Details

For , the system of equations is

where

: time : uninfected host larva density : host adult density : parasitoid adult density : rate of host larvae maturating to host adults : attack rate of parasitoids : rate of larva production for host adults : death rate of host adults : death rate of parasitoids : maturation delay for noninfected larvae : maturation delay for infected larvae

To start the system, we set constant adult host and parasitoid densities (for )

, ,

and calculate the beginning larva density consistently (for )

.

A note on the implementation: in order to numerically solve the system of delay differential equations with the Mathematica built-in function NDSolve, we calculate the integral in the formula for by introducing another state variable

,

which we define for by

and for by

.

Reference

[1] W. W. Murdoch, R. M. Nisbet, S. P. Blythe, W. S. C. Gurney, and J. D. Reeve, "An Invulnerable Age Class and Stability in Delay-Differential Parasitoid-Host Models," The American Naturalist, 129(2), 1987 pp. 263–282. www.jstor.org/stable/2462003.