Mathematics Colloquia and Seminars

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I present a model of the interactions of the entomopathogenic nematode, Heterorhabditis marelatus, and its insect host, Hepialus californicus. The nematode lives in the soil and attacks the larval stage of the host. The free-living infective juveniles of the nematode enters a host through a spiracle or other opening and then regurgitates a symbiotic bacterium, Photorhabdus luminascens, into the inside of the host. The bacteria kills the host and provides a protected environment for the nematode to reproduce. After a period of about six weeks several new infective juveniles emerge from the host cadaver. Young, small hosts give rise to few if any new infective juveniles, while large hosts can produce up to 1,000,000 new individuals. The hosts are univoltine, where as the nematodes reproduce continuously throughout the wet season when the host caterpillars are present in the soil. These two different modes of reproduction lead to a model which uses a continuous time description within a year to obtain a discrete map describing year to year dynamics. A system of delay-differential equations are integrated over a fixed period of time (i.e. the length of the wet season) to generate the discrete return map. The adult moth stage of the host disperses widely, so it is assumed that a constant number of hosts hatch each year. We model the hosts' growth in size, and assume the number of new infective juveniles emerging from a host cadaver is proportional to its size. The dynamics of the model are explored for various levels of initial host density, individual host growth rates, and nematode infectivities.