The evolution of host manipulation by parasites: a game theory analysis

Abstract

Many parasites are known to manipulate the behaviour of intermediate hosts in order to increase their probability of transmission to definitive hosts. This manipulation must have costs. Here we explore the combined effects of three such costs on the amount of effort a parasite should expend on host manipulation. Manipulation can have direct costs to future reproductive success due to energy expended to manipulate the host. There may also be indirect costs if other parasites infect the host and profit from the manipulation without paying the cost of manipulation. These “free riders” may impose a third cost by competing with manipulators for resources within the host. Using game theory analysis and several different competition models we show that intrahost competition will decrease the investment that a parasite should make in manipulation but that manipulation can, under some circumstances, be a profitable strategy even in the presence of non-manipulating competitors. The key determinants of the manipulator’s success and its investment in manipulation are the relatedness among parasites within the host, the ratio of the passive transmission rate to the efficiency of increasing transmission rate and the strength of competitive effects. Manipulation, when exploited by others, becomes an altruistic behaviour. Thus we suggest that our model may be generally applicable to cases where organisms can exploit the investment of others (possibly kin) while also competing with the organism whose investment they exploit.

Appendices

Appendix 1: Comparison of analyses based on five different competition models

• (a) Fitness equations used for the competition models

MODEL

$$ {\text{Logistic}}\quad \left( {1 - x - {\raise0.7ex\hbox{$N$} \!\mathord{\left/ {\vphantom {N K}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$K$}}} \right)\left( {p_{0} + bN\overline{x} } \right) $$

$$ {\text{Monod }}\quad \left( {{\frac{K - N}{a + K - N}} - x} \right)\left( {p_{0} + bN\overline{x} } \right) $$

$$ {\text{Ricker }}\quad e^{{\left( {1 - x - {\raise0.7ex\hbox{$N$} \!\mathord{\left/ {\vphantom {N K}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$K$}}} \right)}} \left( {p_{0} + bN\overline{x} } \right) $$

$$ {\text{Hassell }}\quad \left( {{\frac{1}{{\left( {1 + {\frac{{\left( {\lambda - 1} \right)N}}{K}}} \right)^{h} }}} - x} \right)\left( {p_{0} + bN\overline{x} } \right) $$

$$ {\text{Shared}}\,{\text{resources }}\quad \left( {{\frac{1}{N}} - x} \right)\left( {p_{0} + bN\overline{x} } \right) $$

• (b) ESS investment in host manipulation (x*)

MODEL

$$ {\text{Logistic }}\quad {\frac{{\left( {K - N} \right)\left( {1 + r\left( {N - 1} \right)} \right)}}{{K\left( {N + 1 + r\left( {N - 1} \right)} \right)}}} - {\frac{{p_{0} }}{{b\left( {N + 1 + r\left( {N - 1} \right)} \right)}}} $$

$$ {\text{Monod}}\quad {\frac{{\left( {K - N} \right)\left( {1 + r\left( {N - 1} \right)} \right)}}{{\left( {q + K - N} \right)\left( {N + 1 + r\left( {N - 1} \right)} \right)}}} - {\frac{{p_{0} }}{{b\left( {N + 1 + r\left( {N - 1} \right)} \right)}}} $$

$$ {\text{Ricker }}\quad {\frac{{1 + r\left( {N - 1} \right)}}{N}} - {\frac{{p_{0} }}{bN}} $$

$$ {\text{Hassell }}\quad {\frac{{1 + r\left( {N - 1} \right)}}{{\left( {N + 1 + r\left( {N - 1} \right)} \right)H}}} - {\frac{{p_{0} }}{{b\left( {N + 1 + r\left( {N - 1} \right)} \right)}}}\quad {\text{where}}\,H = \left( {1 + {\frac{{N\left( {\lambda - 1} \right)}}{K}}} \right)^{h} $$

$$ {\text{Shared}}\,{\text{resources }}\quad {\frac{{1 + r\left( {N - 1} \right)}}{{N\left( {N + 1 + r\left( {N - 1} \right)} \right)}}} - {\frac{{p_{0} }}{{b\left( {N + 1 + r\left( {N - 1} \right)} \right)}}} $$

