This paper is devoted to study of time-fractional elliptic equations driven by a multiplicative noise. By combining the eigenfunction expansion method for symmetry elliptic operators, the variation of constant formula for strong solutions to scalar stochastic fractional differential equations, Ito's formula and establishing a new weighted norm associated with a Lyapunov–Perron operator defined from

This paper is concerned with a numerical solution of the acoustic scattering by a bounded impenetrable obstacle in three dimensions. The obstacle scattering problem is formulated as a boundary value problem in a bounded domain by using a Dirichlet-to-Neumann (DtN) operator. An a posteriori error estimate is derived for the finite element method with the truncated DtN operator. The a posteriori error

In this paper, we present and study $ C^1 $ Petrov-Galerkin and Gauss collocation methods with arbitrary polynomial degree $ k $ ($ \ge 3 $) for one-dimen\-sional elliptic equations. We prove that, the solution and its derivative approximations converge with rate $ 2k-2 $ at all grid points; and the solution approximation is superconvergent at all interior roots of a special Jacobi polynomial of degree

We present a spatial food chain model on a bounded domain coupled with optimal control theory to examine harvesting strategies. Motivated by the fishery industry in the Black Sea, the anchovy stock and two more trophic levels are modeled using nonlinear parabolic partial differential equations with logistic growth, movement by diffusion and advection, and Neumann boundary conditions. Necessary conditions

In this paper, we analyze the interaction of localized patterns such as traveling wave solutions for reaction-diffusion systems with nonlocal effect in one space dimension. We consider the case that a nonlocal effect is given by the convolution with a suitable integral kernel. At first, we deduce the equation describing the movement of interacting localized patterns in a mathematically rigorous way

To underpin the concern that environmental change can flip an ecosystem from stable persistence to sudden total collapse, we consider a class of so-called ecoepidemic models, predator – prey/host – parasite systems, in which a base species is prey to a predator species and host to a micro-parasite species. Our model uses generalized frequency-dependent incidence for the disease transmission and mass

A complete Lyapunov function characterizes the behaviour of a general discrete-time dynamical system. In particular, it divides the state space into the chain-recurrent set where the complete Lyapunov function is constant along trajectories and the part where the flow is gradient-like and the complete Lyapunov function is strictly decreasing along solutions. Moreover, the level sets of a complete Lyapunov

The coevolution or coexistence of multiple viruses with multiple hosts has been an important issue in viral ecology. This paper is to study the mathematical properties of the solutions of a chemostat model for two host species and two virus species. By virtue of the global dynamics of its submodels and the theories of uniform persistence and Hopf bifurcation, we derive sufficient conditions for the

This paper is aimed to provide a new theory for the formation of the solar surface eruptions and sunspots. The key ingredient of the study is the new anti-diffusive effect of heat, based on the recently developed statistical theory of heat by the authors [4]. The anti-diffusive effect of heat states that due to the higher rate of photon absorption and emission of the particles with higher energy levels

In 1993, Holt and Lawton introduced a stochastic model of two host species parasitized by a common parasitoid species. We introduce and analyze a generalization of these stochastic difference equations with any number of host species, stochastically varying parasitism rates, stochastically varying host intrinsic fitnesses, and stochastic immigration of parasitoids. Despite the lack of direct, host

We present a review of the different techniques available to study a special kind of fractal basins of attraction known as Wada basins, which have the intriguing property of having a single boundary separating three or more basins. We expose several approaches to identify this topological property that rely on different, but not exclusive, definitions of the Wada property.

We consider the ecological and evolutionary dynamics of a reaction-diffusion-advection model for populations residing in a one-dimensional advective homogeneous environment, with emphasis on the effects of boundary conditions and domain size. We assume that there is a net loss of individuals at the downstream end with rate $ b \geq 0 $, while the no-flux condition is imposed on the upstream end. For

We provide a short review of existing models with multiple taxis performed by (at least) one species and consider a new mathematical model for tumor invasion featuring two mutually exclusive cell phenotypes (migrating and proliferating). The migrating cells perform nonlinear diffusion and two types of taxis in response to non-diffusing cues: away from proliferating cells and up the gradient of surrounding

In this paper, we review some recent results on the nonlinear dynamics of delayed differential equation models describing the interaction between tumor cells and effector cells of the immune system, in which the delays represent times necessary for molecule production, proliferation, differentiation of cells, transport, etc. First we consider a tumor-immune system interaction model with a single delay

In this paper, we discretize and analyze a stoichiometric optimal foraging model where the grazer's feeding effort depends on the producer's nutrient quality. We systematically make comparisons of the dynamical behaviors between the discrete-time model and the continuous-time model to study the robustness of model predictions to time discretization. When the maximum growth rate of producer is low,

