Delay-Differential Equations with Constant Lags

Abstract

1 Delayed Effects Ordinary differential equations (ODEs) are widely used in modeling, but it is becoming more common that models take into account effects that have a delayed action. The populations y 1 (t) of prey and y 2 (t) of predator are often modeled by a system of first-order ODEs y 1 (t) = a y 1 (t) + b y 1 (t) y 2 (t) y 2 (t) = c y 1 (t) + d y 1 (t) y 2 (t) with constants a, b, c, d. Hale [1] considers variations of this model that have a resource limitation on the prey and a birth rate of predators that responds to changes in the populations only after a constant time lag. These models have the form y 1 (t) = a y 1 (t) 1 − y 1 (t) m + b y 1 (t) y 2 (t) y 2 (t) = c y 1 (t) + d y 1 (t − τ) y 2 (t − τ) This is a first-order system of delay-differential equations (DDEs). Here τ > 0 is a lag in the effect of population changes and m is another parameter. Only one lag appears in this system, but in general there might be lags τ 1 ,. .. , τ k. In some applications a lag τ j (t, y) might depend on the time t and/or state vector y. New phenomena are possible with these more general lags, so for the sake of simplicity, we consider only constant lags in this article.