Two-dimensional distributed delays for simulating two competing biologicalprocesses

We describe computer algorithms for modeling insect disease and other biolo gical processes which may be incorporated in detailed simulation models of biological control systems. The method uses "time varying delays" (TVDDs) w hich have been used previously to model insect development. A TVDD lags an input and distributes the output according to a selected probability densit y function, which can range from exponential to normal to a discrete pulse. The computational methods for implementing TVDDs are relatively simple, an d the data requirements (e.g., stage-specific developmental times) are ofte n available from the literature, or commonly collected. A TVDD used to mode l a single biological process (commonly insect development), we refer to as a one-dimensional (ID) TVDD. To model two interacting species, we expand t he TVDD concept into a second dimension which allows simulation of two simu ltaneous biological processes; e.g., development of the insect, and incubat ion of the disease. These processes are assumed to be independent, requirin g independent data sets. We provide a theoretical basis for the two-dimensi onal (2D) TVDD, and pseudocode for implementing both 1D- and 2D-TVDDs. We a lso use published examples to illustrate the utility of the 2D-TVDD method for simulating two competitive biological processes. Finally, we discuss a simulation tool, HERMES (Hierarchical Environment for Research Modeling of Ecological Systems), with which biologists can create detailed mechanistic simulation models by manipulating graphical icons representing underlying m athematical modeling components. HERMES provides a variety of simulation co mponents including representations of both 1D- and 2D-TVDDs; a Power Macint osh version of HERMES and a user's manual are available from the authors.