Mathematical Biology Workshop

May 17-18, 2002

University of Nebraska-Lincoln
Oldfather Hall on the City Campus

School of Biological Sciences and
The Department of Mathematics

Featured Participants

Additional Participants and Attendees

Kurt Anderson, Ecology and Evolutionary Biology, UC Santa Barbara
Marian Borgman-Ingwersen, Department of Biology, Wayne State
Steve Cohn, Department of Mathematics, UNL
Bo Deng, Department of Mathematics, UNL
Lynn Erbe, Department of Mathematics, UNL
John Fuelberth, Department of Mathematics, Wayne State
Wendy Hines, Dapartment of Mathematics, UNL
Tony Joern, School of Biological Sciences, UNL
Hem Raj Joshi, Departments of Mathematics and Ecology, University of Tennessee
Glenn Ledder, Department of Mathematics, UNL
David Logan, Department of Mathematics, UNL
Shawn, Pearcy, Department of Biology, Wayne State
Allan Peterson, Department of Mathematics, UNL
Tom Shores, Department of Mathematics, UNL
Dan Van Peursem, Department of Mathematics, University of South Dakota
Bill Wolesensky, Department of Mathematics, College of Saint Mary

Workshop Information

Titles and Abstracts (Scroll to bottom for Schedule)

Round Table Discussion

Education and Quantitative Issues in the Biosciences

This round table discussion, led by Lou Gross and Tony Joern, will focus on current educational and quantitative issues in the life sciences. Some of the topics may include formal mathematics-computer science, statistics training issues for biologists, encouraging mathematics students to move into mathematical and computational biology, and how to deal with cross-departmental training.

Lou Gross

Everglades Restoration: Computing, Ecology and Public Policy

The Everglades region of South Florida offers one of the most complex challenges to natural system management we face in the U.S. The region has been greatly affected by many years of active human intervention to control the dominant environmental factor driving the system (water), with tremendous expenditures on a variety of canals, locks and structures to drain certain portions of the system and control flooding. The effects of this intervention on natural components of the system have been extensive, including major declines in many species populations, greatly enhanced fluxes of certain nutrients, changes in plant community composition, and the release of high levels of toxicants including mercury. A major effort is now underway to plan for restoration of the system. Computational ecology, by combining mathematical and computer models of natural systems with geographically-explicit details of the biotic and abiotic components of the environment, allows us to rapidly compare alternative virtual futures for this natural system to better plan for sustainable ecosystems. I will present the successful efforts of The Institute for Environmental Modeling to provide ecological assessments for long-term planning for the Everglades. Further details are available at http://atlss.org/.

Bo Deng

Food chain chaos

A survey will be given on various types of chaotic dynamics on food chain models. Two new types will be discussed. This work is an attempt to address the ecological paradox that food chain models are rich in chaos, yet few have been found in nature.

Roger Nisbet

From molecules to ecosystems with dynamic energy budget models

General models describing the acquisition of energy by an individual organism, and its utilization for growth, reproduction and survival, have the potential to link to processes at various levels. The talk will highlight five major areas where such links can occur. First, a successful model based on dynamic energy budgets of individuals (hereafter referred to as a DEB model) can provide insight on sub-cellular processes related to energy use. Second, a DEB model constitutes the energetic basis for the dynamics of populations. Third, the models can make a major contribution to life history theory. Fourth, DEB models can be used to derive inter-specific scaling relationships for physiological rates. Fifth, by using the scaling relationships in combination with population models and some assumptions on stoichiometry, it may be possible to develop tractable models of the flow of elemental matter in communities or ecosystems.

Jose Flores

Mathematical modeling for sterile insect techniques (SIT)

The Screwworm (Cochliomyia hominivorax, Diptera:Calliphoridae) were eradicated from US with the help of the Knipling's model that related mating proportions of native and sterile insects to population fertility and predicted populations trends in subsequent generations. The succesful mathematical model of Lawerence F. Knipling (1979) has generated much practical and theoretical research on SIT. We study and analyze these models and present a general framework for their applications.

