The theoretical value of encounters with parasitized hosts for parasitoids

Abstract

A female parasitoid searching for hosts in a patch experiences a diminishing encounter rate with unparasitized and thus suitable hosts. To use the available time most efficiently, it constantly has to decide whether to stay in the patch and continue to search for hosts or to search for and travel to another patch in the habitat. Several informational cues can be used to optimize the searching success. Theoretically, encounters with unparasitized hosts should lead to a prolonged search in a given patch if hosts are distributed contagiously. The results of empirical studies strongly support this hypothesis. However, it has, to date, not been investigated theoretically whether encounters with already parasitized hosts (which usually entail time costs) provide a parasitoid with valuable information for the optimization of its search in depletable patches, although the empirical studies concerning this question so far have produced ambiguous results. Building on recent advances in Bayesian foraging strategies, we approached this problem by modeling a priori searching strategies (which differ in the amount of information considered) and then testing them in computer simulations. By comparing the strategies, we were able to determine whether and how encounters with already parasitized hosts can yield information that can be used to enhance a parasitoid’s searching success.

Appendix

The total number of hosts in a patch (at all times) N _t can be described as (see Table 1 for an explanation of the variables):

$$N_{t} = N_{{p_{0} }} + N_{{u_{0} }} $$

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where $N_{{p_{0} }}$ and $N_{{u_{0} }}$ are the numbers of parasitized and unparasitized hosts, respectively, initially present in the patch. The mean number of host encounters per host present λ can be defined as

$$\lambda = {N_{e} } \mathord{\left/ {\vphantom {{N_{e} } {N_{t} }}} \right. \kern-\nulldelimiterspace} {N_{t} }$$

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where N _e is the total number of host encounters (with unparasitized or parasitized hosts) in the patch.

Assuming a Poisson process for host encounters (random search, equal probability for unparasitized and parasitized hosts to be found), the probability to encounter i hosts in the patch is

$$P{\left( {i\,encounters} \right)} = \frac{{\lambda ^{i} }}{{i!}}e^{{ - \lambda }} $$

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The probability that zero parasitizations (=encounters with unparasitized hosts=ovipositions) have occurred conditional on N _e is

$$P{\left( {0\,parasitizations|N_{e} encounters} \right)} = {\left( {\frac{{N_{{p_{{_{0} }} }} }}{{N_{t} }}} \right)}^{{N_{e} }} $$

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Therefore, the probability of encountering N _e hosts and having zero parasitizations is

$$P{\left( {0\,parasitizations\hat{}N_{e} encounters} \right)} = {\left( {\frac{{N_{{p_{o} }} }}{{N_{t} }}} \right)}^{{Ne}} \frac{{\lambda ^{{Ne}} }}{{N_{e} !}}e^{{ - \lambda }} $$

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Now, irrespective of N _e, the expected probability of encountering zero of the initially present $N_{{u_{o} }}$ unparasitized hosts and therefore having zero parasitizations is

$$\begin{array}{*{20}c} {{{\sum\limits_{i = 0}^\infty {{\left[ {{\left( {\frac{{N_{{p_{0} }} }}{{N_{t} }}} \right)}^{i} \frac{{\lambda ^{i} }}{{i!}}e^{{ - \lambda }} } \right]}} }}} & {{ = e^{{ - \lambda }} {\sum\limits_{i = 0}^\infty {{\left[ {{\left( {\frac{{N_{{p_{0} }} }}{{N_{t} }}\lambda } \right)}^{i} \frac{1}{{i!}}} \right]}} }}} \\ {{}} & {{ = e^{{ - \lambda }} e^{{\frac{{N_{{p_{{_{0} }} }} }}{{N_{t} }}\lambda }} = e^{{ - \lambda {\left( {\frac{{N_{{u_{0} }} }}{{N_{t} }}} \right)}}} }} \\ \end{array} $$

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Thus, the expected number of currently unparasitized hosts N _u is:

$$N_{u} = N_{{u_{0} }} e^{{ - \lambda \frac{{N_{{u_{0} }} }}{{N_{t} }}}} $$

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Because of the obvious relation $N_{u} = N_{{u_{0} }} - N_{a}$, with N _a the number of unparasitized hosts encountered, it is possible to calculate λ without N _e, thus eliminating one of the unknown variables. $N_{u} - N_{a} = N_{{u_{0} }} e^{{ - \lambda \frac{{N_{{u_{0} }} }}{{N_{t} }}}}$ will give us

$$\lambda = \frac{{N_{t} }}{{N_{{u_{0} }} }}ln{\left( {\frac{{N_{{u_{0} }} }}{{N_{{u_{0} }} - N_{a} }}} \right)}$$

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If only one parasitoid is allowed in the patch (number of foraging parasitoids P _t=1), the “attack equation” developed by Rogers (1972) for foragers in depletable patches, by substituting the respective terms, can now be modified as follows (z=search time):

$$\begin{array}{*{20}c} {{N_{a} = N_{{u_{0} }} {\left( {1 - \exp {\left\{ { - a'z} \right\}}} \right)}}} \\ {{ = N_{{u_{0} }} {\left( {1 - \exp {\left\{ { - a'{\left( {T_{t} - N_{a} T_{h} - N_{r} T_{r} } \right)}} \right\}}} \right)}}} \\ {{ = N_{{u_{0} }} {\left( {1 - \exp {\left\{ { - a'{\left( {T_{t} - N_{a} T_{h} - {\left[ {N_{e} - N_{a} } \right]}T_{r} } \right)}} \right\}}} \right)}}} \\ {{ = N_{{u_{0} }} {\left( {1 - \exp {\left\{ { - a'{\left( {T_{t} - N_{a} T_{h} - {\left[ {\lambda N_{t} - N_{a} } \right]}T_{r} } \right)}} \right\}}} \right)}}} \\ {{ = N_{{u_{0} }} {\left( {1 - \exp {\left\{ { - a'{\left( {T_{t} - N_{a} T_{h} - {\left[ {\frac{{N^{2}_{t} }}{{N_{{u_{0} }} }}{\text{ln}}{\left( {\frac{{N_{{u_{0} }} }}{{N_{{u_{0} }} - N_{a} }}} \right)} - N_{a} } \right]}T_{r} } \right)}} \right\}}} \right)}}} \\ \end{array} $$

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