The role of specialist parasites in structuring host communities

Abstract

Natural enemies can be a powerful force when structuring natural communities, and in facilitating or preventing species coexistence depending on the nature of the trophic interaction. In particular, “keystone” predators can promote species coexistence, provided they preferentially attack the competitively dominant species. However, it is not clear whether parasites can play a similar structuring role; parasites typically form chronic associations with their victims, reducing their fitness (i.e., fecundity) rather than survival, and allowing infected hosts to remain viable competitors within the community. Therefore the density-dependent suppression of the host is likely to be more subtle than that due to predation. Using a series of simple population-dynamic models we show that specialist parasites can facilitate species coexistence, although possibly less so than predators. These results contrast with those typically found with models of generalist parasites, which can reduce the likelihood of species coexistence through apparent competition. In addition, we show that the likelihood of parasite-facilitated species coexistence depends greatly on the specific type of parasite. In particular, macroparasites (e.g., parasitic helminths) may be less likely to facilitate species coexistence than microparasites (e.g., viruses or bacteria) due to their typically highly aggregated distribution amongst their hosts. Furthermore, species coexistence is more likely if the parasite is relatively benign to its host. Parasitism by apparently “harmless” specialist parasites may provide an important but overlooked factor in the maintenance of species diversity, facilitating species invasions into new communities and the emergence of novel infectious diseases.

Appendix

Analysis of the models used in this paper follow standard lines, and here we present in detail the mathematical details for the analysis of the simple two-host-one-parasite model (model 1). Solving model 1 reveals there are six possible equilibria:

• E0 = (0,0,0)

• E1 = (K ₁,0,0)

• E2 = (0,K ₂,0)

• E3 = (U ^1*₁ ,N ^1*₂ ,0)

• E4 = (U ^2*₁ ,0,I ^*₁ )

• E5 = (U ^C*₁ ,N ^C*₂ ,I ^C*₁ )

where

$$ \begin{aligned}{} U^{{1*}}_{1} & = \frac{{K_{1} - \alpha _{{12}} K_{2} }} {{1 - \alpha _{{21}} \alpha _{{12}} }},\quad N^{{1*}}_{2} = \frac{{K_{2} - \alpha _{{21}} K_{1} }} {{1 - \alpha _{{21}} \alpha _{{12}} }} \\ U^{{2*}}_{1} & = \frac{{\gamma _{1} }} {{\beta _{1} }},\quad I^{*}_{1} = \frac{{r_{1} (\beta _{1} K_{1} - \gamma _{1} )}} {{\beta _{1} (\beta _{1} K_{1} + r_{1} )}} \\ U^{{C*}}_{1} & = \frac{{\gamma _{1} }} {{\beta _{1} }},\quad N^{{C*}}_{2} = \frac{{K_{1} (\beta _{1} K_{2} - \gamma _{1} \alpha _{{21}} ) + r_{1} (K_{2} - \alpha _{{21}} K_{1} )}} {{r_{1} (1 - \alpha _{{21}} \alpha _{{12}} ) + \beta _{1} K_{1} }},\quad I^{{C*}}_{1} = \frac{{r_{1} (\beta _{1} (K_{1} - \alpha _{{12}} K_{2} ) - \gamma _{1} (1 - \alpha _{{21}} \alpha _{{12}} ))}} {{\beta _{1} (r_{1} (1 - \alpha _{{21}} \alpha _{{12}} ) + \beta _{1} K_{1} )}} \\ \end{aligned} $$

The Jacobian of model 1 is

$$ {\left( {\begin{array}{*{20}c} {{ - \beta _{1} I_{1} - \frac{{U_{1} r_{1} }} {{K_{1} }} + r_{1} {\left( {1 - \frac{{U_{1} + I_{1} + \alpha _{{12}} N_{2} }} {{K_{1} }}} \right)} - \Omega }} & {{ - \frac{{\alpha _{{12}} U_{1} r_{1} }} {{K_{1} }}}} & {{ - \beta _{1} U_{1} - \frac{{U_{1} r_{1} }} {{K_{1} }}}} \\ {{ - \frac{{\alpha _{{21}} N_{2} r_{2} }} {{K_{2} }}}} & {{ - \frac{{N_{2} r_{2} }} {{K_{2} }} + r_{2} {\left( {1 - \frac{{N_{2} + \alpha _{{21}} (U_{1} + I_{1} )}} {{K_{2} }}} \right)} - \Omega }} & {{ - \frac{{\alpha _{{21}} N_{2} r_{2} }} {{K_{2} }}}} \\ {{\beta _{1} I_{1} }} & {0} & {{ - \Omega }} \\ \end{array} } \right)} $$

where the Ωs are the eigenvalues of the system.

