates natural growth factor decay or uptake in the capillary growth application, the threshold of global solvability for the localized model is dependent upon strength of absorption in a way that is not parallel to our result for the memory-driven model. We conclude by proving blow-up results in the radially symmetric case and observing that future studies are needed to complete the full characterization of global solvability.

In this paper, we consider an age-structured population model with diffusion. We first establish a comparison principle. We then apply the comparison principle to show the existence and uniqueness of solutions by constructing monotone sequences of weak upper and lower solutions. We also use the comparison principle to study the long-time behavior of the solution and present extinction and boundedness results under certain conditions on the parameters in the model. Additionally, as an example of its application, we apply the model to describe the dynamics of a forest population.

In this paper, we develop a hybrid parabolic and hyperbolic equation model, in which a reaction-diffusion equation governs the random movement and settlement of dispersal individuals, while a first-order hyperbolic equation describes the growth of stationary individuals with age structure. We prove the existence and uniqueness of the solution of the model using the monotone method based on a comparison principle. We study the population persistence criteria in terms of four related measures. We numerically investigate how the interplay between population dispersal, reproduction, settlement, and habitat boundary affects the population persistence.

In this paper, we consider an SIR reaction-diffusion model with a linear external source in spatially heterogeneous environment. We first study the global stability of the disease-free equilibrium in spatially heterogeneous environment and the global stability of the endemic equilibrium in spatially homogeneous environment. We then investigate the asymptotic profiles of the endemic equilibrium in spatially heterogeneous environment for small and large diffusion rates.

In this paper, we study the long-time behavior of a size-structured population model. We define a basic reproduction number R and show that the population dies out in the long run if R < 1. If R > 1, the model has a unique positive equilibrium, and the total population is uniformly strongly persistent. Most importantly, we show that there exists a subsequence of the total population converging to the positive equilibrium.

In the development of mathematical models for topics of practical importance, many nonlocal problems for partial differential equations have arisen from engineering, physics and life sciences. In this series of lectures, we will introduce various methods to study the existence/uniqueness and asymptotic behavior of solutions for such model problems.

We study global existence and blow up in finite time for a one‐dimensional fast diffusion equation with memory boundary condition. The problem arises out of a corresponding model formulated from tumor‐induced angiogenesis. We obtain necessary and sufficient conditions for global existence of solutions to the problem. Copyright © 2016 John Wiley & Sons, Ltd.

We study a susceptible–infected–susceptible reaction–diffusion model with spatially heterogeneous disease transmission and recovery rates. A basic reproduction number is defined for the model. We first prove that there exists a unique endemic equilibrium if . We then consider the global attractivity of the disease-free equilibrium and the endemic equilibrium for two cases. If the disease transmission and recovery rates are constants or the diffusion rate of the susceptible individuals is equal to the diffusion rate of the infected individuals, we show that the disease-free equilibrium is globally attractive if , while the endemic equilibrium is globally attractive if .

We consider a two-strain pathogen model described by a system of reaction-diffusion equations. We define a basic reproduction number $R_0$ and show that when the model parameters are constant (spatially homogeneous), if $R_0 >1$ then one strain will outcompete the other strain and drive it to extinction, but if $R_0 \le 1$ then the disease-free equilibrium is globally attractive. When we assume that the diffusion rates are equal while the transmission and recovery rates are heterogeneous, then there are two possible outcomes under the condition $R_0 >1$: 1) Competitive exclusion where one strain dies out. 2) Coexistence between the two strains. Thus, spatial heterogeneity promotes coexistence.

In a recent study, a lower bound is established on the blow up time for solutions of a chemotaxis system, with nonlinear chemotactic sensitivity u(u+1)m−1, set in the three-dimensional unit ball. Here, u is the density of a cell or organism that produces a chemical, with density v, and moves preferentially toward regions of higher concentration of v according to the flux −∇u+χu(u+1)m−1∇v. With χ>0, v is referred to as a “chemoattractant” and, in the case m=1, the system reduces to a version of the Keller–Segel model. Solutions that blow up in finite time have been previously established for the system on a ball in Rn provided n≥2, m>2/n. For technical reasons, the lower bound proven for the blow up time applies in such cases when n=3 and m≤2. We extend the analysis and resulting lower bound to such a model in general convex domains, with n≥2 and any m.

In this paper, we consider a nonlinear size-structured population model with vital rates depending on the total population. We derive sensitivity partial differential equations for the sensitivities of the solution with respect to the reproduction and mortality rates. We also present numerical results to illustrate the use of these sensitivity equations.

In this paper, we study the long-time behavior of solutions to the fast diffusion equation with a memory boundary condition. The problem corresponds to a model introduced in previous studies of tumor-induced angiogenesis. We establish global existence and finite time blow-up results for the problem.