• (c) Proportional decrease in x* at p ₀ = 0 due to competition

MODEL

$$ {\text{Logistic }}\quad {\frac{N}{K}} $$

$$ {\text{Monod }}\quad {\frac{q}{q + K - N}} $$

$$ {\text{Ricker }}\quad {\frac{{1 + r\left( {N - 1} \right)}}{N}} $$

$$ {\text{Hassell }}\quad 1 - {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 H}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$H$}}\quad {\text{where }}\,H = \left( {1 + {\frac{{N\left( {\lambda - 1} \right)}}{K}}} \right)^{h} $$

$$ {\text{Shared}}\,{\text{resources }}\quad {\frac{N - 1}{N}} $$

(Note: Almost all of these proportions are positive which suggests that competition decreases investment when the passive transmission rate is 0).

• (d) dx*/dN (The effect of parasite load on investment in host maipulation) (When the differential is negative, investment should decrease as parasite load increases.)

MODEL

$$ \begin{aligned} {\text{Logistic}} & \quad {\frac{{\left( {1 + r} \right)p_{0} }}{{b\left( {N + 1 + r\left( {N - 1} \right)} \right)^{2} }}} - {\frac{{K\left( {1 - r} \right) + r\left( {2N^{2} - 1} \right) + r*\left( {1 - r} \right)\left( {N - 1} \right)^{2} }}{{K\left( {N + 1 + r\left( {N - 1} \right)} \right)^{2} }}} \\ & {\text{which}}\,{\text{is}}\,{\text{negative}}\quad {\text{when}}\,{\raise0.7ex\hbox{${p_{0} }$} \!\mathord{\left/ {\vphantom {{p_{0} } b}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$b$}} < {\frac{1 - r}{1 + r}} + {\frac{{\left( {1 - r} \right)^{2} }}{{K\left( {1 + r} \right)}}} + {\frac{{rN\left( {N\left( {1 + r} \right) + 2\left( {1 - r} \right)} \right)}}{{K\left( {1 + r} \right)}}} \\ \end{aligned} $$

$$ \begin{aligned} {\text{Monod}} & \quad {\frac{{p_{0} \left( {1 + r} \right)}}{{b\left( {N + 1 + r\left( {N - 1} \right)} \right)}}} + {\frac{{\left( {q + K - N} \right)\left( {\left( {r\left( {K - 2N} \right) - \left( {1 - r} \right)} \right) - \left( {K - N} \right)\left( {1 + r} \right)\left( {1 + r\left( {N - 1} \right)} \right)} \right) + \left( {K - N} \right)\left( {1 + r\left( {N - 1} \right)} \right)}}{{\left( {N + 1 + r\left( {N - 1} \right)} \right)}}} \\ & {\text{which}}\,{\text{is}}\,{\text{negative}}\quad {\text{when}}\,{\raise0.7ex\hbox{${p_{0} }$} \!\mathord{\left/ {\vphantom {{p_{0} } b}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$b$}}\,<\,{\frac{{\left( {1 - r} \right)K^{2} - \left( {1 - r} \right)\left( {2N - q} \right)K + \left( {1 + r} \right)rN^{2} + \left( {1 - r} \right)2rN + \left( {1 - r} \right)^{2} }}{{\left( {q + K - N} \right)\left( {1 + r} \right)}}} \\ \end{aligned} $$

$$ \begin{aligned} {\text{Ricker}} & \quad {\frac{{p_{0} - b + br}}{{bN^{2} }}} \\ & {\text{which}}\,{\text{is}}\,{\text{negative}}\quad {\text{when }}\,{\raise0.7ex\hbox{${p_{0} }$} \!\mathord{\left/ {\vphantom {{p_{0} } b}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$b$}} < 1 - r \\ \end{aligned} $$

