محاسبات قابل اطمینان از دولت های تعادل و شاخه شدن در تجزیه و تحلیل سیستم های زیست محیطی

A problem of frequent interest in analyzing nonlinear ODE models of ecological systems is the location of equilibrium states and bifurcations. Interval-Newton techniques are explored for identifying, with certainty, all equilibrium states and all codimension-one and codimension-two bifurcations of interest within specified model parameter intervals. The methodology is applied to a tritrophic food chain in a chemostat (Canale's model), and a modification of thereof. This modification aids in elucidating the nonlinear effects of introducing a hypothetical contaminant into a food chain.

Ecological systems, including food chains and food webs, are often modeled using systems of nonlinear ordinary differential equations (ODEs). Of particular interest here is the modeling of food chains, which provides challenges in the fields of both theoretical ecology and applied mathematics. Food chain models are descriptive of a wide range of behaviors in the environment, and are useful as a tool to perform ecological risk assessments (Naito, Miyamoto, Nakanishi, Masunaga, & Bartell, 2002). These models are often simple, but display rich mathematical behavior, with varying numbers and stability of equilibria and limit cycles that depend on the model parameters (e.g., Gragnani, De Feo, & Rinaldi, 1998; Moghadas & Gumel, 2003). Many different model formulations are possible, depending on the number of species analyzed, the predation responses used, whether age or fertility structure is of interest for a given species, and how resources are being modeled for the basal species. Analysis of food chain models is often performed by examining the parameter space of the model in one or more variables. This approach is referred to as bifurcation analysis, and it provides a powerful tool for concisely representing a large amount of information regarding both the number and stability of equilibrium states (steady states) in a model. In a two-parameter bifurcation diagram, the shape of bifurcation curves can elucidate the dependence, or lack there of, between model parameters, which in turn can provide information on their ecological relevance. Furthermore, both the shape and the order of bifurcation curves in a diagram can be used to make comparisons between different food chain models. We will focus on one particular food chain model here, namely Canale's chemostat model, as described in detail below. We will also develop and study a version of the model that incorporates an ecosystem contaminant. Determining the equilibrium states and bifurcations of equilibria in a nonlinear dynamical system is often a challenging problem, and great effort can be expended in analyzing even a relatively simple food chain model with nonlinear functional responses. For simple systems, or specific parts of more complex ones, analytic techniques and isocline analysis may be useful. However, for more complex problems, numerical continuation methods are the predominant computational tools, with packages such as AUTO (Doedel et al., 2002), MATCONT (Dhooge, Govaerts, & Kuznetsov, 2003) and others being particularly popular in this context. Continuation methods can be quite reliable, especially in the hands of an experienced user. However, in general, continuation methods are initialization dependent and provide no guarantee that all equilibrium states and all bifurcations of equilibria will be found. Thus, effective use of continuation methods may require some a priori understanding of system behavior in order to reliably create an accurate bifurcation diagram. Gwaltney, Styczynski, and Stadtherr (2004) described an alternative approach, based on interval mathematics, and applied it to a simple tritrophic Rosenzweig–MacArthur model, and variations thereof. We will explore the use of the same approach here, but apply it to more complex models. This computational method uses an interval-Newton approach combined with generalized bisection, and provides a mathematical and computational guarantee that all equilibrium states and bifurcations of equilibria will be located, without need for initializations or a priori insights into system behavior. There are other dynamical features of interest in food chain models, such as limit cycles (and their bifurcations); however, our attention here will be limited to equilibrium states and their bifurcations. Interval methodologies have been successfully applied to the problem of locating equilibrium states and singularities in traditional chemical engineering problems, such as reaction and reactive distillation systems. Examples of these applications are given by Schnepper and Stadtherr (1996), Gehrke and Marquardt (1997), Bischof, Lang, Marquardt, and Mönnigmann (2000), and Mönnigmann and Marquardt (2002). Our interest in ecological modeling is motivated by its use as one tool in studying the impact on the environment of the industrial use of newly discovered materials. Clearly it is preferable to take a proactive, rather than reactive, approach when considering the safety and environmental consequences of using new compounds. Of particular interest is the potential use of room temperature ionic liquid (IL) solvents in place of traditional solvents (Brennecke & Maginn, 2001). IL solvents have no measurable vapor pressure (i.e., they do not evaporate) and thus, from a safety and environmental viewpoint, have several potential advantages relative to the traditional volatile organic compounds (VOCs) used as solvents, including elimination of hazards due to inhalation, explosion and air pollution. However, ILs are, to varying degrees, soluble in water; thus if they are used industrially on a large scale, their entry into the environment via aqueous waste streams is of concern. The effects of trace levels of ILs in the environment are today not well known and thus must be further studied. Ecological modeling provides a means for studying the impact of such perturbations on a localized environment by focusing not just on single-species toxicity information, but rather on the larger impacts on the food chain and ecosystem. Of course, ecological modeling is just one part of a much larger suite of tools, including toxicological (e.g., Bernot, Brueseke, Evans-White, & Lamberti, 2005; Bernot, Kennedy, & Lamberti, 2005; Ranke et al., 2004; Stepnowski, Skladanowski, Ludwiczak, & Laczynska, 2004), microbiological (e.g., Docherty & Kulpa, 2005; Pernak, Sobaszkiewicz, & Mirska, 2003) and other (e.g., Gorman-Lewis & Fein, 2004; Ropel, Belvèze, Aki, Stadtherr, & Brennecke, 2005) studies, that must be used in addressing this issue.

We have demonstrated here the utility of an interval-Newton approach for the computationally rigorous and reliable computation of all equilibrium states and bifurcations of equilibria (fold, transcritical, Hopf, double-fold and fold-Hopf) in nonlinear models of ecosystem dynamics, with focus on a model that includes the effect of a contaminant. Using this methodology one can easily, without any need for initialization or a priori insight into system behavior, generate complete solution branch and bifurcation diagrams. The ability to easily and reliably analyze nonlinear food chain models can expose unexpected and counterintuitive behavior. The knowledge provided by this sort of analysis may be quite useful in managing risk in the complex and highly interdependent nonlinear systems found in our environment.