Asymptotic convergence of p‐Laplace equationswith constraint as p tends to 1

In this paper we treat the Euler–Lagrange equation of a functional including the p‐Laplacian for 1<p<2. The limiting equation as corresponds to a steady state problem of a modified Stefan–Gibbs–Thomson problem. We investigate the structure of the limit set of all solutions as . Furthermore, we clarify the relationship between the limit set and the solution set of the limiting equation. Copyright © 2002 John Wiley & Sons, Ltd.     

Stability of non‐constant steady‐state solutions for non‐isentropic Euler–Maxwell system with a temperature damping term

This work is concerned with the periodic problem for compressible non‐isentropic Euler–Maxwell systems with a temperature damping term arising in plasmas. For this problem, we prove the global in time existence of a smooth solution around a given non‐constant steady state with the help of an induction argument on the order of the mixed time‐space derivatives of solutions in energy estimates. Moreover, we also show the convergence of the solution to this steady state as the time goes to the infinity. This phenomenon on the charge transport shows the essential relation of the systems with the non‐isentropic Euler–Maxwell and the isentropic Euler–Maxwell systems. Copyright © 2015 John Wiley & Sons, Ltd.     

Travelling waves of a predator–prey model with a spatiotemporal delay

This paper is concerned with the existence of traveling waves to a predator–prey model with a spatiotemporal delay. By analyzing the corresponding characteristic equations, the local stability of a positive steady state and each of boundary steady states are established, and the existence of Hopf bifurcation at the positive steady state is also discussed. By constructing a pair of upper–lower solutions and by using the cross‐iteration method as well as the Schauder's fixed‐point theorem, the existence of a traveling wave solution connecting the semi‐trivial steady state and the positive steady state is proved. Numerical simulations are carried out to illustrate the main theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

Dynamics of a spatiotemporal model on populations in a polluted river

In this paper, we investigate the dynamics for a reaction–diffusion–advection system which models populations in a polluted river. More precisely, we study the stability of steady states, which yields sufficient conditions that lead to population persistence or extinction. Furthermore, some dependence of the stability of the toxicant-only steady state and the population-toxicant coexistence steady state on the model parameters are given.     

Global stability and the Hopf bifurcation for some class of delay differential equation

In this paper, we present an analysis for the class of delay differential equations with one discrete delay and the right‐hand side depending only on the past. We extend the results from paper by U. Fory? (Appl. Math. Lett. 2004; 17 (5):581–584), where the right‐hand side is a unimodal function. In the performed analysis, we state more general conditions for global stability of the positive steady state and propose some conditions for the stable Hopf bifurcation occurring when this steady state looses stability. We illustrate the analysis by biological examples coming from the population dynamics. Copyright © 2007 John Wiley & Sons, Ltd.     

Parameter studies of modal vibrations associated with impurity pinned dislocations in a metal single crystal

Koehler's model [1–2] of motion for edge‐type dislocations in a metal single crystal that are pinned down by impurity atoms is studied. An exact solution can be found, which is composed of a rapidly decaying transient and a steady time‐oscillating, steady state vibration. This solution is used to improve Koehler's [1] approximation to the steady time‐oscillating steady state vibration. General parameter studies of the modes of oscillation are then performed. The present result is of some significance, because it allows insight into the behavior of crystalline solids over a wide parameter range, whereas Koehler's asymptotic approach is valid only for materials that exhibit order‐of‐magnitude variation in system parameters. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 427–439, 2001.     

Dynamics analysis of a reaction–diffusion system with Beddington–DeAngelis functional response and strong Allee effect

In this paper, the dynamics of a diffusive predator–prey model with modified Leslie–Gower term and strong Allee effect on prey under homogeneous Neumann boundary condition is considered. Firstly, we obtain the qualitative properties of the system including the existence of the global positive solution and the local and global asymptotical stability of the constant equilibria. In addition, we investigate a priori estimate and the nonexistence of nonconstant positive steady state solutions. Finally, we establish the existence and local structure of steady state patterns and time-periodic patterns for the system.     