A discrete delay is included to model the time between the capture of the prey and its conversion to viable biomass in the simplest classical Gause type predator-prey model that has equilibrium dynamics without delay. As the delay increases from zero, the coexistence equilibrium undergoes a supercritical Hopf bifurcation, two saddle-node bifurcations of limit cycles, and a cascade of period doublings

Viral dynamics within plant hosts can be important for understanding plant disease prevalence and impacts. However, few mathematical modeling efforts aim to characterize within-plant viral dynamics. In this paper, we derive a simple system of delay differential equations that describes the spread of infection throughout the plant by barley and cereal yellow dwarf viruses via the cell-to-cell mechanism

In the study of equilibrium solutions for partial differential equations there are so many equilibria that one cannot hope to find them all. Therefore one usually concentrates on finding individual branches of equilibrium solutions. On the one hand, a rigorous theoretical understanding of these branches is ideal but not generally tractable. On the other hand, numerical bifurcation searches are useful

We develop in this paper efficient and accurate numerical schemes based on the scalar auxiliary variable (SAV) approach for the generalized Zakharov system and generalized vector Zakharov system. These schemes are second-order in time, linear, unconditionally stable, only require solving linear systems with constant coefficients at each time step, and preserve exactly a modified Hamiltonian. Ample

A general model of incompatible small-strain elasticity is presented and analyzed, based on the linearized strain and its associated incompatibility tensor field. Strain incompatibility accounts for the presence of dislocations, whose motion is ultimately responsible for the plastic behaviour of solids. The specific functional setting is built up, on which existence results are proved. Our solution

We consider a recently introduced formulation for fluid-structure interaction problems which makes use of a distributed Lagrange multiplier in the spirit of the fictitious domain method. In this paper we focus on time integration methods of second order based on backward differentiation formulae and on the Crank–Nicolson method. We show the stability properties of the resulting method; numerical tests

We consider the conductivity equation in a bounded domain in $ \mathbb{R}^{d} $ with $ d\geq3 $. In this study, the medium corresponds to a very contrasted two phase homogeneous and isotropic material, consisting of a unit matrix phase, and an inclusion with high conductivity. The geometry of the inclusion phase is so that the resulting Jacobian determinant of the gradients of solutions $ DU $ takes

We consider the d-dimensional Boussinesq system defined on a sufficiently smooth bounded domain, with homogeneous boundary conditions, and subject to external sources, assumed to cause instability. The initial conditions for both fluid and heat equations are taken of low regularity. We then seek to uniformly stabilize such Boussinesq system in the vicinity of an unstable equilibrium pair, in the critical

We study local bifurcations of periodic solutions to time-periodic (systems of) integrodifference equations over compact habitats. Such infinite-dimensional discrete dynamical systems arise in theoretical ecology as models to describe the spatial dispersal of species having nonoverlapping generations. Our explicit criteria allow us to identify branchings of fold- and crossing curve-type, which include

In the article, some bilinear evolution equations in Hilbert space driven by paths of low regularity are considered and solved explicitly. The driving paths are scalar-valued and continuous, and they are assumed to have a finite $ p $-th variation along a sequence of partitions in the sense given by Cont and Perkowski [Trans. Amer. Math. Soc. Ser. B, 6 (2019) 161–186] ($ p $ being an even positive

We review and discuss various aspects that the modeling of pharmacometric properties has on the structure of optimal solutions in mathematical models for cancer treatment. These include (i) the changes in the interpretation of the solutions as pharmacokinetic (PK) models are added, respectively deleted from the modeling and (ii) qualitative changes in the structures of optimal controls that occur as

In this paper, we consider a class of second-order Hamiltonian systems of the form $ \ddot{u}(t)-L(t) u(t)+\nabla W(t,u(t)) = 0 $

In this paper we study global dynamic aspects of the quadratic system $ \dot x = yz,\quad \dot y = x-y,\quad \dot z = 1-x(\alpha y+\beta x), $

In this paper, an anisotropic bilinear finite element method is constructed for the elliptic boundary layer optimal control problems. Supercloseness properties of the numerical state and numerical adjoint state in a $ \epsilon $-norm are established on anisotropic meshes. Moreover, an interpolation type post-processed solution is shown to be superconvergent of order $ O(N^{-2}) $, where the total number

In this paper, we study some existence results for fractional differential equations subject to some kind of initial conditions. First, we focus on the linear problem and we give an explicit form of solutions by reduction to an integral problem. We analyze some properties of the solutions to the linear problem in terms of its coefficients. Then we provide examples of application of the mentioned properties

