Stability in distribution of solutions (SDS) is important but challenging in stochastic population models with delays since the traditional methods are difficult to apply. This article focuses on a three-species stochastic delay prey-mesopredator-superpredator system and explores its SDS by a new approach. The new approach avoids the difficulties of some existing methods and can also be applied to investigate the SDS of other stochastic delay population models. The study reveals that the complete dynamic scenarios of SDS are characterized by three parameters eta 1 >eta 2 >eta 3: if eta 1 1 >eta 2 >eta 3, then the prey converges weakly to a unique ergodic invariant distribution (UEID) while both the mesopredator and the superpredator are extinct; if eta 1 >eta 2 > 1 >eta 3, then both the prey and the mesopredator converge weakly to a UEID while the superpredator are extinct; if eta 3 > 1, then the distributions of prey-mesopredator-superpredator converge weakly to a UEID.

This article studies the problem for parameter identification of nonlinear dynamical systems (i.e., the Hammerstein-Wiener systems) with additive coloured noises. Based on the gradient search and the key term separation, a generalized extended stochastic gradient (GESG) algorithm is given for estimating the system parameters. To improve the computational efficiency, a data filtering based GESG algorithm and a data filtering based multi-innovation GESG algorithm are derived by applying the data filtering technique and the multi-innovation identification theory. Moreover, the proposed algorithms are proved to be convergent under proper conditions. Finally, the simulation results verify the theoretical analysis.

In this article we investigate a parabolic-parabolic-elliptic two-species chemotaxis system with weak competition and show global asymptotic stability of the coexistence steady state under a smallness condition on the chemotactic strengths, which seems more natural than the condition previously known. For the proof we rely on the method of eventual comparison, which thereby is shown to be a useful tool even in the presence of chemotactic terms.

This work provides a method for the Finite Element implementation of a previously developed large deformation model for porous fibre-reinforced materials with statistically oriented fibres. The model, which includes a description of the elastic properties and the permeability, was implemented in the open source Finite Element package FEBio, and a number of different simulations were performed to verify the correctness and the robustness of the implementation. In a benchmark test, with statistical, anisotropic but homogenous fibre orientation, the simulations showed that the fluid flow along the dominant fibre direction is significantly larger than across the impermeable, reinforcing fibres, as suggested by previous experimental results and predicted by the theory. Furthermore, as an example of practical application, a realistic articular cartilage model, with statistical and depth-dependent fibre orientation was implemented, and an unconfined compression test was simulated. This could be regarded as the first full, realistic model of articular cartilage in which the effect of the statistically oriented collagen fibres was accounted for not only for the elastic properties, but also for the permeability.

Subdiffusion equations with distributed-order fractional derivatives describe some important physical phenomena. In this paper, we consider the time distributed-order and Riesz space fractional diffusions on bounded domains with Dirichlet boundary conditions. Here, the time derivative is defined as the distributed-order fractional derivative in the Caputo sense, and the space derivative is defined as the Riesz fractional derivative. First, we discretize the integral term in the time distributed-order and Riesz space fractional diffusions using numerical approximation. Then the given equation can be written as a multi-term time-space fractional diffusion. Secondly, we propose an implicit difference method for the multi-term time-space fractional diffusion. Thirdly, using mathematical induction, we prove the implicit difference method is unconditionally stable and convergent. Also, the solvability for our method is discussed. Finally, two numerical examples are given to show that the numerical results are in good agreement with our theoretical analysis.

This paper is devoted to the attraction-repulsion chemotaxis system with a logistic source: {u(t) = Delta u - chi del . (u del v) + mu del . (u del w) + R(u), x is an element of Omega, t > 0, rho v(t) = Delta v - alpha(1)v + beta(1)u, x is an element of Omega, t > 0, rho w(t) = Delta w - alpha(2)w + beta(2)u, x is an element of Omega, t > 0, where Omega subset of R-N(N >= 1) is a bounded domain with smooth boundary and R(s) 0, there exist global bounded classical solutions for any logistic damping tau >= 1. When the attraction dominates the repulsion in the sense that mu beta(2) - chi beta(1) 0, we will investigate the similar problem for N = 1 and N = 2. We will also study the regularity of stationary solutions.

