Imbunm Kim

Turing instability for a ratio-dependent predator–prey model with diffusion

Abstract Ratio-dependent predator–prey models have been increasingly favored by field ecologists where predator–prey interactions have to be taken into account the process of predation search. In this paper we study the conditions of the existence and stability properties of the equilibrium solutions in a reaction–diffusion model in which predator mortality is neither a constant nor an unbounded function, but it is increasing with the predator abundance. We show that analytically at a certain critical value a diffusion driven (Turing type) instability occurs, i.e. the stationary solution stays stable with respect to the kinetic system (the system without diffusion). We also show that the stationary solution becomes unstable with respect to the system with diffusion and that Turing bifurcation takes place: a spatially non-homogenous (non-constant) solution (structure or pattern) arises. A numerical scheme that preserve the positivity of the numerical solutions and the boundedness of prey solution will be presented. Numerical examples are also included.

On the Derivation of Highest-Order Compact Finite Difference Schemes for the One- and Two-Dimensional Poisson Equation with Dirichlet Boundary Conditions

The primary aim of this paper is to answer the question, What are the highest-order five- or nine-point compact finite difference schemes? To answer this question, we present several simple derivations of finite difference schemes for the one- and two-dimensional Poisson equation on uniform, quasi-uniform, and nonuniform face-to-face hyperrectangular grids and directly prove the existence or nonexistence of their highest-order local accuracies. Our derivations are unique in that we do not make any initial assumptions on stencil symmetries or weights. For the one-dimensional problem, the derivation using the three-point stencil on both uniform and nonuniform grids yields a scheme with arbitrarily high-order local accuracy. However, for the two-dimensional problem, the derivation using the corresponding five-point stencil on uniform and quasi-uniform grids yields a scheme with at most second-order local accuracy, and on nonuniform grids yields at most first-order local accuracy. When expanding the five-poin...

Higher-order schemes for the Laplace transformation method for parabolic problems

In this paper we solve linear parabolic problems using the three stage noble algorithms. First, the time discretization is approximated using the Laplace transformation method, which is both parallel in time (and can be in space, too) and extremely high order convergent. Second, higher-order compact schemes of order four and six are used for the the spatial discretization. Finally, the discretized linear algebraic systems are solved using multigrid to show the actual convergence rate for numerical examples, which are compared to other numerical solution methods.

A fractional-order model for MINMOD Millennium.

MINMOD Millennium has been widely used to estimate insulin sensitivity (SI) in glucose-insulin dynamics. In order to explain the rheological behavior of glucose-insulin we attempt to modify MINMOD Millennium with fractional-order differentiation of order α ∈ (0, 1]. We show that the new modified model has non-negative, bounded solutions and a stable equilibrium point. Quasi-optimal fractional orders and parameters are estimated by using a nonlinear weighted least-squares method, the Levenberg-Marquardt algorithm, and the fractional Adams-Bashforth-Moulton method for several subjects (normal subjects and type 2 diabetic patients). The numerical results confirm that SI is significantly lower in diabetics than in non-diabetics. In addition, we explain the new factor (τ(1 - α)) determining glucose tolerance and the relation between SI and τ(1 - α).

Estimation of parameter functions in ordinary differential equations with a stage structure: a linear case

In many mathematical models with complex parameter functions, it is critical to suggest a strategy for estimating those functions. This paper considers the problem of fitting parameter functions to given data by using a finite number of piecewise functions. Here, the primary aim is to propose several necessary and sufficient conditions that can guarantee the existence of parameter functions. With these criteria, the paper provides an algorithm for solving the parameter estimation problem efficiently. Numerical examples verify that the proposed algorithm is effective in estimating parameter functions in differential equations.