Glenn Ledder † and Brigitte Tenhumberg ‡† † Department of Mathematics ‡ School of Biological Sciences University of Nebraska-Lincoln *funded by NSF grant DUE An “Experimental” Interdisciplinary Course in Mathematical Ecology*

Overview DESIGN Goal Design Issues Pedagogical Principles The Research Topic IMPLEMENTATION Laboratory Activities “Lecture” Activities A Virtual Laboratory Results

Goals Project Goal: Prepare a cadre of young scientists who can do interdisciplinary research in mathematics/biology Course Goal: Introduce interdisciplinary research in mathematics/biology to talented students at an early stage in their careers.

Design Issues The course must be self-contained. –We cannot assume knowledge of calculus, statistics, or any specific biology topic. –We cannot assume laboratory experience.

Design Issues The course must be self-contained. –We cannot assume knowledge of calculus, statistics, or any specific biology topic. –We cannot assume laboratory experience. The course must be integrated at different levels. –Math and biology –Theory and experiment –Research design, conduct, and dissemination

Pedagogical Principles The experiments must all contribute to a coherent body of theory. The math must always be motivated by the experiments. Background material must be presented when needed. Students must be asked to assemble the pieces into an integrated whole.

The Research Topic General Theme: biological pest control

The Research Topic General Theme: biological pest control Specific Research Questions: –How does an aphid population grow? The theory must incorporate the aphid life cycle. The theory must be formulated as a mathematical model. The theory must be quantitatively realistic.

The Research Topic General Theme: biological pest control Specific Research Questions: –How does an aphid population grow? The theory must incorporate the aphid life cycle. The theory must be formulated as a mathematical model. The theory must be quantitatively realistic. –How does predation by lady beetles affect aphid populations? The theory must be formulated as a mathematical model.

The Greenbug Aphid: S. graminum –Many greenbug colonies consist of viviparous (born live) females that reproduce asexually. Sexual reproduction occurs only when conditions require overwintering.

The Greenbug Aphid: S. graminum –Many greenbug colonies consist of viviparous (born live) females that reproduce asexually. Sexual reproduction occurs only when conditions require overwintering. –Greenbugs have 5 developmental stages: 4 instars of juveniles plus adult.

The Greenbug Aphid: S. graminum –Many greenbug colonies consist of viviparous (born live) females that reproduce asexually. Sexual reproduction occurs only when conditions require overwintering. –Greenbugs have 5 developmental stages: 4 instars of juveniles plus adult. –Plant damage results from toxic saliva rather than loss of nutrients. Severe crop damage occurs before the aphids become short of food. (we can ignore density effects)

Theoretical Population Dynamics Discrete linear stage-structured model: x t+1 = M x t, where x is a vector giving the populations of the different stages and M is a matrix of parameters

Theoretical Population Dynamics Discrete linear stage-structured model: x t+1 = M x t, where x is a vector giving the populations of the different stages and M is a matrix of parameters Research tasks: –construct the model –estimate the parameters –predict population growth –test the predictions

Laboratory Activities Clip-cage experiments: –Put 1 newborn aphid in a clip-cage on a live sorghum leaf. –Check its development stage daily by looking for exuvies. –After it becomes an adult, count its daily offspring. Population growth experiments: –Put 1 adult aphid in a large cage with a sorghum plant. –Count the aphids each day.

“Lecture” Activities 1.Mathematical modeling a.Develop model b.Determine parameter values from clip-cage experiments c.Run Matlab simulation with model d.Compare simulation results with population growth experiment data

“Lecture” Activities 1.Mathematical modeling a.Develop model b.Determine parameter values from clip-cage experiments c.Run Matlab simulation with model d.Compare simulation results with population growth experiment data 2.Discovery of model behavior

“Lecture” Activities 1.Mathematical modeling a.Develop model b.Determine parameter values from clip-cage experiments c.Run Matlab simulation with model d.Compare simulation results with population growth experiment data 2.Discovery of model behavior 3.Model analysis a.Find eigenvalues and eigenvectors b.Determine growth rate and proportions

Teaching Modeling Skills Modeling process: a.Develop model b.Determine parameter values c.Run Matlab simulation d.Check simulation results Issues: –Data takes a long time to collect. –The real model is 5-dimensional.

Presenting Bugbox, a simple computer simulation for structured population dynamics! Because Bugbox is a simulation, its behavior doesn’t necessarily match any real insect population. It functions as a biology lab for a virtual world. Boxbugs are simpler than real insects: –They don’t move. –Development rate is chosen by the experimenter. –Each life stage has a distinctive appearance. larva pupa adult