Comparison Principles and Asymptotic Behavior of Delayed Age-Structured Neuron Models

Random diffusive age-structured population models have been studied by many researchers. Though nonlocal diffusion processes are more applicable to many biological and physical problems compared with random diffusion processes, there are very few theoretical results on age-structured population models with nonlocal diffusion. In this paper our objective is to develop basic theory for age-structured population dynamics with nonlocal diffusion. In particular, we study the semigroup of linear operators associated to an age-structured model with nonlocal diffusion and use the spectral properties of its infinitesimal generator to determine the stability of the zero steady state. It is shown that (i) the structure of the semigroup for the age-structured model with nonlocal diffusion is essentially determined by that of the semigroups for the age-structured model without diffusion and the nonlocal operator when both birth and death rates are independent of spatial variables; (ii) the asymptotic behavior can be determined by the sign of spectral bound of the infinitesimal generator when both birth and death rates are dependent on spatial variables; (iii) the weak solution and comparison principle can be established when both birth and death rates are dependent on spatial variables and time; and (iv) the above results can be generalized to an age-size structured model. In addition, we compare our results with the age-structured model with Laplacian diffusion in the first two cases (i) and (ii).

Numerical analysis of age-structured HIV model with general transmission mechanism

Spatial dynamics of a juvenile-adult model with impulsive harvesting and evolving domain

This paper is concerned with the nonlinear stability of traveling wavefronts for a single species population model with nonlocal dispersal and age structure. By using the weighted energy method together with the comparison principle, we prove that the traveling wavefront is exponentially stable, when the initial perturbation around the wavefronts decays exponentially at –∞, but it can be arbitrarily large in other locations. In particular, our result implies that the time delay is harmless for stability of traveling wavefronts of the model.

Analysis of age and spatially dependent population model: Application to forest growth

This paper is dedicated to investigate the propagation dynamics of a three-species reaction-diffusion system with nonlocal diffusion which includes diffusion kernel functions symmetry and asymmetry. First off, we give the asymptotic spreading speeds by using its partial commonality with the minimum propagation spreadings and construct a set $ \Pi $ to obtain their signs which just shows it is fundamentally different from local diffusion. The following main focus on the asymptotic behaviour with different initial values. The key point for asymmetry kernel is to construct suitable upper and lower solutions and obtain long-time asymptotic behavior by using comparison principle and our improved 'forward-backward spreading' method which first proposed by Xu et al. (J Funct Anal 280(2021)108957). Especially, transforming the square scale, introducing new variables and using linear programming to obtain the existence of variables and other techniques are our improvement of this method which ensures it works for multi-population systems. Accordingly, the asymptotic behavior and some monotone property results with the symmetric kernel are all obtained by comparison principle.

An age-structured fish model with birth and harvesting pulses is established, where birth pulses are responsible for increasing the amount of fish due to the constant multiple placement of juveniles, and harvesting pulses describe the decrease of fish due to fishing activities. The principal eigenvalue as a threshold value depending on the harvesting and birth intensity is firstly investigated by three different ways. The asymptotic behavior of the population is fully investigated and sufficient conditions for the species to be extinct or persist are given. Numerical simulations suggest that interaction between negative harvesting intervention and positive birth perturbation decides extinction and persistence of the species. It is possible to transform between expansion and extinction of species for sustainable development of fishery resources by choosing appropriate pulse intensities and perturbation timing.

On the existence in the large of solutions to the one-dimensional, isentropic hydrodynamic equations in a bounded domain.- Initial-boundary value and scattering problems in mathematical physics.- On shape optimization of a turbine blade.- Free boundary problems for the Navier-Stokes equations.- A geometric maximum principle, plateau's problem for surfaces of prescribed mean curvature, and the two dimensional analogue of the catenary.- Finite Elements for the Beltrami operator on arbitrary surfaces.- Comparison principles in capillarity.- Remarks on diagonal elliptic systems.- Quasiconvexity, growth conditions and partial regularity.- The monotonicity formula in geometric measure theory, and an application to a partially free boundary problem.- Isoperimetric problems having continua of solutions.- Harmonic maps - Analytic theory and geometric significance.- Asymptotic behavior of solutions of some quasilinear elliptic systems in exterior domains.- Decomposition theorems and their application to non-linear electro- and magneto-static boundary value problems.- Initial boundary value problems in thermoelasticity.- Applications of variational methods to problems in the geometry of surfaces.- Open problems in the degree theory for disc minimal surfaces spanning a curve in ?3.- On a modified version of the free geodetic boundary-value problem.

