Turing patterns and spatiotemporal patterns in a tritrophic food chain model with diffusion

In this paper, the temporal, spatial, and spatiotemporal patterns of a tritrophic food chain reaction–diffusion model with Holling type II functional response are studied. Firstly, for the model with or without diffusion, we perform a detailed stability and Hopf bifurcation analysis and derive criteria for determining the direction and stability of the bifurcation by the center manifold and normal form theory. Moreover, diffusion-driven Turing instability occurs, which induces spatial inhomogeneous patterns for the reaction–diffusion model. Then, the existence of positive non-constant steady-states of the reaction–diffusion model is established by the Leray–Schauder degree theory and some a priori estimates. Finally, numerical simulations are presented to visualize the complex dynamic behavior.     

Spatiotemporal patterns induced by delay and cross-fractional diffusion in a predator-prey model describing intraguild predation

In this article, we study a reaction-diffusion predator-prey model that describes intraguild predation. We mainly consider the effects of time delay and cross-fractional diffusion on dynamical behavior. By using delay as the bifurcation parameter, we perform a detailed Hopf bifurcation analysis and derive the algorithm for determining the direction and stability of the bifurcating periodic solutions. We also demonstrate that proper cross-fractional diffusion can induce Turing pattern, and the smaller the order of fractional diffusion is, the more easily Turing pattern is able to occur.     

Stability and Hopf bifurcation of a diffusive predator‐prey model with hyperbolic mortality

The dynamics of a reaction‐diffusion predator‐prey model with hyperbolic mortality and Holling type II response effect is considered. The stability of the positive equilibrium and the existence of Hopf bifurcation are investigated by analyzing the distribution of eigenvalues without diffusion. We also study the spatially homogeneous and nonhomogeneous periodic solutions through all parameters of the system which are spatially homogeneous. To verify our theoretical results, some numerical simulations are also presented. © 2015 Wiley Periodicals, Inc. Complexity 21: 34–43, 2016     

Global solutions with shock waves to the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws II

This work is a continuation of our previous work. In the present paper, we study the existence and uniqueness of global piecewise C¹ solutions with shock waves to the generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws with linear damping in the presence of a boundary. It is shown that the generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws with linear damping with nonlinear boundary conditions in the half space {(t, x) | t ≥ 0, x ≥ 0} admits a unique global piecewise C¹ solution u = u (t, x) containing only shock waves with small amplitude and this solution possesses a global structure similar to that of a self‐similar solution u = U (x /t) of the corresponding homogeneous Riemann problem, if each characteristic field with positive velocity is genuinely nonlinear and the corresponding homogeneous Riemann problem has only shock waves but no rarefaction waves and contact discontinuities. This result is also applied to shock reflection for the flow equations of a model class of fluids with viscosity induced by fading memory. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Compact difference scheme for two-dimensional fourth-order nonlinear hyperbolic equation

High-order compact finite difference method for solving the two-dimensional fourth-order nonlinear hyperbolic equation is considered in this article. In order to design an implicit compact finite difference scheme, the fourth-order equation is written as a system of two second-order equations by introducing the second-order spatial derivative as a new variable. The second-order spatial derivatives are approximated by the compact finite difference operators to obtain a fourth-order convergence. As well as, the second-order time derivative is approximated by the central difference method. Then, existence and uniqueness of numerical solution is given. The stability and convergence of the compact finite difference scheme are proved by the energy method. Numerical results are provided to verify the accuracy and efficiency of this scheme.     

Existence of solutions for linear hyperbolic initial-boundary value problems in a rectangle

In a recent article, we achieved the well-posedness of linear hyperbolic initial and boundary value problems (IBVP) in a rectangle via semigroup method, and we found that there are only two elementary modes called hyperbolic and elliptic modes in the system. It seems that, there is only one set of boundary conditions for the hyperbolic mode, while there are infinitely many sets of boundary conditions for the elliptic mode, which can lead to well-posedness. In this article, we continue to consider linear hyperbolic IBVP in a rectangle in the constant coefficients case and we show that there are also infinitely many sets of boundary conditions for hyperbolic mode which will lead to the existence of a solution. We also have uniqueness in some special cases. The boundary conditions satisfy the reflection conditions introduced in Section 3, which turn out to be equivalent to the strictly dissipative conditions.     

Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi‐symmetric vessels

In this paper, we present a mathematical analysis of the quasilinear effects arising in a hyperbolic system of partial differential equations modelling blood flow through large compliant vessels. The equations are derived using asymptotic reduction of the incompressible Navier–Stokes equations in narrow, long channels. To guarantee strict hyperbolicity we first derive the estimates on the initial and boundary data which imply strict hyperbolicity in the region of smooth flow. We then prove a general theorem which provides conditions under which an initial–boundary value problem for a quasilinear hyperbolic system admits a smooth solution. Using this result we show that pulsatile flow boundary data always give rise to shock formation (high gradients in the velocity and inner vessel radius). We estimate the time and the location of the first shock formation and show that in a healthy individual, shocks form well outside the physiologically interesting region (2.8m downstream from the inlet boundary). In the end we present a study of the influence of vessel tapering on shock formation. We obtain a surprising result: vessel tapering postpones shock formation. We provide an explanation for why this is the case. Copyright © 2003 John Wiley & Sons, Ltd.     

Turing instability and Hopf bifurcation in a diffusive Leslie–Gower predator–prey model

In this paper, a reaction‐diffusion predator–prey system that incorporates the Holling‐type II and a modified Leslie‐Gower functional responses is considered. For ODE, the local stability of the positive equilibrium is investigated and the specific conditions are obtained. For partial differential equation, we consider the dissipation and persistence of solutions, the Turing instability of the equilibrium solutions, and the Hopf bifurcation. By calculating the normal form, we derive the formulae, which can determine the direction and the stability of Hopf bifurcation according to the original parameters of the system. We also use some numerical simulations to illustrate our theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Delay induced subcritical Hopf bifurcation in a diffusive predator-prey model with herd behavior and hyperbolic mortality

In this paper, we consider the dynamics of a delayed diffusive predator-prey model with herd behavior and hyperbolic mortality under Neumann boundary conditions. Firstly, by analyzing the characteristic equations in detail and taking the delay as a bifurcation parameter, the stability of the positive equilibria and the existence of Hopf bifurcations induced by delay are investigated. Then, applying the normal form theory and the center manifold argument for partial functional differential equations, the formula determining the properties of the Hopf bifurcation are obtained. Finally, some numerical simulations are also carried out and we obtain the unstable spatial periodic solutions, which are induced by the subcritical Hopf bifurcation.     

Determining the conductivity for a nonautonomous hyperbolic operator in a cylindrical domain

This paper is devoted to the reconstruction of the conductivity coefficient for a nonautonomous hyperbolic operator an infinite cylindrical domain. Applying a local Carleman estimate, we prove the uniqueness and a Hölder stability in the determination of the conductivity using a single measurement data on the lateral boundary. Our numerical examples show good reconstruction of the location and contrast of the conductivity function in 3 dimensions.     

Stability analysis in a diffusional immunosuppressive infection model with delayed antiviral immune response

In this paper, the diffusion is introduced to an immunosuppressive infection model with delayed antiviral immune response. The direction and stability of Hopf bifurcation are effected by time delay, in the absence of which the positive equilibrium is locally asymptotically stable by means of analyzing eigenvalue spectrum; however, when the time delay increases beyond a threshold, the positive equilibrium loses its stability via the Hopf bifurcation. The stability and direction of the Hopf bifurcation is investigated with the norm form and the center manifold theory. The stability of the Hopf bifurcation leads to the emergence of spatial spiral patterns. Numerical calculations are performed to illustrate our theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.     

Effect of herd shape in a diffusive predator-prey model with time delay

In this paper, we deal with the effect of the shape of herd behavior on the interaction between predator and prey. The model analysis was studied in three parts. The first, The analysis of the system in the absence of spatial diffusion and the time delay, where the local stability of the equilibrium states, the existence of Hopf bifurcation have been investigated. For the second part, the spatiotemporal dynamics introduce by self diffusion was determined, where the existence of Hopf bifurcation, Turing driven instability, Turing-Hopf bifurcation point have been proved. Further, the order of Hopf bifurcation points and regions of the stability of the non trivial equilibrium state was given. In the last part of the paper, we studied the delay effect on the stability of the non trivial equilibrium, where we proved that the delay can lead to the instability of interior equilibrium state, and also the existence of Hopf bifurcation. A numerical simulation was carried out to insure the theoretical results.     