The asymptotic behavior of a model for 2D incompressible stochastic micropolar fluid flows with rough noise on a Poincaré domain is investigated. First, the existence and uniqueness of solutions to an evolution equation arising from the underlying stochastic micropolar fluid model is established via the Galerkin method and energy method. Then the existence of a random attractor is studied by using

In this work we estimate the convergence rate for time stepping schemes applied to nonlocal dynamic fracture modeling. Here we use the nonlocal formulation given by the bond based peridynamic equation of motion. We begin by establishing the existence of $ H^2 $ peridynamic solutions over any finite time interval. For this model the gradients can become large and steep slopes appear and localize when

A van der Pol damped SD oscillator, which was proposed by Ruilan Tian, Qingjie Cao and Shaopu Yang (2010, Nonlinear Dynamics, 59, 19-27), is studied. By improving the criterion function of determining the lowest upper bound of the number of zeros of Abelian Integrals, we show that the number of zeros of Abelian integrals of this SD oscillator is two which is sharp.

In this paper, we consider the initial-boundary value problem to the non-isothermal incompressible liquid crystal system with both variable density and temperature. Global well-posedness of strong solutions is established for initial data being small perturbation around the equilibrium state. As the tools in the proof, we establish the maximal regularities of the linear Stokes equations and parabolic

A reaction-diffusion predator-prey system with prey-taxis and predator-taxis describes the spatial interaction and random movement of predator and prey species, as well as the spatial movement of predators pursuing prey and prey evading predators. The spatial pattern formation induced by the prey-taxis and predator-taxis is characterized by the Turing type linear instability of homogeneous state and

This paper deals with the 3D incompressible Navier-Stokes equations with density-dependent viscosity in the whole space. The global well-posedness and exponential decay of strong solutions is established in the vacuum cases, provided the assumption that the bound of density is suitably small, which extends the results of [Nonlinear Anal. Real World Appl., 46:58-81, 2019] to the global one. However

In this paper we study the rate of convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria. We first analyze a three-species system with boundary equilibria in some stoichiometric classes, and whose right hand side is bounded above by a quadratic nonlinearity in the positive orthant. We prove similar results on the convergence to the positive

We solve a conjecture on multiple nondegenerate steady states, and prove bistability for sequestration networks. More specifically, we prove that for any odd number of species, and for any production factor, the fully open extension of a sequestration network admits three nondegenerate positive steady states, two of which are locally asymptotically stable. In addition, we provide a non-empty open set

We investigate the main phase space properties of the QR-flow when restricted to upper Hessenberg matrices. A complete description of the linear behavior of the equilibrium matrices is given. The main result classifies the possible $ \alpha $- and $ \omega $-limits of the orbits for this system. Furthermore, we characterize the set of initial matrices for which there is convergence towards an equilibrium

A new explicit Milstein-type scheme for SDE driven by Lévy noise is proposed where both drift and diffusion coefficients are allowed to grow super-linearly. The strong rate of convergence (in $ \mathcal{L}^2 $-sense) is shown to be arbitrarily close to one which is consistent with the corresponding result on the classical Milstein scheme obtained for coefficients satisfying global Lipschitz conditions

In this article we generalize the convolution quadrature (CQ) method, which aims at approximating the fractional calculus, to the case for the distributed order calculus. Our method is a natural expansion that the approximation formulas, convergence results and correction technique reduce to the cases for the CQ method if the weight function $ \mu(\alpha) $ is defined by $ \delta(\alpha-\alpha_0) $

In this paper we consider a selection-mutation model with an advection term formulated on the space of finite signed measures on $ \mathbb{R}^d $. The selection-mutation kernel is described by a family of measures which allows the study of continuous and discrete kernels under the same setting. We rescale the selection-mutation kernel to obtain a diffusively rescaled selection-mutation model. We prove

In the last years, several authors studied a class of continuous-time semi-Markov processes obtained by time-changing Markov processes by hitting times of independent subordinators. Such processes are governed by integro-differential convolution equations of generalized fractional type. The aim of this paper is to develop a discrete-time counterpart of such a theory and to show relationships and differences

This paper investigates the Cauchy problem to a class of stochastic non-autonomous evolution equations of parabolic type governed by noncompact evolution families in Hilbert spaces. Combining the theory of evolution families, the fixed point theorem with respect to convex-power condensing operator and a new estimation technique of the measure of noncompactness, we established some new existence results

For FitzHugh-Nagumo lattice dynamical systems (LDSs) many authors studied the existence of global attractors for deterministic systems [4,34,41,43] and the existence of global random attractors for stochastic systems [23,24,27,48,49], where for non-autonomous cases, the nonlinear parts are considered of the form $ f\left( u\right) $. Here we study the existence of the uniform global attractor for a