A direct method for solving block‐Toeplitz with near‐circulant‐block systems with applications to hybrid manufacturing systems

In this paper, we present a direct method for solving linear systems of a block‐Toeplitz matrix with each block being a near‐circulant matrix. The direct method is based on the fast Fourier transform (FFT) and the Sherman–Morrison–Woodbury formula. We give a cost analysis for the proposed method. The method is then applied to solve the steady‐state probability distribution of a hybrid manufacturing system which consists of a manufacturing process and a re‐manufacturing process. Copyright © 2005 John Wiley & Sons, Ltd.     

Stability in predator – prey models and discretization of a modified Volterra – Lotka model

We consider n  2 populations of animals that are living in mutual predator – prey relations or are pairwise neutral to each other. We assume the temporal development of the population densities to be described by a system of differential equations which has an equilibrium state solution. We derive sufficient conditions for this equilibrium state to be stable by Lyapunov's method. The results supplement those published elsewhere.

Further we consider a modification of the Volterra – Lotka model which admits an asymptotically stable steady state solution. This model is discretized in two ways and we investigate how small the time step size has to be chosen in order to guarantee that the steady state solution is an attractive fixed point of the discretized model. This investigation is connected with the determination of the model parameters from given data.     

Hopf bifurcation in a delayed reaction–diffusion–advection population model

In this paper, we investigate a reaction–diffusion–advection model with time delay effect. The stability/instability of the spatially nonhomogeneous positive steady state and the associated Hopf bifurcation are investigated when the given parameter of the model is near the principle eigenvalue of an elliptic operator. Our results imply that time delay can make the spatially nonhomogeneous positive steady state unstable for a reaction–diffusion–advection model, and the model can exhibit oscillatory pattern through Hopf bifurcation. The effect of advection on Hopf bifurcation values is also considered, and our results suggest that Hopf bifurcation is more likely to occur when the advection rate increases.     

Optimal fishery with coastal catch

In many spatial resource models, it is assumed that an agent is able to harvest the resource over the complete spatial domain. However, agents frequently only have access to a resource at particular locations at which a moving biomass, such as fish or game, may be caught or hunted. Here, we analyze an infinite time‐horizon optimal control problem with boundary harvesting and (systems of) parabolic partial differential equations as state dynamics. We formally derive the associated canonical system, consisting of a forward–backward diffusion system with boundary controls, and numerically compute the canonical steady states and the optimal time‐dependent paths, and their dependence on parameters. We start with some one‐species fishing models, and then extend the analysis to a predator–prey model of the Lotka–Volterra type. The models are rather generic, and our methods are quite general, and thus should be applicable to large classes of structurally similar bioeconomic problems with boundary controls. Recommedations for Resource Managers

• Just like ordinary differential equation‐constrained (optimal) control problems and distributed partial differential equation (PDE) constrained control problems, boundary control problems with PDE state dynamics may be formally treated by the Pontryagin's maximum principle or canonical system formalism (state and adjoint PDEs).
• These problems may have multiple (locally) optimal solutions; a first overview of suitable choices can be obtained by identifying canonical steady states.
• The computation of canonical paths toward some optimal steady state yields temporal information about the optimal harvesting, possibly including waiting time behavior for the stock to recover from a low‐stock initial state, and nonmonotonic (in time) harvesting efforts.
• Multispecies fishery models may lead to asymmetric effects; for instance, it may be optimal to capture a predator species to protect the prey, even for high costs and low market values of the predators.

     

Parameter uniform approximations for time‐dependent reaction‐diffusion problems

Using discrete Green's functions techniques, we present a classification of fitted mesh methods for time‐dependent reaction diffusion problems, based on the analyses of Linß (Linß, Numer Algor 40 (2005), 23–32) for the analogous steady‐state problem and of Kopteva (Kopteva, Computing 66 (2001), 179–197) for time‐dependent convection‐diffusion problems. As examples of how to apply the analysis, we derive error estimates for the fitted meshes of Shishkin and Bakhvalov, and provide supporting numerical results. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Effects of the self- and cross-diffusion on positive steady states for a generalized predator–prey system

This paper deals with a generalized predator–prey system with cross-diffusion and homogeneous Neumann boundary condition, where the cross-diffusion is included in such a way that the prey runs away from the predator. We first give a priori estimate for positive steady states to the system. Then we obtain the non-existence result of non-constant positive steady states. Finally, we investigate the stability of constant equilibrium point and the existence of non-constant positive steady states. It is shown that the system admits a non-constant positive steady state provided that either of the self-diffusions is large or the cross-diffusion is small.     