Relaxed stability conditions for continuous‐time Takagi–Sugeno fuzzy systems based on a new upper bound inequality

The important issue of reducing the conservatism of feasible stability criteria for continuous‐time Takagi–Sugeno fuzzy systems is studied in this article. In order to obtain more advanced result than previous ones, a new upper bound inequality is proposed and thus the properties of the normalized fuzzy weighting functions' time derivatives can be better used than the previous ones. In particular, the so‐called “redundant terms” considered in previous literature can be converted to “useful terms” which play a positive role in the underlying analysis process. Moreover, some useless additional variables and their derived inequalities are removed for enhancing the efficiency. Finally, an illustrative example is given to show the effectiveness of the proposed method. © 2016 Wiley Periodicals, Inc. Complexity 21: 289–295, 2016     

Pattern formation of a diffusive predator-prey model with herd behavior and nonlocal prey competition

In this paper, we study the influence of the nonlocal interspecific competition of the prey population on the dynamics of the diffusive predator-prey model with prey social behavior. Using the linear stability analysis, the conditions for the positive constant steady state at which undergoes Hopf bifurcation, T-H bifurcation (Turing-Hopf bifurcation) are investigated. The Turing patterns occur in the presence of the nonlocal competition and cannot be found in the original system. For determining the dynamical behavior near T-H bifurcation point, the normal form of the T-H bifurcation has been used. Some graphical representations are provided to illustrate the theoretical results.     

Global structure instability of Riemann solutions for general quasilinear hyperbolic systems of conservation laws in the presence of a boundary

This work is a continuation of our previous work [Z.-Q. Shao, D.-X. Kong, Y.-C. Li, Shock reflection for general quasilinear hyperbolic systems of conservation laws, Nonlinear Anal. TMA 66 (1) (2007) 93-124]. In this paper, we study the global structure instability of the Riemann solution containing shocks, at least one rarefaction wave for general n×n quasilinear hyperbolic systems of conservation laws in the presence of a boundary. We prove the nonexistence of global piecewise C¹ solution to a class of the mixed initial-boundary value problem for general n×n quasilinear hyperbolic systems of conservation laws on the quarter plane. Our result indicates that this kind of Riemann solution mentioned above for general n×n quasilinear hyperbolic systems of conservation laws in the presence of a boundary is globally structurally unstable. Some applications to quasilinear hyperbolic systems of conservation laws arising from physics and mechanics are also given.     

Numerical and theoretical study of weak Galerkin finite element solutions of Turing patterns in reaction–diffusion systems

In this paper, we introduce numerical schemes and their analysis based on weak Galerkin finite element framework for solving 2‐D reaction–diffusion systems. Weak Galerkin finite element method (WGFEM) for partial differential equations relies on the concept of weak functions and weak gradients, in which differential operators are approximated by weak forms through the Green's theorem. This method allows the use of totally discontinuous functions in the approximation space. In the current work, the WGFEM solves reaction–diffusion systems to find unknown concentrations (u, v) in element interiors and boundaries in the weak Galerkin finite element space WG(P₀, P₀, RT₀) . The WGFEM is used to approximate the spatial variables and the time discretization is made by the backward Euler method. For reaction–diffusion systems, stability analysis and error bounds for semi‐discrete and fully discrete schemes are proved. Accuracy and efficiency of the proposed method successfully tested on several numerical examples and obtained results satisfy the well‐known result that for small values of diffusion coefficient, the steady state solution converges to equilibrium point. Acquired numerical results asserted the efficiency of the proposed scheme.     

A mathematical model for Babesiosis disease in bovine and tick populations

In this paper, we analyze the Babesiosis transmission dynamics on bovine and tick populations. Ticks play a role of infectious agents and vector of the protozoan Babesia hemo‐parasite. In this sense, we set out a mathematical model with constant size population for the evolution of the infected bovines with Babesiosis and analyze its qualitative dynamics. Statistical data are used to estimate some of the parameters of the model. Numerical simulations of the model varying the parameters show different scenarios about the spread of the disease. Copyright © 2011 John Wiley & Sons, Ltd.