Global dynamics of Filippov-type plant disease models with an interaction ratio threshold

A Filippov-type plant disease model is developed by introducing a interaction ratio threshold, the number of susceptible plants infected by per diseased plant, which determines whether control measures including replanting or roguing are carried out. The main purpose of this paper is to give a completely qualitative analysis of the model. By employing Poincaré maps, our analysis reveals rich dynamics including a global attractor bounded by a touching closed orbit, which is convergent in finite time from its outside, a global attractor bounded by two touching closed orbits and a pseudo-saddle, and a globally asymptotically stable pseudo-node. Moreover, we give biological implications of our results in implementing control strategies for plant diseases.     

Dynamical analysis of chemotherapy models with time-dependent infusion

Classical mathematical models for chemotherapy assume a constant infusion rate of the chemotherapy agent. However in reality the infusion rate usually varies with respect to time, due to the natural (temporal or random) fluctuation of environments or clinical needs. In this work we study a non-autonomous chemotherapy model where the injection rate and injection concentration of the chemotherapy agent are time-dependent. In particular, we prove that the non-autonomous dynamical system generated by solutions to the non-autonomous chemotherapy system possesses a pullback attractor. In addition, we investigate the detailed interior structures of the pullback attractor to provide crucial information on the effectiveness of the treatment. The main analytical tool used is the theory of non-autonomous dynamical systems. Numerical experiments are carried out to supplement the analysis and illustrate the effectiveness of different types of infusions.     

Buckling and nonlinear dynamics of elastically coupled double-beam systems

This paper deals with damped transverse vibrations of elastically coupled double-beam system under even compressive axial loading. Each beam is assumed to be elastic, extensible and supported at the ends. The related stationary problem is proved to admit both unimodal (only one eigenfunction is involved) and bimodal (two eigenfunctions are involved) buckled solutions, and their number depends on structural parameters and applied axial loads. The occurrence of a so complex structure of the steady states motivates a global analysis of the longtime dynamics. In this regard, we are able to prove the existence of a global regular attractor of solutions. When a finite set of stationary solutions occurs, it consists of the unstable manifolds connecting them.     

Anisotropic strange quark star in Finch-Skea geometry and its maximum mass for non-zero strange quark mass (ms ≠ 0)

A class of relativistic astrophysical compact objects is analyzed in the modified Finch-Skea geometry described by the MIT bag model equation of state of interior matter, \begin{document}$ p=\dfrac{1}{3}\left(\rho-4B\right) $\end{document}, where B is known as the bag constant. B plays an important role in determining the physical features and structure of strange stars. We consider the finite mass of the strange quark (\begin{document}$ m_{s} \neq 0 $\end{document}) and study its effects on the stability of quark matter inside a star. We note that the inclusion of strange quark mass affects the gross properties of the stellar configuration, such as maximum mass, surface red-shift, and the radius of strange quark stars. To apply our model physically, we consider three compact objects, namely, (i) VELA X-1, (ii) 4U 1820-30, and (iii) PSR J 1903+327, which are thought to be strange stars. The range of B is restricted from 57.55 to \begin{document}$B_{\rm stable}$\end{document} (\begin{document}$\rm MeV/fm^{3}$\end{document}), for which strange matter might be stable relative to iron (\begin{document}$^{56}{\rm Fe}$\end{document}). However, we also observe that metastable and unstable strange matter depend on B and \begin{document}$ m_{s} $\end{document}. All energy conditions hold well in this approach. Stability in terms of the Lagrangian perturbation of radial pressure is studied in this paper.     

Asymptotic behavior of a generalized Cahn–Hilliard equation with a mass source

We consider in this article a generalized Cahn–Hilliard equation with mass source (nonlinear reaction term) which has applications in biology. We are interested in the well-posedness and the study of the asymptotic behavior of the solutions (and, more precisely, the existence of finite-dimensional attractors). We first consider the usual Dirichlet boundary conditions and then Neumann boundary conditions. The latter require additional assumptions on the mass source term to obtain the dissipativity. Indeed, otherwise, the order parameter u can blow up in finite time. We also give numerical simulations which confirm the theoretical results.     

Existence and estimation of the Hausdorff dimension of attractors for an epidemic model

We prove the existence of pullback and uniform attractors for the process associated to a non‐autonomous SIR model, with several types of non‐autonomous features. The Hausdorff dimension of the pullback attractor is also estimated. We illustrate some examples of pullback attractors by numerical simulations. Copyright © 2016 John Wiley & Sons, Ltd.     

Two types of upper semi‐continuity of bi‐spatial attractors for non‐autonomous stochastic p‐Laplacian equations on

We consider the long time behavior of solutions for the non‐autonomous stochastic p‐Laplacian equation with additive noise on an unbounded domain. First, we show the existence of a unique ‐pullback attractor, where q is related to the order of the nonlinearity. The main difficulty existed here is to prove the asymptotic compactness of systems in both spaces, because the Laplacian operator is nonlinear and additive noise is considered. We overcome these obstacles by applying the compactness of solutions inside a ball, a truncation method and some new techniques of estimates involving the Laplacian operator. Next, we establish the upper semi‐continuity of attractors at any intensity of noise under the topology of . Finally, we prove this continuity of attractors from domains in the norm of , which improves an early result by Bates et al.(2001) who studied such continuity when the deterministic lattice equations were approached by finite‐dimensional systems, and also complements Li et al. (2015) who discussed this approximation when the nonlinearity f(·,0) had a compact support. Copyright © 2017 John Wiley & Sons, Ltd.     

Discretization of Asymptotically Stable Stationary Solutions of Delay Differential Equations with a Random Stationary Delay

We prove the existence of a stationary random solution to a delay random ordinary differential system, which attracts all other solutions in both pullback and forwards senses. The equation contains a one-sided dissipative Lipschitz term without delay, while the random delay appears in a globally Lipschitz one. The delay function only needs to be continuous in time. Moreover, we also prove that the split implicit Euler scheme associated to the random delay differential system generates a discrete time random dynamical system, which also possesses a stochastic stationary solution with the same attracting property, and which converges to the stationary solution of the delay random differential equation pathwise as the stepsize goes to zero.