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Computation of Equilibrium States in Food Networks Using Interval Analysis

Courtney R. Gwaltney, Roger A. Schmitz, and Mark A. Stadtherr. Department of Chemical and Biomolecular Engineering, University of Notre Dame, 182 Fitzpatrick Hall, Notre Dame, IN 46556

Ecosystem models are increasingly being used as tools to aid in the management of environmental risks, especially the impacts of pollution (Pastorok et al., 2003). These models consist of a set of balance equations that model the population or biomass of a particular species or a group of species within a food network. Specific models of growth and mortality can be assigned for each species or group. Nutrients and resources can be modeled either implicitly, by using a logistic growth function, or explicitly, by using a chemostat-type model or by including a detritus pool and a sub-web of detritivores. The balance equations are often given as a system of continuous-time, nonlinear ordinary differential equations (ODEs). Our interest in ecosystem modeling is motivated by its use as one tool in studying the impact on the environment of the industrial use of newly discovered materials, such as ionic liquids (Brennecke and Maginn, 2001).

Analysis of food network models often involves the determination of equilibrium states (steady states), and the study of how the number and stability of equilibrium states change as some model parameter is varied (bifurcation analysis). Locating all steady states in a complex food network can be a significant computational challenge, especially as the number of species in a model increases. For an n-species model with linear growth functions, the number of steady states may be as large as 2n, though it will typically be less than this because not all zero-nonzero combinations of species biomass yield feasible solutions (nonnegative biomass values). Furthermore, if the growth function is nonlinear, then there may be additional feasible steady-state solutions. In general, the number of equilibrium states will be unknown a priori, and may vary significantly as model parameters change. For example, in one system considered here the number of equilibrium states ranges from 1 to 324 as the total mass of the system is varied.

Continuation-based strategies are often used to locate equilibrium states in a dynamic model (Kuznetsov, 1991). However, these methods are initialization dependent and provide no guarantee that all steady states will be found. In previous work (Gwaltney et al., 2004, Gwaltney and Stadtherr, 2006) with several small food chain models (three or four state variables), we found that methods based on interval analysis (e.g., Kearfott, 1996), specifically an interval-Newton strategy, were very effective for finding all equilibrium states. In this presentation, we will explore the effectiveness of this approach on larger food web models.

Interval methods will be used in conjunction with two different strategies to solve for all of the steady states in a food web model. The first strategy is a simultaneous approach in which the entire model is solved simultaneously for all the equilibrium states. The second is a sequential approach in which the problem is converted into a sequence of smaller subproblems, each involving one particular zero-nonzero combination of species biomasses. Both approaches used provide a mathematical and computational guarantee that all steady states will be found.

We will consider as examples two theoretical ecosystem models, both of which are significantly larger and more complex than problems solved previously with interval methods. The first ecosystem model considered is a food web involving seven species. The second ecosystem model examined is a food web involving twelve species (4 producers, 5 consumers, 3 detritivores) and four nutrients, and is a closed, nonlinear system. These examples are used to demonstrate the effectiveness of the interval-Newton approach in the analysis of food network models.


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