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Modeling Parasitoid-Host Dynamics with Delay Differential Equations

A parasitoid is an organism that inhabits a host organism. Unlike a true parasite, however, it ultimately sterilizes or consumes the host, a more dire prognosis for the host.
This Demonstration shows the population dynamics of a parasitoid-host model. Parasitoids attack host larvae, which then develop into new parasitoids instead of adult hosts. Maturation delays for host and parasitoid are explicitly taken into account by implementing delay differential equations. The model is adopted from [1] in a slightly simplified form.

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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.
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