Simulation of structured populations in chemically stressed environments
Thomas G. Hallam Qingping Deng
Mathematical Biosciences & Engineering 2006, 3(1): 51-65 doi: 10.3934/mbe.2006.3.51
A heterogenous environment usually impacts, and sometimes determines, the structure and function of organisms in a population. We simulate the effects of a chemical on a population in a spatially heterogeneous environment to determine perceived stressor and spatial effects on dynamic behavior of the population. The population is assumed to be physiologically structured and composed of individuals having both sessile and mobile life history stages, who utilize energetically-controlled, resource-directed, chemical-avoidance advective movements and are subjected to random or density dependent diffusion. From a modeling perspective, the presence of a chemical in the environment requires introduction of both an exposure model and an effects module. The spatial location of the chemical stressor determines the exposure levels and ultimately the effects on the population while the relative location of the resource and organism determines growth. We develop a mathematical model, the numerical analysis for this model, and the simulation techniques necessary to solve the problem of population dynamics in an environment where heterogeneity is generated by resource and chemical stressor. In the simulations, the chemical is assumed to be a nonpolar narcotic and the individuals respond to the chemical via both physiological response and by physical movement. In the absence of a chemical stressor, simulation experiments indicate that despite a propensity to move to regions of higher resource density, organisms need not concentrate in the vicinity of high levels of resource. We focus on the dynamical variations due to advection induced by the toxicant. It is demonstrated that the relationship between resource levels and toxicant concentrations is crucial in determining persistence or extinction of the population.
keywords: life history physiology energetics organism movement toxicology numerical techniques.
A nonoverlapping domain decomposition method for nonconforming finite element problems
Qingping Deng
Communications on Pure & Applied Analysis 2003, 2(3): 297-310 doi: 10.3934/cpaa.2003.2.297
A nonoverlapping domain decomposition method for nonconforming finite element problems of second order partial differential equations is developed and analyzed. In particular, its convergence is demonstrated and convergence rate is estimated. The method is based on a Robin boundary condition as its transmission condition together with a derivative-free transmission data updating technique on the interfaces. The method is directly presented to finite element problems without introducing any Lagrange multipliers. The method can be naturally applied to general multi-subdomain decompositions and implemented on parallel machines with local communications. The method also allows choosing subdomains very flexibly, which can be even as small as an individual element. Therefore, the method can be regarded as a bridge connecting between direct methods and iterative methods for linear systems. Finally, some numerical experiments are also presented to demonstrate the effectiveness of the method.
keywords: Robin boundary condition parallel transmission condition domain decomposition iterative nonconforming Nonoverlapping finite element.

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