## Summer 2003 Program in Modeling Complex Ecosystems

**Mentored by Jay Walton and Tom Wehrly**

**Student Participants:** from REU: Susan Ficken, Byron Van
Gorp, Liz Lang, Stephanie Green, Theresa Johnston, Jennifer Owens,
Charlie O'Connor, Brian Sullivan, Peter Horn, Nicholas Hawkins, Ted
Lai; VIGRE participants: BELCHER, JEB EVERETT; CHEN, XIANJIN ;
DOSTERT, PAUL FRANCIS; FUSELIER, EDWARD JULES; GREEN, STEPHANIE
RAE; JIANG, LIJIAN; KHALMANOVA, DINARA KHA; MARCHANKA, YULIYA;
SENDOVA, TSVETANKA BOZ; WESTBROOK, RYAN SCOTT; ZHANG, ZHIGANG

**General Description.** In this interdisciplinary seminar we
will discuss some mathematical aspects of ecosystems modeling. Its
main emphasis will lie on the impact of changes in the landscape
topology on populations, including its distribution and dynamics.
Two subareas will be emphasized - deterministic modeling using
differential equations and stochastic modeling, which uses
techniques from statistics.

The seminar is intended for (graduate and advanced undergraduate) students from both Mathematics and Biological Sciences programs. Students in the Mathematics program will be introduced to a variety of situations and problems related to ecosystems where basic concepts of topology, geometry, statistics and differential equations will be applied. Students in Biological Sciences programs will learn new mathematical tools and concepts that will form a solid basis for modeling ecosystems. In either situation, the students will broaden their education and have a glimpse of some of the many contributions that Mathematics can provide to the study of biological systems.

Throughout the course, various modeling problems will be presented, and students will learn how to use the concepts presented in the seminar to solve these problems. Student participation will be a key ingredient in the seminar, as an attempt to approach the solutions to proposed problems via a broad discussion among the participants.

One topic we will examine carefully is the habitat fragmentation and re-connection question, which is concerned with the following issue. Decades of world wide wilderness land use policy decisions have resulted in many wilderness areas being broken into chains of isolated patches of national parks, forests, refuges or other protected zones. While the total area of these patches might seem considerable, their fragmented structure into relatively small patches might not provide viable habitats for many species, especially large predators. Many ecologists argue that the health of the predatory species at the top of the food chain provide a reliable barometer of the health of the entire ecological system. Consequently, there are large scale programs being proposed and in some cases already under way to connect fragmented habitats via corridors which will permit certain species, especially large predators, to travel between formerly isolated patches. The mathematical problem is to model this process in order to try to gauge the likelihood that it will be effective in making for more robust wildlife habitats, or whether it could cause a negative disruption to a habitat system and make matters worse.

The biodiversity/eco-stability debate addresses the assertion that greater biodiversity leads to increased stability of an ecosystem, i.e. the more diverse the gene pool, the more robust the ecosystem is to perturbations. This is a long term politically and ethically charged debate in need of rigorous quantitative scientific input.

Among the mathematical tools to be used are notions from finite dimensional dynamical systems, partial differential equations and differential geometry. Although sophisticated models could involve difficult concepts from these subjects, simplified models can be investigated by bright undergraduates with minimal background (but with a healthy enthusiasm to learn). The students are introduced to a variety of deterministic approaches to modeling ecosystems beginning with the classical Lotka/Volterra system of interacting species or predator/prey models. Very elementary topological ideas are used to quantify the fragmented structure of habitats from the point of view of individual species. More specifically, different species might see a given habitat system as having different connected components or patches. For example, certain bird species might see a habitat as connected whereas some plant or small, crawling animal species might see it as highly fragmented. Thus, the first task in studying an ecosystem is to determine its patches, or connected components, from the point of view of each of the species selected for study. Then one introduces interacting species models on the landscape.

The simplest level of modeling of species interaction dynamics on a fragmented habitat with corridors involves using ordinary differential equations for the densities of the species on the system of patches permitting migration of certain species between certain patches. These problems are ideal for undergraduate students to tackle, and preliminary results from the summer 2002 program suggest that selective migration can indeed qualitatively alter the dynamics over isolated patches without migration. Some groups of REU/VIGRE students will be given projects extending these ODE models with migration to more than two patches and to more than two interacting species. Among the many possible questions to pursue is whether or not it matters which of the species has the ability to migrate? For example, in predator/prey type interactions, does one see similar dynamics if just the predator can migrate or if just the prey can migrate?

Other groups of students will be given projects modeling the spatial distribution and movement of species within patches as well as between patches. These models will be of two types, continuous time partial differential equation models (the simplest being of diffusion type) and discrete time, convolution integral models (as proposed recently by Alan Hastings). Some groups will use simple diffusion operators to model the movement of species around a flat landscape, while other groups will study generalizations of these operators to non-flat surfaces using ideas from differential geometry.

As indicated above, past ecosystem modeling in the REU/VIGRE seminars was focussed exclusively upon deterministic models. We propose to have a subset of the students pursue statistical approaches to ecosystem modeling. Again, many avenues are open for these projects including stochastic population models. These are particularly useful for modelling the transient behavior of populations and also the behavior of small populations.

The basic ideas of stochastic population models will be introduced. The methodology is based on Markov models. Basic assumptions about the population lead to a system of Kolmogorov differential equations for the probability functions. Generating functions are useful in obtaining the moments or cumulants of the population size. The Kolmogorov differential equations lead naturally using the generating function approach to systems of differential equations for the cumulants. These can be solved analytically or numerically to describe the population. This methodology can be applied to models for a single population including both linear birth-immigration-death models and nonlinear birth-immigration-death models.

Our goal is to extend the basic ideas illustrated in single-population models to model multiple populations such as predators and prey. We also want to bring spatial aspects into the modelling by allowing for several locations. This leads us to developing models for multiple populations. Joint moments and cumulants for the sizes of multiple populations are introduced. In a manner analogous to that for a single population, bivariate Kolmogorov differential equations lead naturally using the generating function approach to systems of differential equations for the bivariate cumulants. This approach is illustrated by applying the methodology to multiple population models that can include births, deaths, immigration, and migration.

Much of our interest will be on the bivariate distribution of the number of predator and prey animals over the course of time. The above methodology enables us to obtain the cumulants of the bivariate distribution for any time. To approximate the corresponding bivariate density, such methods as saddlepoint approximations and series expansions will be used. Such research is helpful in the formulation of management strategies. Researchers can apply the stochastic models to determine the effects of various control procedures such as reducing birth rate, increasing a death rate, and restricting immigration on the bivariate distribution of the numbers of predators and prey.

The students' research will be concerned with the development and application of these stochastic methods to the analysis of models for various predator-prey combinations. Matis and Kiffe [MK] were successful in modelling bee/mite populations using this approach. Students will obtain data for other animal pairs from the literature and will investigate the applicability of the population models to these data. The bivariate distributions of predator/prey counts will be approximated using cumulants and saddlepoint methods. Computer simulations will demonstrate the variability inherent in these stochastic models. Another proposal is to develop models that incorporate spatial compartments. Stochastic predator-prey models with spatial compartments have not yet been developed.