Kurt Anderson

Predicting the effects of local environmental variation on Population dynamics at larger spatial scales in stream ecosystems

(With Roger M. Nisbet and Sebastian Diehl) Recent empirical studies of streams have elucidated the effects of manipulating resource levels (e.g., nutrients and irradiance), prey items (e.g. algae, insects), and toxicants on the population dynamics of organisms within small sections of stream. Theory has developed concomitantly to interpret these types of experiments, and key results emphasize the importance of consumer behavior, particularly immigration and emigration behavior in response to environmental variability, in determining the short-term and local population dynamics of stream organisms. However, available theory lacks the ability to "scale up" the results gleaned from examining local population dynamics, thus creating a gap in understanding how these local processes relate to patterns observed over larger reaches of stream. I submit a discrete-patch model of an aquatic insect consumer inhabiting a stream composed of a series of patches that are linked to one another by dispersal. The consumer’s dynamics are described using both the flexible behavioral responses to environmental variation as well as the passive downstream dispersal that are characteristic of aquatic insects. Numerical simulations reveal that, for a large range of specific model forms, the system of linked patches settles at equilibrium to an analytically predictable spatially homogeneous steady-state value in the absence of environmental variation. When local spatial environmental variation is introduced, the length over which the effects of these disturbances decay downstream can also be predicted analytically. I will relate this spatial length scale of decay to specific behavioral and life-history parameters. In addition, I will interpret the results of the model using data collected from high altitude streams in the eastern Sierra Nevada.

Hem Raj Joshi

Solving a parabolic identification problem by optimal control methods

An unknown coefficient of the interaction term of a parabolic system with a Neumann boundary condition in a multi-dimensional bounded domain is identified. The solution of the system represents the concentrations of prey and predator populations. Given partial (perhaps noisy) observations of a true solution in a subdomain, we seek to ``identify" the coefficient of the interaction term using an optimal control technique, involving Tikhonov's regularization. The existence and uniqueness of the optimal control approximating the desired coefficient are obtained, an optimality system is derived, the identification problem is discussed and an example illustrating how to find a solution numerically is presented.

Thomas P. Witelski

Intermediate asymptotics for driven population dynamics in a finite region

Population dynamics described by a nonlinear convection-diffusion are studied using perturbation methods. For the problem of introduction of a new population into a finite, empty region, the dynamics can be described in several stages using similarity solutions, traveling waves, and asymptotic expansions. For regions with finite carrying capacities (due to limited resources), saturation effects will be considered.

J. David Logan

Wave fronts in population models with limited flux

We study the existence of traveling wave fronts in a population model that includes growth, migration, removal, and diffusion. Unlike the classical Fickian model, the diffusive flux is assumed to be a nonlinear function of the gradient, and it remains bounded. The model gives rise to both smooth wave fronts and shock-like fronts, depending upon the parameter values.

Will Wilson

Consuming and Grouping: Resource-mediated animal aggregation

We demonstrate that a simplistic foraging rule for a consumer in a spatially explicit resource environment leads to consumer grouping. Although consumer groups sweeping through the renewing resource environment represents the model's dynamical attractor, for short time scales (represented by a constant total consumer population) three different distributions emerge. At low consumer density, population distributions are variable and spatially fixed, but not grouped. Moving groups erupt at intermediate consumer densities. At high consumer density, there is no spatial variability in the resource and consumer densities. Similar results have been observed in a variety of empirical systems.

Bill Wolesensky

Reactor Models of Digestion Modulation

We develop a chemical reactor model (a batch reactor coupled in series to a plug flow reactor) to link digestion processes and foraging behavior and food quality for certain insects (e.g., grasshoppers and locusts). The model contains feedbacks based on nutrient concentration levels in the haemolymph to determine intermeal delays, i.e., the onset and cessation of feeding.

Glenn Ledder

Dynamic Energy Budget Models with Predation

We formulate a pair of dynamic energy budget models for the growth and reproduction of individual organisms. Both models feature a predation-based survival probability assumption that incorporates the effect of changes in organism size on predation risk. The models differ in that one assumes a net assimilation allocation rule while the other assumes a net production allocation rule. Each of the models is used to define a life history problem with a goal of finding a life history strategy that optimizes the expected lifetime reproductive energy. A central issue is whether the models can correctly predict the possibility that an optimal strategy might be for an individual to grow to a large size in spite of an arbitrarily small probability of survival to maturity. This is counterintuitive in the context of dynamic energy budgets, and yet it clearly happens for many species of fish. Analysis of the models shows that the net production model obtains this result when the predation risk is significantly greater for small individuals than large ones, while the net assimilation model can not yield this result.

Friday, May 17

Saturday, May 18