$$ \begin{aligned} {\text{Hassell}} & \quad {\frac{{2ph\left( {\lambda - 1} \right)}}{{b\left( {N + 1 + r\left( {N - 1} \right)} \right)\left( {K + N\left( {\lambda - 1} \right)} \right)}}} - {\frac{{\left( {1 + r\left( {N - 1} \right)} \right)h\left( {\lambda - 1} \right)}}{{H\left( {N + 1 + r\left( {N - 1} \right)} \right)\left( {K + N\left( {\lambda - 1} \right)} \right)}}} - {\frac{r}{{H\left( {N + 1 + r\left( {N - 1} \right)} \right)}}} + {\frac{{p\left( {1 + r} \right)}}{{b\left( {N + 1 + r\left( {N - 1} \right)} \right)^{2} }}} - {\frac{{\left( {1 + r} \right)\left( {1 + r\left( {N - 1} \right)} \right)}}{{H\left( {N + 1 + r\left( {N - 1} \right)^{2} } \right)}}} \\ & {\text{which}}\,{\text{is}}\,{\text{negative}}\quad {\text{when}}\,{\raise0.7ex\hbox{${p_{0} }$} \!\mathord{\left/ {\vphantom {{p_{0} } b}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$b$}} < {\frac{{\left( {N + 1 + r\left( {N - 1} \right)} \right)\left( {1 + r\left( {N - 1} \right)} \right)h\left( {\lambda - 1} \right) + \left( {1 - r} \right)\left( {K + N\left( {\lambda - 1} \right)} \right)}}{{H\left( {1 + r} \right)\left( {K + N\left( {\lambda - 1} \right)} \right)}}} \\ \end{aligned} $$

$$ \begin{aligned} {\text{Shared}}\,{\text{resources}} & \quad {\frac{{p_{0} \left( {1 + r} \right)}}{{b\left( {N + 1 + r\left( {N - 1} \right)} \right)^{2} }}} - {\frac{{r^{2} \left( {N - 1} \right)^{2} + \left( {N^{2} - 2} \right)r + \left( {2N + 1} \right)}}{{\left( {N + 1 + r\left( {N - 1} \right)} \right)^{2} N^{2} }}} \\ & {\text{which}}\,{\text{is}}\,{\text{negative}}\quad {\text{when }}\,\left( {{\raise0.7ex\hbox{${p_{0} }$} \!\mathord{\left/ {\vphantom {{p_{0} } b}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$b$}} < {\frac{{r^{2} \left( {N - 1} \right)^{2} + \left( {N^{2} - 2} \right)r + \left( {2N + 1} \right)}}{{N^{2} \left( {1 + r} \right)}}}} \right) \\ \end{aligned} $$

• (e) dx*/dr (The effect of relatedness on investment)

MODEL

$$ {\text{Logistic }}\quad {\frac{{N\left( {K - N} \right)\left( {N - 1} \right)}}{{K\left( {N + 1 + r\left( {N - 1} \right)} \right)^{2} }}} - {\frac{{p_{0} \left( {N - 1} \right)}}{{b\left( {N + 1 + r\left( {N - 1} \right)} \right)^{2} }}}\quad{\text{which}}\,{\text{is}}\,{\text{positive}}\quad {\text{when}}\,N < K $$

$$ {\text{Monod }}\quad {\frac{{\left( {N - 1} \right)\left( {bN\left( {K - N} \right) + p_{0} \left( {q + K - N} \right)} \right)}}{{b\left( {q + K - N} \right)\left( {N + 1 + r\left( {N - 1} \right)} \right)^{2} }}}\quad{\text{which}}\,{\text{is}}\,{\text{positive}}\quad {\text{when}}\,N < K $$

