## Ecology

Properties of organisms change during their lifetimes: their birth and death rates may change as they age, they may grow larger, and they may pass through multiple life cycle stages. As such changes occur, individuals may affect the overall population growth differently. Thus, in many instances in order to understand population growth, it is necessary to keep track of the population composition, or structure, through time. Accounting for structure leads to a population model that is a partial differential equation for the population density function. At the same time, properties of individuals can be affected by the total population size. For example, death rates might increase, or birth and growth rates decrease, in a crowded population. Allowing demographic rates to depend on the total population size makes the partial differential equation model nonlinear.

A main question in ecology is whether a population will persist. Mathematically, are equilibrium solutions stable? For the general model, even this fairly simple question is currently impossible to answer analytically. In work with Richard Vance and William Newman [1], we addressed this issue when the demographic rates depend only on population size. The model reduced to nonlinear ordinary differential equations and could be analyzed using standard techniques. In subsequent work [2], we considered nonlinear age-structured population models with individuals belonging to one of two age classes, juveniles or adults. Within each class the demographic rates were constant but varied between the classes. As a consequence of this assumption, the governing equation reduced to a system of ordinary delay differential equations. These equations were analyzed using a combination of analytical and numerical methods. Contrary to a prevalent view in theoretical ecology that time delays are destabilizing, we found that for a wide range of parameters a stable age distribution and stable total population size were reached, independent of the length of the time delay. The principal difference in our analysis was that a full accounting was made of the number of individuals in each class as a function of the length of the time delay. That is, with all else held fixed, just increasing the length of the juvenile phase could result in an increase in the number of juveniles. The dynamic consequences of varying the length of the delay were not included in most other studies.

For a general model, numerical methods provide the only means for determining stability of equilibria. Of course, numerical methods also allow study of transient effects such as the recovery of a stressed population back to equilibrium, and other more complicated effects currently beyond the scope of analysis. The partial differential equations of structured population models have rarely been treated numerically. In two papers [3-4] methods that are widely used in the study of compressible fluid flow were adapted to the study of age or mass structured populations. The common link is that both problems involve a hyperbolic conservation law. The methods have proven to be reliable and efficient on a wide range of possible dynamic behaviors that arise in models of blowflies, squirrels and mosquitofish.

# References:

• [1] Richard R. Vance, William I. Newman and Deborah Sulsky, The Demographic Meanings of the Classical Population Growth Models of Ecology, Theor. Pop. Biol., 33: 199-225 (1988).
• [2] Deborah Sulsky, Richard R. Vance and William I. Newman, Time Delays in Age Structured Populations, J. Theor. Biol., 141: 403-422 (1989).
• [3] Deborah Sulsky, Numerical Solution of Structured Population Models I. Age-Structure, J. Math. Biol., 31: 817-839 (1993).
• [4] Deborah Sulsky, Numerical Solution of Structured Population Models II. Mass-Structure, J. Math. Biol., 32: 491-514 (1994).