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Mutual Interference
The Nicholson-Bailey model shows unstable host-parasite (prey-predator) dynamics. Later authors began to examine aspects of host-parasite interactions that could provide stability to the model -- and presumably reflect the assumed stability of host-parasite systems in Nature. In 1969 two British researchers, M. P. Hassell and C. G. Varley modified the Nicholson-Bailey model to incorporate parasite density effects --what they termed "mutual interference". In essence, mutual interference occurs, because searching parasites, upon encountering each other (or cues that another parasite is/has been present) cause one or both parasites to stop searching (and possibly leave the area). The effects of mutual interference would be magnified as parasite density increases, thus potentially providing a "density-dependent" effect that could stabilize the Nicholson-Bailey model. Hassell and Varley noted that a plot of the relationship between the "area of discovery", a, and parasite density (on a log-log scale) showed that as parasite density increases, the area of discovery decreases (Fig. 1).
Figure 1 The equation that describes these data is:
Log a = log a0 - M log P [1.]
where;
a = area of discovery
a0 = area of discovery when P= 1 (e.g., log P = 0)
M = slope of the line
P = number of parasites
Solving equation 1 (e.g., taking antilogs) gives:
a = Q P-m [2.]
Hassell and Varley then re-defined a0 as "Q", the "quest" constant and M to be the mutual interference constant between searching parasites. Inserting equation 2 into the Nicholson-Bailey description of the probability of a host escaping attack (e-aPt) gives:
e-QP-m . P [3.]
which simplifies to;
e-QP(1-m) [4.]
Inserting equation 4 into the Nicholson-Bailey model gives the final form, which has subsequently named the "Hassell-Varley model":
Nt+1 = Nt e-QPt
Pt+1 = Nt [1 - e-QPt(1-m)]
The dynamics of the Hassell-Varley model are different than the Nicholson-Bailey model, as the Hassell-Varley model provides for stable dynamics between parasites and host populations (Fig. 2).
Figure 2 Analysis of the model has shown that stability increases as "m" increases (to a point). Later authors suggested that the effect of mutual interference was not directly due to parasite-parasite encounters per se, but due to the parasites distributing themselves in an aggregative manner with among a patchily distributed host -- an effect called termed "pseudo-interference". Critics of the Hassell-Varley model pointed out that most of the evidence for interference came from laboratory data, where parasite densities and contacts between searching females would be much higher than would occur in Nature.
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Comments? Questions?
Please send feedback to bob_oneil@entm.purdue.edu Copyright © Midwest Institute for Biological Control, 2000 This page was last updated 08.21.00 www.biocontrol/theoriesmodels/mutualinter.html |
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