$$ {\text{Ricker }}\quad {\frac{N - 1}{N}}\quad{\text{which}}\,{\text{is}}\,{\text{always}}\,{\text{positive}} $$

$$ {\text{Hassell }}\quad {\frac{{\left( {N + 1} \right)\left( {{\raise0.7ex\hbox{${bN}$} \!\mathord{\left/ {\vphantom {{bN} H}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$H$}} + p_{0} } \right)}}{{b\left( {N + 1 + r\left( {N - 1} \right)} \right)^{2} }}}\quad{\text{which}}\,{\text{is}}\,{\text{always}}\,{\text{positive}} $$

$$ {\text{Shared}}\,{\text{resources }}\quad {\frac{{\left( {N - 1} \right)\left( {b + p_{0} } \right)}}{{b\left( {N + 1 + r\left( {N - 1} \right)} \right)^{2} }}}\quad{\text{which}}\,{\text{is}}\,{\text{always}}\,{\text{positive}} $$

• (f) dx*/db (The effect of conversion efficiency on investment in host manipulation)

MODEL

$$ {\text{Logistic }}\quad {\frac{{p_{0} }}{{b^{2} \left( {N + 1 + r\left( {N - 1} \right)} \right)}}}\quad {\text{which}}\,{\text{is}}\,{\text{always}}\,{\text{positive}} $$

$$ {\text{Monod}}\quad {\frac{{p_{0} }}{{b^{2} \left( {N + 1 + r\left( {N - 1} \right)} \right)}}}\quad {\text{which}}\,{\text{is}}\,{\text{always}}\,{\text{positive}}\quad {\text{if}}\,p_{0} > 0 $$

$$ {\text{Ricker }}\quad {\frac{{p_{0} }}{{b^{2N} }}}\quad{\text{which}}\,{\text{is}}\,{\text{always}}\,{\text{positive}} $$

$$ {\text{Hassell }}\quad {\frac{{p_{0} }}{{b^{2} \left( {N + 1 + r\left( {N - 1} \right)} \right)}}}\quad{\text{which}}\,{\text{is}}\,{\text{always}}\,{\text{positive}} $$

$$ {\text{Shared}}\,{\text{resources }}\quad {\frac{{p_{0} }}{{b^{2} \left( {N + 1 + r\left( {N - 1} \right)} \right)}}}\quad{\text{which}}\,{\text{is}}\,{\text{always}}\,{\text{positive}} $$

• (g) dx*/dp ₀ (The effect of passive transmission rate on investment in host manipulation)

MODEL

$$ {\text{Logistic }}\quad {\frac{ - 1}{{b\left( {N + 1 + r\left( {N - 1} \right)} \right)}}}\quad{\text{which}}\,{\text{is}}\,{\text{always}}\,{\text{negative}} $$

$$ {\text{Monod }}\quad {\frac{ - 1}{{b\left( {N + 1 + r\left( {N - 1} \right)} \right)}}}\quad{\text{which}}\,{\text{is}}\,{\text{always}}\,{\text{negative}} $$

$$ {\text{Ricker }}\quad {\frac{ - 1}{{b\left( {N + 1 + r\left( {N - 1} \right)} \right)}}}\quad{\text{which}}\,{\text{is}}\,{\text{always}}\,{\text{negative}} $$

$$ {\text{Hassell }}\quad {\frac{ - 1}{{b\left( {N + 1 + r\left( {N - 1} \right)} \right)}}}\quad{\text{which}}\,{\text{is}}\,{\text{always}}\,{\text{negative}} $$

$$ {\text{Shared}}\,{\text{resources }}\quad {\frac{ - 1}{{b\left( {N + 1 + r\left( {N - 1} \right)} \right)}}}\quad{\text{which}}\,{\text{is}}\,{\text{always}}\,{\text{negative}} $$

• (h) dx*/dK (The effect of reducing competition intensity (increasing K) on investment in host manipulation)

