Individual-based ecological models: Spatial models for undergraduate investigation Louis J. Gross Departments of Mathematics and Ecology and Evolutionary Biology University of Tennessee, Knoxville, TN 37996-1300 gross@math.utk.edu Handout for Mathematics for Life Sciences Students Conference, Iowa State University, May 17-18, 1996 An ongoing discussion amongst those who work on theory in biology deals with the relative benefits of reductionist versus holistic viewpoints. Reductionists tend to argue for breaking systems down to their smallest feasible objects of interest, trying to build up from mechanisms at this small scale to larger scale phenomena. Those with a more holistic view prefer to deal with aggregated properties of systems first, and then add to it complications imposed by smaller scale structure within the system. These contrasting views of the process of theory construction are mirrored in the general process of model construction in any field. There are those who much prefer to build highly simplified models first, and then add new variables and constructs in order to increase the realism, while others prefer to start by including every aspect of the system which may be important, perhaps eventually discarding much of it as the process proceeds. The alternative views may thus converge towards similar models (as measured by the number of state variables say), but not necessarily. My experience is that, at least in theoretical biology, those trained as mathematicians tend towards the holistic end of this spectrum (and it is indeed more of a continuum than just two distinct contrasting views), while those with more of an engineering background tend towards the reductionist view. A prime example of the contrast between these approaches occurs in population biology. The vast majority of theory in population biology has started from very simple differential equations in which a single variable represents population density, solutions of these are analyzed mathematically, and potentially compared to abundance estimates from field or lab observations. Although these models have had a great influence on theory in ecology, their aggregated form is particularly difficult to relate to observational biology. As this aggregated view of a population is highly simplified, a wide variety of extensions have been made to incorporate (1) size, age or physiological structure (leading to coupled systems of differential equations or partial differential equations); (2) space (leading to metapopulation models in which a population is broken down into distinct patches, or in the continuous space case to partial differential equations); (3) discrete generations (leading to difference equation models and matrix models); (4) stochastic effects (leading to birth and death processes and stochastic differential equations). In all these cases, as extensions are added, the models become less analytically tractable, and become considerably more difficult for students to follow, particularly life science students with a typically minimal exposure to college level mathematics. A consequence of the above is that undergraduate life science students, if exposed to any mathematical models in ecology at all, will generally see only a few cases of very general models, which even they realize are caricatures only minimally related to real biology. Alternatively, they may be exposed to computer programs (such as Populus) which calculate the mathematical behavior of more complicated models, though they typically have little ability to follow the mathematics which underlies the models. Happily, there are now some options which allow undergraduates to deal personally with much more realistic models based on more of a reductionist world view. Exposing students to these models not only allows them to individually investigate some quite fascinating ecological phenomena, but also shows them first hand that modern theory construction is not bound by the limits of analytical mathematics. Computational ecology will continue to grow in importance, and it is now quite feasible to expose undergraduates to its promise (as well as its limitations). This further illustrates for students that there is not one single "correct" approach to science, and that there are a diverse array of mathematical and computational tools available - they are not necessarily limited by the smattering of calculus and basic discrete math that is all the quantitative training most students obtain. Other advantages of exposing students to these computational approaches is that it gives them control over "an experiment" - due to it's stochastic nature no two students (or lab groups) will obtain exactly the same results, and each can be very free to investigate the effects of a wide variety of different effects. This can also of course be a disadvantage, as it gives the potential for students to control experiments with many degrees of freedom. Some constraints and guidance in the labs is therefore required. One recent reductionist approach in ecology analyzes systems based upon the actions of individuals. These individual-based models track the behavior, growth, reproduction and death of individuals, from which they build up the dynamics of aggregated units such as populations and communities. These individual-based models are beginning to have significant impact on a variety of theoretical and practical questions in ecology, and there are now a few such models which are available for teaching purposes. The program we will use in this Workshop is EcoBeaker, written by Eli Meir and published by Sinauer Press, which has associated with it a variety of ecological scenarios preprogrammed to illustrate key ecological concepts (see the WWW site http://www.zoology.washington.edu/ecobeaker/ecobeaker.html). Other individual-based modeling resources include those at the National Micropolution Simulation Resource (with a biomedical emphasis - see http://www.nmsr.labmed.umn.edu/nmsr/NMSR.html), the Swarm program based at the Sante Fe Institute (see http://www.santafe.edu/projects/swarm/), David Griffeath's Ecomachine project (see his Primordial Soup Kitchen Page at http://math.wisc.edu/~griffeat/kitchen.html), and Rick Durrett's stochastic spatial models (see http://www.tc.cornell.edu/er94/ff02spring/ff04models.html). A basic, but now outdated, reference is DeAngelis, D. L. and L. J. Gross (1992) (editors) (1992) Individual-Based Models and Approaches in Ecology, Routledge, Chapman and Hall, New York. There are several ways to construct individual-based models. My purpose here is to just mention the basics for models with explicit spatial structure. In this case, some type of spatial grid is set up, in which each location (or pixel in a computer map) corresponds to a spatial location with a certain spatial extent for each grid cell. Each grid cell then is characterized by a state, with each state corresponding to (a) presence or absence of a given species, (b) number of a given species present in the cell, (c) numbers of each of several species present in the cell, etc. There are then two basic approaches: (i) model how each grid cell changes based upon some set of rules and the states of surrounding grid cells, or (ii) model how each individual organism being considered moves among the grid cells and how the organisms state changes through time. Type (i) is the cellular automata approach, in which each cell is in one of a relatively small number of states and the rules for changes in state depend on the current state and that of nearest neighbors. The program SimLife from Maxis is of this type, in which each grid location consists of a single individual of a given species. Type (ii) is a truly individual-based approach, in which individual organisms can move anywhere across the spatial set of grid cells, change their respective states (e.g. size, fat content, number of offspring, etc.) and have their location attached to them as just another state variable. In this approach, each grid cell just combines the individuals located in it at any particular time. The program EcoBeaker allows one to combine both of the above approaches. It allows allows one to set up changes in states of a grid cell as a simple transition matrix (possibly dependent on neighboring cells) as for example used in the Situation File to illustrate the Intermediate Disturbance Hypothesis. In this File, each grid cell is assigned to a state based upon the species present (e.g. Grasses, Blackberry Bushes, Oak Trees, Fire, etc.), with a certain probability of transition to a new state the following year. Alternatively, the program allows species to be set up as Individualistic, in which they have certain movement rules, and in which there can be several individuals of different species extant in a given grid cell at any particular time. The Situation File to Illustrate Competitive Exclusion is one example of this, in which the various Rabbit species have differing daily energy requirements, and movement rules. In addition to allowing students to change the parameters associated with any particular species, to add new species, and to code entirely new scenarios (admittedly this is something very few students would attempt), EcoBeaker also allows students to investigate the effects of different sampling methods. While EcoBeaker is structured to be a very easily applied tool, the complexity of model output is also instructive for students. They have to decide what are the appropriate output variables to plot, and to come up with their own methods to summarize results. Though reminiscent of real field experiments, the program output is obviously more constrained. It does serve to illustrate however that, though such models are not that difficult to construct, the interpretation of results is non-trivial, particularly when trying to pick apart the effects of different model parameters. This is of course just as true for more realistic individual-based models, designed not for education but for application to ecosystem management. On this point, see L. J. Gross (1994) Limitations of reductionist approaches in ecological modeling: model evaluation, model complexity and environmental policy. In Wildlife Toxicology and Population Modeling: Integrated Studies of Agroecosystems (Eds. R. J. Kendall and T. E. Lacher), pp. 509-518. Lewis Publishers and CRC Press, Boca Raton.