Peter Y. H. Pang

Qualitative analysis of a ratio-dependent predator–prey system with diffusion

Ratio-dependent predator–prey models are favoured by many animal ecologists recently as they better describe predator–prey interactions where predation involves a searching process. When densities of prey and predator are spatially homogeneous, the so-called Michaelis–Menten ratio-dependent predator–prey system, which is an ordinary differential system, has been studied by many authors. The present paper deals with the case where densities of prey and predator are spatially inhomogeneous in a bounded domain subject to the homogeneous Neumann boundary condition. Its main purpose is to study qualitative properties of solutions to this reaction-diffusion (partial differential) system. In particular, we will show that even though the unique positive constant steady state is globally asymptotically stable for the ordinary-differential-equation dynamics, non-constant positive steady states exist for the partial-differential-equation model. This demonstrates that stationary patterns arise as a result of diffusion.

Non-constant positive steady states of a predator-prey system with non-monotonic functional response and diffusion

This paper deals with non-constant positive steady-state solutions of a predator-prey system with non-monotonic functional response, also called Holling type-IV interaction terms, and diffusion under the homogeneous Neumann boundary condition. We first establish positive upper and lower bounds for such solutions, and then study their non-existence, global existence and bifurcation.

Qualitative Analysis of a Prey-Predator Model with Stage Structure for the Predator

In this paper, we propose a diffusive prey-predator model with stage structure for the predator. We first analyze the stability of the nonnegative steady states for the reduced ODE system and then study the same question for the corresponding reaction-diffusion system with homogeneous Neumann boundary conditions. We find that a Hopf bifurcation occurs in the ODE system, but no Turing pattern happens in the reaction-diffusion system. However, when a natural cross diffusion term is included in the model, we can prove the emergence of stationary patterns (i.e., nonconstant positive stationary solutions) for this system; moreover, these stationary patterns do not exist in the considered parameter regime when there is no cross diffusion.

Asymptotic Behavior of a Class of Nonlinear Delay Difference Equations

In this paper, we shall study the asymptotic behavior of solutions of difference equations of the form x n +1 = x n p f ( x n m k 1 , x n m k 2 ,…, x n m k r ), n =0,1,…, where p is a positive constant and k 1 ,…, k r are (fixed) nonnegative integers. In particular, permanence and global attractivity will be discussed.

Monotone iterative methods for a general class of discrete boundary value problems

Abstract In this paper we shall offer comparison results as well as monotone iterative schemes for the construction of solutions to a very general class of discrete boundary value problems. The discrete system we consider includes in particular the n th order prototype systems, finite as well as infinite discrete delay equations, and discrete integral equations. Further, the boundary conditions we consider include the initial, terminal, periodic and transport type problems. Numerical examples illustrating the usefulness of the proposed schemes to a variety of boundary value problems are also included.

Global asymptotic stability of positive steady states of a diffusive ratio-dependent prey–predator model

Abstract In this work, we study a ratio-dependent prey–predator model with diffusion and homogeneous Neumann boundary condition. We prove that the unique positive constant steady state is locally and uniformly stable, and is globally asymptotically stable under some assumptions. The proof uses the iteration method.

Global solutions for a general strongly coupled prey-predator model

Abstract This work investigates global solutions for a general strongly coupled prey–predator model that involves (self-)diffusion and cross-diffusion, where the cross-diffusion is of the form v / ( 1 + u l ) with l ≥ 1 . Very few mathematical results are known for such models, especially in higher spatial dimensions.

Inverse obstacle scattering with two scatterers

In this paper, we discuss the inverse scattering of two obstacles D11 and D12 with D11 C D11. For the sound-soft problem, we establish the uniqueness of the interior obstacle D11

On discrete boundary value problems arising in transport phenomena

For the discrete boundary value problems arising in transport processes we provide comparison results. These results are used to develop monotone iterative methods for the construction of the maximal and minimal solutions in a sector. The advantage of this technique is that the successive approximations are the solutions of the initial and terminal value problems. Numerical illustration showing the sharpness as well the importance of the obtained results is also included.

Sharp discrete inequalities inn independent variables

Abstract We offer very general Opial- and Wirtinger-type inequalities for discrete functions ofn independent variables. Some particular cases of our results improve several recent results.