MODEL

$$ {\text{Logistic }}\quad \left\{ {{\frac{{N\left( {rN + 1 - r} \right)}}{{K^{2} \left( {N + 1 + r\left( {N - 1} \right)} \right)}}}} \right\}\quad {\text{which}}\,{\text{is}}\,{\text{always}}\,{\text{positive}} $$

$$ {\text{Monod }}\quad {\frac{{q\left( {1 + r\left( {N - 1} \right)} \right)}}{{\left( {N + 1 + r\left( {N - 1} \right)} \right)\left( {q + K - N} \right)^{2} }}}\quad {\text{which}}\,{\text{is}}\,{\text{always}}\,{\text{positive}} $$

$$ {\text{Ricker}}\quad 0\,\left( {{\text{The}}\,{\text{strength}}\,{\text{of}}\,{\text{competition}}\,{\text{has}}\,{\text{no}}\,{\text{effect}}\,{\text{on}}\,{\text{investment}}\,{\text{in}}\,{\text{host}}\,{\text{manipulation}}.} \right) $$

$$ {\text{Hassell }}\quad {\frac{{Nh\left( {\lambda - 1} \right)}}{{HK\left( {K + N\left( {\lambda - 1} \right)\left( {N + 1 + r\left( {N - 1} \right)} \right)} \right)}}}\quad {\text{which}}\,{\text{is}}\,{\text{positive}}\quad {\text{when}}\,N < K $$

Shared resources (Does not apply)

Appendix 2: Evaluation of non-linear models

We explore possible effects of a non linear version of Eq. 1 using:

$$ W = \left( {1 - x - {\raise0.7ex\hbox{$N$} \!\mathord{\left/ {\vphantom {N K}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$K$}}} \right)\left( {p_{0} + bN^{a} \overline{x} } \right) $$

(A1)

where a is a parameter which describes the shape of the combined effects of all the parasites in the host on the host’s susceptibility to predation by the parasite’s definitive host. When a exceeds one the combined effect of the parasites exceeds the sum of the individual effects producing a certain synergy; when a is less than one, the combined effect is less than the sum of the individual effects. When a = 1, Eq. A1 becomes Eq. 1.

Most of the inferences drawn from Eq. 1 are also supported by analysis of Eq. A1. For all positive values of a, increases in competition should decrease the investment in host manipulation and the proportional decrease is described by Eq. 4. As in the linear model, investment should increase as relatedness, r, and as manipulation efficiency, b, increase; investment should decrease as the passive transmission rate, p ₀, increases. Analysis of the non linear model differs from the linear model only with respect to the effect of increases in parasite load. In this case, Eq. 5 describing the conditions under which investment should increase as parasite load increases becomes

$$ {\raise0.7ex\hbox{${p_{0} }$} \!\mathord{\left/ {\vphantom {{p_{0} } b}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$b$}} > {\raise0.7ex\hbox{${\left( {1 - r} \right)N^{a} }$} \!\mathord{\left/ {\vphantom {{\left( {1 - r} \right)N^{a} } {\left( {\left( {1 + r} \right)aN - \left( {1 - a} \right)\left( {1 - r} \right)} \right)}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\left( {\left( {1 + r} \right)aN - \left( {1 - a} \right)\left( {1 - r} \right)} \right)}$}} . $$

(A2)

This condition differs greatly from Eq. 5 because of the presence of N on the right hand side of the equation. If a < 1, indicating that the combined effect of several parasites is less than the sum of their individual effects, investment should decrease with increased parasite load at all passive transmission rates. Further, the range of passive transmission rates and parasite loads for which manipulation is profitable is greatly reduced compared to the linear model.

On the other hand, when a > 1, lower passive transmission rates (compared to the linear model) can generate increased manipulation as a function of increased parasite load while manipulation will be profitable under a wider range of passive transmission rates and parasite loads.