In this paper, the spatial, temporal and spatiotemporal dynamics of a reaction-diffusion predator-prey system with mutual interference described by the Crowley-Martin-type functional response, under homogeneous Neumann boundary conditions, are studied. Preliminary analysis on the local asymptotic stability and Hopf bifurcation of the spatially homogeneous model based on ordinary differential equations is presented. For the reaction-diffusion model, firstly the invariance, uniform persistence and global asymptotic stability of the coexistence equilibrium are discussed. Then it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Next it is proved that the model exhibits Hopf bifurcation which produces temporal inhomogeneous patterns. Furthermore, at the points where the Turing instability curve and Hopf bifurcation curve intersect, it is demonstrated that the model undergoes Turing-Hopf bifurcation and exhibits spatiotemporal patterns. Finally, the existence and non-existence of positive non-constant steady states of the reaction-diffusion model are established. Numerical simulations are presented to verify and illustrate the theoretical results.

In this article, we investigate two free boundary problems for a Lotka-Volterra competition system in a higher space dimension with sign-changing coefficients. One may be viewed as describing how two competing species invade if they occupy an initial region; the other describes the dynamical process of a new competitor invading into the habitat of a native species. For simplicity, it is assumed that the environment is radially symmetric. The main purpose of this article is to understand the asymptotic behaviour of competing species spreading via a free boundary. We derive some sufficient conditions for species spreading success and spreading failure. Moreover, for the case of successful spreading, we provide the long-time behaviour of solutions to free boundary problems.

We consider localized bulging/necking in an inflated hyperelastic membrane tube with closed ends. We first show that the initiation pressure for the onset of localized bulging is simply the limiting pressure in uniform inflation when the axial force is held fixed. We then demonstrate analytically how, as inflation continues, the initial bulge grows continually in diameter until it reaches a critical size and then propagates in both directions. The bulging solution before propagation starts is of the solitary-wave type, whereas the propagating bulging solution is of the kink-wave type. The stability, with respect to axially symmetric perturbations, of both the solitary-wave type and the kink-wave type solutions is studied by computing the Evans function using the compound matrix method. It is found that when the inflation is pressure controlled, the Evans function has a single non-negative real root and this root tends to zero only when the initiation pressure or the propagation pressure is approached. Thus, the kink-wave type solution is probably stable but the solitary-wave type solution is definitely unstable.

In this paper, we study the global dynamics of an susceptible, vaccinated, infectious and recovered epidemiological model with infection-age structure. Biologically, we assume that effective contacts between vaccinated individuals and infectious individuals are less than that between susceptible individuals and infectious individuals. Using Lyapunov functions, we show that the global stability of each equilibrium is completely determined by the basic reproduction number R-0: if R-0 1, then there exists a unique endemic equilibrium which is globally asymptotically stable.

A new transform approach for solving mixed boundary value problems for the biharmonic equation in simply and multiply connected circular domains is presented. This work is a sequel to Crowdy (2015, IMA J. Appl. Math., 80, 1902-1931) where new transform techniques were developed for boundary value problems for Laplace's equation in circular domains. A circular domain is defined to be a domain, which can be simply or multiply connected, having boundaries that are a union of circular arc segments. The method provides a flexible approach to finding quasi-analytical solutions to a wide range of problems in fluid dynamics and plane elasticity. Three example problems involving slow viscous flows are solved in detail to illustrate how to apply the method; these concern flow towards a semicircular ridge, a translating and rotating cylinder near a wall as well as in a channel geometry.

Abstract In analytical and numerical studies on bubbles in liquids, often the Rayleigh initial condition of a spherical bubble at maximum radius is used: the Rayleigh case. This condition cannot be realized in practice, instead the bubbles need first to be generated and expanded. The energy-deposit case with its initial condition of a small, spherical bubble of high internal pressure that expands into water at atmospheric pressure is studied for comparison with the Rayleigh case. From the many possible configurations, a single bubble near a flat solid boundary is chosen as this is a basic configuration to study erosion and cleaning phenomena. The bubble contains a small amount of non-condensable gas obeying an adiabatic law. The water is compressible according to the Tait equation. The Euler equations in axial symmetry are solved with the help of the open source software package OpenFOAM, based on the finite volume method. The volume of fluid method is used for interface capturing. Rayleigh bubbles of $R_\mathrm{max} = 500\,\mu $m and energy-deposit bubbles that reach $R_\mathrm{max} = 500\,\mu $m after expansion in an unbounded liquid are compared with respect to microjet velocity, microjet impact pressure and microjet impact times, when placed or being generated near a flat solid boundary. Velocity and pressure fields from the impact zone are given to demonstrate the sequence of phenomena from axial liquid microjet impact via annular gas-jet and annular liquid-nanojet formation to the Blake splash and the first torus-bubble splitting. Normalized distances $D^{\ast } = D/R_\mathrm{max}$ (D = initial distance of the bubble centre from the boundary) between 1.02 and 1.5 are studied. Rayleigh bubbles show a stronger collapse with about 50% higher microjet impact velocities and also significantly higher microjet impact pressures.