yong Xiao

Efficient single-step preconditioned HSS iteration methods for complex symmetric linear systems

Abstract We propose a single-step preconditioned variant of HSS (SPHSS) and an efficient parameterized SPHSS (PSPHSS) iteration method for solving a class of complex symmetric linear systems. Under suitable conditions, we analyze the convergence properties of the SPHSS and PSPHSS iteration methods. Theoretical analysis shows that the minimal upper bounds for the spectral radius of the SPHSS and PSPHSS iteration matrices are less than those of the SHSS and PSHSS iteration matrices when using the optimal parameters, respectively. Numerical results show that the PSPHSS iteration method has comparable advantage over several other iteration methods whether the experimental optimal parameters are used or not.

Efficient parameterized HSS iteration methods for complex symmetric linear systems

Abstract Based on the new HSS (NHSS) iteration method proposed by Pour and Goughery (2015) and the efficient PSHSS iteration method by Zeng and Ma (2016), we introduce an efficient parameterized HSS (PNHSS) and a parameterized single-step HSS (PS*HSS) iteration methods for solving a class of complex symmetric linear systems. Convergence properties of the PNHSS and the PS*HSS iteration methods are studied, which show that the iterative sequences are convergent to the unique solution of the linear system for any initial guess under a loose restriction on the parameter ω . Furthermore, we derive an upper bound for the spectral radius of the PNHSS iteration matrix, and the quasi-optimal parameters α ∗ and ω ∗ which minimize the above upper bound are also considered. Both theoretical and numerical results show that the PNHSS and the PS*HSS iteration methods outperform the NHSS and the SHSS iteration methods. Little difference about the computational efficiency from the point of view of the CPU times between the PS*HSS, the PNHSS and the PSHSS iteration methods is justified by using the experimental optimal parameters. However, sometimes the PS*HSS and the PNHSS iteration methods are more efficient than the PSHSS iteration method when the experimental optimal parameters are not used.

Efficient preconditioned NHSS iteration methods for solving complex symmetric linear systems

Abstract Based on the new HSS (NHSS) iteration method introduced by Pour and Goughery (2015), we propose a preconditioned variant of NHSS (P*NHSS) and an efficient parameterized P*NHSS (PPNHSS) iteration methods for solving a class of complex symmetric linear systems. The convergence properties of the P*NHSS and the PPNHSS iteration methods show that the iterative sequences are convergent to the unique solution of the linear system for any initial guess when the parameters are properly chosen. Moreover, we discuss the quasi-optimal parameters which minimize the upper bounds for the spectral radius of the iteration matrices. Numerical results show that the PPNHSS iteration method is superior to several iteration methods whether the experimental optimal parameters are used or not.

Pattern analysis of a modified Leslie–Gower predator–prey model with Crowley–Martin functional response and diffusion

Abstract In this paper, we study a modified Leslie–Gower predator–prey model with Crowley–Martin functional response and spatial diffusion under homogeneous Neumann boundary condition, and obtain some important qualitative properties, including the existence of the global positive solution, the dissipation and persistence of the two species, the local and global asymptotic stability of the constant equilibria, and Hopf bifurcation around the interior constant equilibrium. In addition, we establish the existence and nonexistence of nonconstant positive steady states.

pth moment stability in stochastic neutral Volterra-Levin equation with Lévy noise and variable delays

In this paper, we study a class of stochastic neutral Volterra-Levin equations which are equipped with Lévy noise and variable delay and we obtain p th moment exponential stability. Some well-known results are improved and generalized.

Turing Patterns in a Predator-Prey System with Self-Diffusion

For a predator-prey system, cross-diffusion has been confirmed to emerge Turing patterns. However, in the real world, the tendency for prey and predators moving along the direction of lower density of their own species, called self-diffusion, should be considered. For this, we investigate Turing instability for a predator-prey system with nonlinear diffusion terms including the normal diffusion, cross-diffusion, and self-diffusion. A sufficient condition of Turing instability for this system is obtained by analyzing the linear stability of spatial homogeneous equilibrium state of this model. A series of numerical simulations reveal Turing parameter regions of the interaction of diffusion parameters. According to these regions, we further demonstrate dispersion relations and spatial patterns. Our results indicate that self-diffusion plays an important role in the spatial patterns.

A simple and efficient method with high order convergence for solving systems of nonlinear equations

We propose an m + 1 -step modified Newton method of convergence order m + 2 to solve systems of nonlinear equations which are third Frechet differentiable in a convex set containing the zero. Computational efficiency in the general form for a positive integer m is discussed, which shows that the efficiency increases with m when applied to large systems. Moreover, a comparison between the efficiency of this technique and some existing efficient methods is made, which implies that the present method is more efficient particularly for solving large systems of equations. Theoretical results about order of convergence and computational efficiency are largely verified in numerical examples.

Mathematical analysis on a multidimensional model of morphogen transport with receptor synthesis

In biological development, morphogens are locally produced and spread to other regions in organs, forming gradients that control the inter-related pattern and growth of developing organs. Mechanisms of morphogen transport were built and investigated by numerical simulations in [A. D. Lander, Q. Nie and F. Y. M. Wan, Do morphogen gradients arise by diffusion? Developmental Cell2 (2002) 785–796]. In that paper, model C, which considers endocytosis, exocytosis and receptor synthesis and degradation, is in a one-dimensional spatial region and couples a partial differential equation with ordinary differential equations. Here, this model is promoted to an arbitrary dimension bounded region. We prove existence, uniqueness and non-negativity of a global solution for this advanced model, of its steady-state solution and linear stability of steady state by operator semigroup, the Schauder theorem and local perturbation method. Our results improve previous results for this model in a one dimension region.

STABILITY OF REGULATORY PROTEIN GRADIENTS INDUCED BY MORPHOGEN DPP IN DROSOPHILA WING DISC

In the development of Drosophila wing disc, morphogen Dpp, which is a signaling molecule from a local region and disperses into anterior and posterior compartments, builds up a gradient with precise pattern information. Experiments have demonstrated that the key genes (brk, dad, omb and sal) and phosphorylated protein (pMad), which are activated by Dpp signaling molecules and form the gradients of the corresponding proteins of these genes, direct and control the spatial pattern of the wing disc. However, the regulatory network of these genes are in complex and nonlinear interaction with upstream regulators and downstream targets. In this paper, the mathematical model is built according to the regulatory relationships of these key genes. The stabilities of the gradients of these corresponding proteins are investigated. Furthermore, numerical simulations show that these gradients are robust with respect to some major reaction rates in this regulatory network.

Analysis of a Lévy-diffusion Leslie-Gower predator-prey model with nonmonotonic functional response

In this paper, a Levy-diffusion Leslie-Gower predator-prey model with a nonmonotonic functional response is studied. We show the existence, uniqueness and attractiveness of the globally positive solution to this model. Moreover, to its corresponding steady-state model, we obtain the stability of the semi-trivial solutions, the existence and nonexistence of coexistence states by the method of topological degree, the uniqueness and stability of coexistence state, and the multiplicity and stability of coexistence states by Grandall-Rabinowitz bifurcation theorem. In addition, to get these results, we study the property of the Levy diffusion operator, and give out the comparison principle of the generalized parabolic Levy-diffusion differential equation, as well as the existence and stability of the solution for the steady-state Logistic equation with Levy diffusion. Furthermore, we obtain the comparison principle of the steady-state Levy-diffusion equation. As far as we know, these results are new in the ecological model.