It is clear that the trivial solution E0 is always unstable. The characteristic equation at E1 is

$$ {\left( { - r_{1} - \Omega } \right)}{\left( {r_{2} - \frac{{\alpha _{{21}} K_{1} r_{2} }} {{K_{2} }} - \Omega } \right)}{\left( { - \gamma _{1} + \beta _{1} K_{1} - \Omega } \right)} = 0. $$

Therefore, for stability of E1 we require:

$$ \gamma \; > \; \beta _{1} {\text{ }}K_{1} \quad {\text{and}}\quad K_{1} \; > \; K_{2} /\alpha _{{21}} $$

(criterion 1)

Similarly, the characteristic equation at E2 is

$$ {\left( { - r_{2} - \Omega } \right)}{\left( {r_{1} - \frac{{\alpha _{{12}} K_{2} r_{1} }} {{K_{1} }} - \Omega } \right)}{\left( { - \gamma _{1} - \Omega } \right)} = 0, $$

and so stability of E2 requires:

$$ K_{1} \; < \; K_{2} \alpha _{{12}} .$$

(criterion 2)

Criteria 1 and 2 are the stability criteria for the simple Lotka-Volterra model with no parasite. If one or the other is fulfilled then that equilibrium is stable and the corresponding species wins. However, these criteria are not mutually exclusive—if both are fulfilled (i.e., 1/α ₂₁ < K ₁/K ₂ < α ₁₂) then bistability occurs, and the winner is determined by their initial conditions. Hence, the full criteria for stability of E1 and E2 are K ₁/K ₂ > Max(1/α ₂₁, α ₁₂) and K ₁/K ₂ < Min(α ₁₂, 1/α ₂₁), respectively. Finally, in terms of stability in the absence of the parasite, examining the definitions for U ^1*₁ and N ^1*₂ shows that for E3 to be biologically possible we need 1/α ₂₁ > K ₁/K ₂ > α ₁₂ and 1 > α ₂₁ α ₁₂, which define stable species coexistence in the absence of the parasite. However, further examination of the Jacobian evaluated at E3 reveals an additional requirement for stability of E3:

$$ \beta (K_{1} - \alpha _{{12}} K_{2} )\; < \;\gamma _{1} (1 - \alpha _{{12}} \alpha _{{21}} ).$$

(criterion 3)

If this criterion is fulfilled, then the parasite is unable to invade and the two species coexist alone.

Examining the expression for P* at E4 shows that for this state to be biologically plausible requires β ₁ K ₁  > γ ₁. In other words, for the parasite to be able to persist, its maximum transmission rate (β ₁ K ₁) must exceed its rate of loss (γ ₁). In addition, examining the Jacobian evaluated at E4 reveals an additional requirement for stability:

$$ K_{1} \frac{{(\gamma _{1} + r_{1} )}} {{(\beta _{1} K_{1} + r_{1} )}} \; > \; \frac{{K_{2} }} {{\alpha _{{21}} }}.$$

(criterion 4)

This is effectively the criterion for species 1 to out-compete species 2 in the absence of the parasite (i.e., K ₁ > K ₂/α ₂₁), adjusted to take into account the burden of being infected by the parasite.