The present analysis introduces a system of cooperative species formulated with a high order parabolic operator, a Fisher-KPP reaction and a linear advection. Firstly, the oscillatory behaviour of solutions is shown to exist with a shooting method approach. It is to be highlighted that the existence of oscillatory patterns (also called instabilities) is an inherent property of high order operators. Afterwards, existence and uniqueness results are provided. The most remarkable result, obtained during the existence exercise, is related with the finding of a particular time-degenerate bound for the advection term that ensures positivity of solutions. This is one of the main results as such positivity property does not hold for high order operators in general. Indeed, high-order operators provide oscillatory solutions that may induce such solutions to be negative in the proximity of the null state introduced by the Fisher-KPP reaction term. As a consequence, a comparison principle does not hold as formulated in order two operators. Further, a positive maximal kernel with similar asymptotic behaviour compared to the high order kernel has been shown to exist and a precise assessment has been done with a computational exercise. Eventually, such a positive maximal kernel permits to show the existence of a comparison principle.

The objective of this paper is to study asymptotic behavior of a class of higher-order delay differential equations with a p-Laplacian like operator. Symmetry ideas are often invisible in these studies, but they help us decide the right way to study them, and show us the correct direction for future developments. New oscillation criteria are obtained by employing a refinement of the generalized Riccati transformations and comparison principles. This new theorem complements and improves a number of results reported in the literature. Some examples are provided to illustrate the main results.

This paper is concerned with invasion entire solutions of a monostable time periodic Lotka-Volterra competition-diffusion system. We first give the asymptotic behaviors of time periodic traveling wave solutions at infinity by a dynamical approach coupled with the two-sided Laplace transform. According to these asymptotic behaviors, we then obtain some key estimates which are crucial for the construction of an appropriate pair of sub-super solutions. Finally, using the sub-super solutions method and comparison principle, we establish the existence of invasion entire solutions which behave as two periodic traveling fronts with different speeds propagating from both sides of x-axis. In other words, we formulate a new invasion way of the superior species to the inferior one in a time periodic environment.

This paper is concerned with traveling waves and entire solutions of one epidemic model with asymmetric dispersal kernel function arising from the spread of an epidemic by oral-faecal transmission. The asymmetry of the kernel function will have an influence on two aspects: (ⅰ) the minimal wave speed of traveling wave fronts may be nonpositive, but we give a new restrictive condition on the kernel function to guarantee it is positive; (ⅱ) the two traveling wave solutions with the same speed spreading from right and left of $x$-axis may be different in shape, which further makes that the entire solutions with five or four parameters may be asymmetric and the entire solutions with three parameters increasing in $x$ may be different from those decreasing in $x$ in shape. As for traveling wave solutions, we get the existence, asymptotic behavior and uniqueness of the two traveling wave solutions spreading from right and left of $x$-axis, respectively. We further construct three new entire solutions with five, four or three parameters. Two comparison principles also be established.

In this paper, we study several aspects related with solutions of nonlocal problems whose prototype is [Formula: see text] where we take, as the most important instance, [Formula: see text] with [Formula: see text] as well as [Formula: see text], [Formula: see text] is a smooth symmetric function with compact support and [Formula: see text] is either a bounded smooth subset of [Formula: see text], with nonlocal Dirichlet boundary condition, or [Formula: see text] itself. The results deal with existence, uniqueness, comparison principle and asymptotic behavior. Moreover, we prove that if the kernel is rescaled in a suitable way, the unique solution of the above problem converges to a solution of the deterministic Kardar–Parisi–Zhang equation.

Existence and uniqueness of rigidly rotating spiral waves by a wave front interaction model

By refining the standard Riccati substitution technique, integral averaging technique and comparison principle, we obtain new oscillation and asymptotic behavior for a class of third-order neutral differential equations with discrete and distributed delay. These criteria dealing with some cases have not been covered by the existing results in the literature. We present many sufficient conditions and related examples in order to illustrate the main results.