10.5. Host-Parasitoid Models

   Parasitoids are insect species which larvae develop as parasites on other insect
   species. Parasitoid larvae usually kill its host (some times the host is
   paralyzed by ovipositing parasitoid female) whereas adult parasitoids are
   free-living insects (see [1]images of parasitoids). Most of parasitoid species
   are either wasps or flies.

   Parasitoids and their hosts often have synchronized life-cycles, e.g., both have
   one generation per year (monovoltinous). Thus, host-parasite models usually use
   discrete time steps that correspond to generations (years).

Model of Thompson (1922)

   The model assumes that female parasitoids lay their eggs randomly on host
   individuals and do not distinguish between healthy and already parasitized hosts.
   In this case, the number of parasitoid eggs laid on one host should have a
   poisson distribution:

                                      [eq20.gif]

   where p(i) is the proportion of hosts that get i parasitoid eggs, and M is the
   mean number of parasitoid eggs per one host.

   Survived hosts are those which get 0 parasitoid eggs. The proportion of survived
   hosts is equal to p(0) = exp(-M).

   Variables:
     * P = Density of parasitoid females
     * H = Density of hosts

   Parameter
     * F = Parasitoid fecundity (no. of eggs laid by 1 female)

   PF = Density of eggs laid by all parasitoid females per unit area
   [eq21.gif] = Average no. of eggs per host individual
   Then, host survival is

                                      [eq22.gif]

   The full model is:

                                      [eq23.gif]

   The first equation describes host survival and reproduction. The numbers of
   survived hosts are multiplied by Ro which means reproduction.

   In the second equation, each parasitized host produce one adult parasitoid in the
   next generation. P is the density of females only. Thus, the numbers of
   parasitoids is multiplied by the proportion of females = q.

   In the model of Thompson, it is assumed that parasites always lay all their eggs.
   Thus, realized fecundity equals potential fecundity. This assumption implies
   unlimited search abilities of parasitoids. In nature, parasites often do not
   realize their potential fecundity just because they can not find enough hosts.
   Thus, the model of Thompson may overestimate parasitism rates especially if host
   density is low.

Model of Nicholson and Bailey (1935)

   This model is more realistic than the Thompson's model and is widely used by
   ecologists. It assumes that parasitoid female is able to examine area a ("area of
   discovery") during its life time. When a host is found, parasitoid lays only one
   egg in it. However, the same host can be found again later and then the parasite
   will lay another egg in it because we assume that parasites do not distinguish
   between healthy hosts and already parasitized hosts.

   Because each encounter with the host results in depositing 1 egg, the realized
   fecundity equals the product of the area of discovery and host density: F = aH.
   Substituting this value of F into the Thompson model we get:

                                      [eq24.gif]

   In the Nicholson and Bailey model, the potential fecundity of parasites is not
   limited. Parasites lay an egg at every encounter with the host even if the number
   of encounters is very large (e.g., if host density is high). Thus, this model may
   overestimate parasitism rates at high host density.

Model of Rogers (1972)

   The model of Rogers applies the model of Holling, which was originally developed
   for predator-prey systems, to host- parasite systems. It assumes two kinds of
   limitations in host-parasitoid interactions: limited parasitoid fecundity (as in
   the model of Thompson) and limited search rate (as in the model of Nicholson and
   Bailey).

   We will use the Holling's disc equation (see [2]section 10.3) to model the
   functional response of parasitoids. The number of hosts attacked by one
   parasitoid female is equal to

                                      [eq24a.gif]

   We can modify this equation by setting T=1 because search rate is considered per
   life time of parasitoid female. Life time can be coded as 1 because the time step
   is equal to 1 generation. The ratio [eq26.gif] is the maximum fecundity of
   parasitoid female. Then:

                                      [eq25.gif]

   When parasitoid female attacks a host it lays an egg. Thus, realized fecundity F
   = Ha. Substituting this value of F into the Thompson's model we get:

                                      [eq27.gif]

   In the model of Rogers, realized fecundity is different from the potential
   fecundity whereas in previous models this distinction was not present.

   All models of host-parasitoid system are unstable: they generate oscillations
   with increasing amplitude.

                                    [gnichdyn.gif]

   This is the dynamics of the model of Nicholson and Bailey.

   However, in nature host-parasitoid population never show oscillations with
   infinitely increasing amplitude. This is not because the models do not capture
   the mechanisms of host-parasitoid interactions, but because additional ecological
   processes (e.g., intraspecific competition in hosts or in parasitoids) can
   partially or completely stabilize the system. It was also shown that spatial
   heterogeneity and parasitoid dispersal among host patches may also stabilize the
   population system.

References:

   Nicholson, A. J., and V. A. Bailey. 1935. The balance of animal populations.
   Proceedings of the Zool. Soc. of London. 1: 551-598.
   Rogers, D. J. 1972. Random search and insect population models. J. Animal Ecol.
   41: 369-383.
   Thompson, W. R. 1929. On the relative value of parasites and predators in the
   biological control of insect pests. Bull. Entomol. Res. 19: 343-350.

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   [6]Alexei Sharov 1/12/96

References

   1. http://www.forst.uni-muenchen.de/LST/ZOO/PHERODIP/DIPRIONIDAE/PARASITOIDS/parasitoid_images.html
   2. http://alexei.nfshost.com/PopEcol/lec10/funcresp.html
   3. http://alexei.nfshost.com/PopEcol/lec10/fullmod.html
   4. http://alexei.nfshost.com/PopEcol/lec10/predat.html
   5. http://alexei.nfshost.com/PopEcol/lec10/pathogen.html
   6. http://alexei.nfshost.com/~sharov/alexei.html