Mathematical Models with Delay: An Introduction

LIDIA SZCZUPAK; ROCÍO BALDERRAMA; HORACIO ROTSTEIN

Jornada; Distinguished young researcher talk.; 2018

The use of differential equations to model biological systemshas a long history, for example, Malthus and Verhulst employedthem in models of population and Lotka and Volterra in theprey-predator model.These models are used to understand complicatedphenomena, it is clear that the simplest models cannot capturethe variety of dynamics observed in natural systems.There are many possible approaches to dealing with thesecomplexities:One can construct larger systems of differential equations.These systems can be quite good at approximating observedbehavior, but they contain many parameters.Obtain an intuitivesense of which components are most important in determininga behavior regime can be quite difficult.Another approach is the inclusion of time delay terms in thedifferential equations. The delays can represent gestationtimes, incubation periods, transport delays, or can simply lumpcomplicated biological processes together, accounting only forthe time required for these processes to occur.delay models have the advantage of combining a simple,intuitive derivation with a wide variety of possible behaviorregimes for a single systemthese models have the disadvantage of hide much of thedetailed workings of complex biological systems, and it issometimes precisely these details which are of interest. Inaddition, from the mathematical point of view, delay modelsare more complex to study.We would like to understand. . .whether the introduction of time delays might enrich thedynamics of the models, or whether their behavior isessentially the same as the ordinary differential equationsmodels they modify.when the introduction of time delays introduce ?artificial?behavior to the model.