Nonexistence of coexisting steady‐state solutions of Dirichlet problem for a cross‐diffusion model

We study a special case of Shigesada–Kawasaki–Teramoto (SKT) model for two competing species with the Dirichlet boundary conditions. In our case, one of the species is not influenced by self‐diffusion or cross‐diffusion. We specify the explicit range of parameters by contradiction such that there are no coexisting steady‐state solutions to the model.     

Bivariate spline method for numerical solution of steady state Navier–Stokes equations over polygons in stream function formulation

We use the bivariate spline finite elements to numerically solve the steady state Navier–Stokes equations. The bivariate spline finite element space we use in this article is the space of splines of smoothness r and degree 3r over triangulated quadrangulations. The stream function formulation for the steady state Navier–Stokes equations is employed. Galerkin's method is applied to the resulting nonlinear fourth‐order equation, and Newton's iterative method is then used to solve the resulting nonlinear system. We show the existence and uniqueness of the weak solution in H²(Ω) of the nonlinear fourth‐order problem and give an estimate of how fast the numerical solution converges to the weak solution. The Galerkin method with C¹ cubic splines is implemented in MATLAB. Our numerical experiments show that the method is effective and efficient. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 147–183, 2000     

Extension of high‐order compact schemes to time‐dependent problems

In this article, we present an extension of our previous approaches for steady‐state higher‐order compact (HOC) difference methods to time‐dependent problems. The formulation also provides a framework for similar treatment of other HOC spatial schemes. A stability analysis is provided for transient convection‐diffusion in 1D and transient diffusion in 2D. Supporting numerical experiments are included to illustrate stability and accuracy as well as oscillatory and dissipative behavior. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 657–672, 2001     

HUMAN FERTILITY DECISIONS AND COMMON PROPERTY RESOURCES: A DYNAMIC ANALYSIS

ABSTRACT. In rural areas of developing countries, parental decisions on number of offspring may be made on the basis of the role of children in harvesting local common property renewable resources. It has been argued that this may lead to a cycle of human over‐population and resource over‐exploitation. To investigate the plausibility of this argument, we present a discrete dynamic model with two state variables representing human population level N and resource stock level S. The model is similar to one given by Nerlove and Meyer but differs in several important respects. It is assumed that, in each over‐lapping generation of parents and children, parents decide how many children to have based on their resulting share of the local resource harvest and the costs associated with child‐rearing. Using simulation and analytical methods, the long term steady state population and resource stock levels for this dynamic noncooperative game are contrasted with the steady state when parental fertility decisions are made in a cooperative manner.     

On the dynamics of axially moving strings

We investigate an axially moving cable with large sag modelled as string and calculate the steady state solution between two rolls. Alternatively we model the cable as beam with very small bending stiffness and compare this solution with the string solution. The validity of the analytically found boundary layer solution is verified by numerical methods. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability Analysis of a Diffusive Predator-prey Model with Hunting Cooperation

In this paper, we are concerned with the dynamics of a diffusive predator-prey model that incorporates the functional response concerning hunting cooperation. First, we investigate the stability of the semi-trivial steady state. Then, we investigate the influence of the diffusive rates on the stability of the positive constant steady state. It is shown that there exists diffusion-driven Turing instability when the diffusive rate of the predator is smaller than the critical value, which is dependent on the diffusive rate of the prey, and the semi-trivial steady state and the positive constant steady state are both locally asymptotically stable when the diffusive rate of the predator is larger than the critical value. Finally, the nonexistence of nonconstant steady states is discussed.     

Finite‐dimensional Realizations of Regime‐switching HJM Models

This paper studies Heath–Jarrow–Morton‐type models with regime‐switching stochastic volatility. In this setting the forward rate volatility is allowed to depend on the current forward rate curve as well as on a continuous time Markov chain y with finitely many states. Employing the framework developed by Björk and Svensson we find necessary and sufficient conditions on the volatility guaranteeing the representation of the forward rate process by a finite‐dimensional Markovian state space model. These conditions allow us to investigate regime‐switching generalizations of some well‐known models such as those by Ho–Lee, Hull–White, and Cox–Ingersoll–Ross.