Keng Deng

Blow-up rates for parabolic systems

Let Ω ⊂ ℝn be a bounded domain andBR be a ball in ℝn of radiusR. We consider two parabolic systems: ut=Δu +f(υ), υi=Δυ +g(u) in Ω × (0,T) withu=v=0 on δΩ × (0,T) andut=Δu, vt=Δv inBr × (0,T) withδe/δv=f (v), δe/δv=g(u) onδBR × (0,T). Whenf(v) andg(u) are power law or exponential functions, we establish estimates on the blow-up rates for nonnegative solutions of the systems.

A finite difference approximation for a coupled system of nonlinear size-structured populations

Abstract : We study a quasilinear nonlocal hyperbolic initial-boundary value problem that models the evolution of N size-structured subpopulations competing for common resources. We develop an implicit finite difference scheme to approximate the solution of this model. The convergence of this approximation to a unique bounded variation weak solution is obtained. The numerical results for a special case of this model suggest that when subpopulations are closed under reproduction, one subpopulation survives and the others go to extinction. Moreover, in the case of open reproduction, survival of more than one population is possible.

A NONAUTONOMOUS JUVENILE-ADULT MODEL: WELL-POSEDNESS AND LONG-TIME BEHAVIOR VIA A COMPARISON PRINCIPLE ∗

A nonautonomous nonlinear continuous juvenile-adult model where juveniles and adults depend on different resources is developed. It is assumed that juveniles are structured by age, while adults are structured by size. Existence-uniqueness results are proved using the monotone method based on a comparison principle established in this paper. Conditions on the model parameters that lead to extinction or persistence of the population are obtained via the upper-lower solution technique.

Monotone method for first order nonlocal hyperbolic initial-boundary value problems

In this paper, the monotone method is extended to nonlocal hyperbolic intial-boundary value problems of first order. A comparison principle is established, and linear convergence of monotone sequences of upper and lower solutions is shown.

A monotone approximation for a nonlinear nonautonomous size-structured population model

We develop a monotone approximation, based on upper and lower solutions technique, for solving a nonlinear nonautonomous size-structured model with competition between individuals. Such an approximation results in the existence and uniqueness of the solution for the model problem. It also provides a practicable and efficient numerical scheme. Furthermore, we establish a first order convergence result and present a numerical example.

Quenching for solutions of a plasma type equation

Here 0 l), the so called “slow diffusion” case. The analysis will be carried out in the manner of [5], in which Levine gave a complete qualitative study of problem (I) with m = 1. The plan of our paper is as follows: in the next section, we present the comparison theorem and local existence of solutions for (I); in the third section, we characterize the set of stationary solutions of (I); finally, in the last section, we establish stability and quenching results for problem (I).

Fitting a Structured Juvenile–Adult Model for Green Tree Frogs to Population Estimates from Capture–Mark–Recapture Field Data

We derive point and interval estimates for an urban population of green tree frogs (Hyla cinerea) from capture–mark–recapture field data obtained during the years 2006–2009. We present an infinite-dimensional least-squares approach which compares a mathematical population model to the statistical population estimates obtained from the field data. The model is composed of nonlinear first-order hyperbolic equations describing the dynamics of the amphibian population where individuals are divided into juveniles (tadpoles) and adults (frogs). To solve the least-squares problem, an explicit finite difference approximation is developed. Convergence results for the computed parameters are presented. Parameter estimates for the vital rates of juveniles and adults are obtained, and standard deviations for these estimates are computed. Numerical results for the model sensitivity with respect to these parameters are given. Finally, the above-mentioned parameter estimates are used to illustrate the long-time behavior of the population under investigation.

Dynamical behavior of solutions of a semilinear heat equation with nonlocal singularity

The heat equation with a nonlocal nonlinearity

Dynamics of a susceptible–infected–susceptible epidemic reaction–diffusion model

u_t = u_{xx} + {{\varepsilon \| {u( \cdot ,t)} \|^q } / {(1 - u)}},0 0

Stochastic juvenile–adult models with application to a green tree frog population

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