Finally, examining the definitions of N ^C*₂ and I ^C*₁ at E5 reveals the following criteria need to be fulfilled for stability of E5 (species coexistence in the presence of the parasite):

$$ \beta _{1} (K_{1} - \alpha _{{12}} K_{2} )\; > \;\gamma _{1} (1 - \alpha _{{12}} \alpha _{{21}} ) $$

and

$$ K_{1} \; < \; \frac{{K_{2} }} {{\alpha _{{21}} }}\frac{{(\beta _{1} K_{1} + r_{1} )}} {{(\gamma _{1} + r_{1} )}} $$

which are the reverse of criteria 3 and 4, effectively dictating that the parasite is able to invade but species 1 is not able to exclude species 2. Analysis of the corresponding Jacobian confirms that these criteria define the region of species coexistence in the presence of the parasite. In addition, stability of E5 requires that AB > C where A, B and C are coefficients of the characteristic equation Ω³ + AΩ² + BΩ + C = 0 of the Jacobian evaluated at E5. Numerical simulations show that if this criterion is broken then the system undergoes sustained limit cycles.

Analysis of the host–macroparasite model

Analysis of the other models presented in this paper follow similar lines to those described above. We omit the details of those analyses for brevity, but provide here a brief outline of the analysis of the host–macroparasite model (model 2). Solving model 2 reveals there are five possible equilibria:

• E1 = (K ₁,0,0)

• E2 = (0,K ₂,0)

• E3 = (N ^1*₁ ,N ^1*₂ ,0)

• E4 = (N ^2*₁ ,0,P ^*₁ )

• E5 = (N ^C*₁ ,N ^C*₂ , P ^C*₁ )

where

$$ \begin{aligned}{} & N^{{1*}}_{1} = \frac{{K_{1} - \alpha _{{12}} K_{2} }} {{1 - \alpha _{{21}} \alpha _{{12}} }},\quad N^{{1*}}_{2} \; = \;\frac{{K_{2} - \alpha _{{21}} K_{1} }} {{1 - \alpha _{{21}} \alpha _{{12}} }} \\ & N^{{2*}}_{1} = \;\frac{{K_{1} \omega (\Gamma + r_{1} \theta )}} {{\beta _{1} \lambda K_{1} + r_{1} \omega \theta }},\quad P^{*}_{1} \; = \;\frac{{N^{{2*}}_{1} r_{1} (\beta _{1} \lambda K_{1} - \omega \Gamma )}} {{\nu (\beta _{1} \lambda K_{1} + r_{1} \omega \theta )}} \\ & N^{{C*}}_{1} \; = \;\frac{{r_{1} \theta (K_{1} - \alpha _{{12}} K_{2} ) + K_{1} \Gamma }} {{\Phi K_{1} + r_{1} \theta (1 - \alpha _{{21}} \alpha _{{12}} )}},\quad N^{{C*}}_{2} \; = \;K_{2} - \alpha _{{21}} N^{{C*}}_{1} ,\\& P^{{C*}}_{1} \; = \;\frac{{N^{{C*}}_{1} (\Phi N^{{C*}}_{1} - \Gamma )}} {{\nu \theta }} \\ \end{aligned} $$

where θ = (1 + k)/k and φ = β₁λ/ω. The Jacobian of model 2 is:

$$ {\left( {\begin{array}{*{20}c} {{ - \frac{{N_{1} r_{1} }} {{K_{1} }} + r_{1} {\left( {1 - \frac{{N_{1} + \alpha _{{12}} N_{2} }} {{K_{1} }}} \right)} + \Omega }} & {{ - \frac{{\alpha _{{12}} N_{1} r_{1} }} {{K_{1} }}}} & {{ - \nu }} \\ {{ - \frac{{\alpha _{{21}} N_{2} r_{2} }} {{K_{2} }}}} & {{ - \frac{{N_{2} r_{2} }} {{K_{2} }} + r_{2} {\left( {1 - \frac{{\alpha _{{21}} N_{1} + N_{2} )}} {{K_{2} }}} \right)} - \Omega }} & {0} \\ {{\frac{{\beta _{1} \lambda P}} {\omega } + \frac{{P^{2} \nu \theta }} {{N_{1} ^{2} }}}} & {0} & {{\frac{{\beta _{1} \lambda N_{1} }} {\omega } - \frac{{2P\nu \theta }} {{N_{1} }} - \Gamma - \Omega }} \\ \end{array} } \right)} $$

where the Ωs are the eigenvalues of the system. We then obtain the boundaries separating the regions of parameter space by evaluating the coefficients of the characteristic equation for each equilibrial state, as described for model 1 above.