We study a terminal value parabolic system with nonlinear-nonlocal diffusions. Firstly, we consider the issue of existence and ill-posed property of a solution. Then we introduce two regularization methods to solve the system in which the diffusion coefficients are globally Lipschitz or locally Lipschitz under some a priori assumptions on the sought solutions. The existence, uniqueness and regularity

The current paper is devoted to the stochastic damped Ostrovsky equation driven by white noise. By establishing the uniform estimates for the solution in $ H^1 $ norm, we prove the global well-posedness and the existence of invariant measure for stochastic damped Ostrovsky equation with random initial value. Moreover, we obtain the ergodicity of stochastic damped Ostrovsky equation with deterministic

The question of whether and how two competing consumers can coexist on a single limiting resource has a long tradition in ecological theory. We build on a recent seasonal (hybrid) model for one consumer and one resource, and we extend it by introducing a second consumer. Consumers reproduce only once per year, the resource reproduces throughout the"summer" season. When we use linear consumer reproduction

We study the well-posedness for the mildly compressible Navier-Stokes-Cahn-Hilliard system with non-constant viscosity and Landau potential in two and three dimensional domains.

Searching saddle points on the potential energy surface is a challenging problem in the rare event. When there exist multiple saddle points, sampling different initial guesses are needed in local search methods in order to find distinct saddle points. In this paper, we present a novel global optimization-based dimer method (GOD) to efficiently search saddle points by coupling ant colony optimization

We consider three matrix models of order 2 with one random entry $ \epsilon $ and the other three entries being deterministic. In the first model, we let $ \epsilon\sim \rm{Bernoulli}\left(\frac{1}{2}\right) $. For this model we develop a new technique to obtain estimates for the top Lyapunov exponent in terms of a multi-level recursion involving Fibonacci-like sequences. This in turn gives a new characterization

This work aims to prove the small time large deviation principle (LDP) for a class of stochastic partial differential equations (SPDEs) with locally monotone coefficients in generalized variational framework. The main result could be applied to demonstrate the small time LDP for various quasilinear and semilinear SPDEs such as stochastic porous medium equations, stochastic $ p $-Laplace equations,

In this paper, we propose a new weak solution to an optimal stopping problem in finance and economics. The main advantage of this new definition is that we do not need the Dynamic Programming Principle, which is critical for both classical verification argument and modern viscosity approach. Additionally, the classical methods in differential equations, e.g. penalty method, can be used to derive some

We consider a force-based model for cell motion which models cell forces using Hooke's law and a random outreach from the cell center. In previous work this model was simplified to track the centroid by setting the relaxation time to zero, and a formula for the expected velocity of the centroid was derived. Here we extend that formula to allow for chemotaxis of the cell by allowing the outreach distribution

In this paper we focus on the computation of periodic solutions of Delay Differential Equations (DDEs) with constant delays. The method is based on defining a Poincaré section in a suitable functional space and looking for a fixed point of the flow in this section. This is done by applying a Newton method on a suitable discretisation of the section. To avoid computing and storing large matrices we

This paper proposes an efficient computational framework for longitudinal velocity control of a large number of autonomous vehicles (AVs) and develops a traffic flow theory for AVs. Instead of hypothesizing explicitly how AVs drive, our goal is to design future AVs as rational, utility-optimizing agents that continuously select optimal velocity over a period of planning horizon. With a large number

The ensemble method has been developed for accelerating a sequence of numerical simulations of evolution problems. Its main idea is, by manipulating the time stepping and grouping discrete problems, to make all members in the same group share a common coefficient matrix. Thus, at each time step, instead of solving a sequence of linear systems each of which contains only one right-hand-side vector,

In this paper we present a null controllability result for a degenerate semilinear parabolic equation with first order terms. The main result is obtained after the proof of a new Carleman inequality for a degenerate linear parabolic equation with first order terms.

The classical work by Zwanzig [J. Stat. Phys. 9 (1973) 215-220] derived Langevin dynamics from a Hamiltonian system of a heavy particle coupled to a heat bath. This work extends Zwanzig's model to a quantum system and formulates a more general coupling between a particle system and a heat bath. The main result proves, for a particular heat bath model, that ab initio Langevin molecular dynamics, with

An ergodic control problem is studied for controlled linear stochastic equations driven by cylindrical Lévy noise with unbounded control operator in a Hilbert space. A family of optimal controls is shown to consist of those asymptotically achieving the feedback form that employs the corresponding Riccati equation. The formula for optimal cost is given. The general results are applied to stochastic