10.5. Host-Parasitoid Models

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)

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

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

The full model is:

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)

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:

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)

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

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 is the maximum fecundity of parasitoid female. Then:

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:

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.

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: