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Supporting Information
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Michael Sieber and Frank M. Hilker. Prey, predators, parasites: intraguild predation or
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simpler community modules in disguise?
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Appendix S1. Details of the coordinate transformation
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Here we show how to change the state variables of a differential equation model. The idea is to
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subsume two populations (e.g. total prey) and to keep track of the distribution of the two prey
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populations in another new variable (e.g., ratio of prey B to prey A, A > 0). That is, we transform
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the model according to the map
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 A  A  B   N 
  
  
 B   B / A    i  .
 P  P   P 
  
  
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With the assumption of indiscriminate predation (see main text), the transformed system (6-8) is
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then obtained as follows.
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1. The differential equation for the predators P is easiest to determine, as we keep this state
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variable. We simply have to subsume A+B=N in the original Eq. (3) to give Eq. (8) in the
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main text.
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2. The differential equation for the total prey population N=A+B is obtained by
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dN dA dB


.
dt
dt dt
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The predation terms can be subsumed by using the identity N=A+B. The growth term
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applies only to the susceptible prey A= N /(1+i), and the virulence only to the infected prey
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B=N i /(1+i). This gives Eq. (6).
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3. Finally, the differential equation for the ratio i of prey species (e.g. infecteds to susceptibles)
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requires a bit more care. The reason is that we have to use the quotient rule in order to get
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the time derivative. That is
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'
'
 B  B A  BA
i   
,
A2
 A
'
1
'
2
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where the prime denotes differentiation with respect to time. We can replace B' and A' by
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the right-hand sides of Eqs. (2) and (1), respectively. Then we are left with substituting A
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and B as before. Simplifying terms eventually gives Eq. (7).
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Appendix S2. Basins of attraction in the case of multistability
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In the case of alternative attractors, the initial conditions determine which one of the attractors will
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be eventually approached. This raises the question how the basins of attraction of the two attractors
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are arranged in phase space. Here, the basin of attraction of an attractor denotes the set of all initial
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values for which the corresponding solution asymptotically approaches this attractor. Since we
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consider a three-dimensional model, a high resolution scan of the phase space would require an
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immense amount of computation time. Thus, we restrict the scan to a plane with B = 0.5 fixed in
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order to give an impression of the basins of attraction.
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Figure S1: Basins of attraction for the exploitative competition (EC) model (Armstrong &
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McGehee, 1980). Black indicates initial values, for which the corresponding solution
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approaches the chaotic attractor. White indicates initial values, for which the periodic
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attractor is approached. For this diagram, solutions starting in 46,522 different points on the
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fixed plane B = 0.5 have been investigated for their asymptotic behavior. Model equations are
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(T7)-(T9) in the main text; parameters: r  1, a1  5, a2  1, h  1/ 7, m1  1.7, m2  0.6,  1   2  1.
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The result of the scan is shown in Fig. S1. The parameter set is chosen such that one attractor is
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periodic and the other one is chaotic. Even though only a two-dimensional transect has been
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explored and the resolution of the scan is not high enough to reproduce all details of the basin
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boundaries, the intricate structure of the basins of attraction is nevertheless visible.
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The basins of attraction seem to be made up of interleaved circular regions on the plane. This gives
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a hint of the three-dimensional structure of the basins, which might look like nested tubes.
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Appendix S3. Transformation of IGP into a food chain
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Here we show that an IGP module can be transformed into a food chain when the IG predator and
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prey are similar from the basal prey's point of view. We start with the following IGP model studied
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by Tanabe & Namba (2005), where A, B, and P are the basal prey, IG prey and IG predator,
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respectively:
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dA
 r (1  A) A


dt 
logistic growth
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 a AP

predation by P
dB
  a AB

dt consumptio
n of A
dP
  a AP

dt consumptio
n of A
 a AB ,

(S1)
predation by B
BP

predation by P
 mB B ,

natural mortality
  BP


consumption of B
(S2)
 mP P .

(S3)
natural mortality
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Note that the two predators have equal preferences for and attack rates of the basal prey. This
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satisfies the condition of indiscriminate predation and yields a special case of the original Tanabe-
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Namba model ( a12  a13 and a 21  a31 in their formulation). Introducing the total predator
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population C=P+B and the ratio i=P/B of IG predator to IG prey leads to
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dA
 r (1  A) A


dt 
logistic growth
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dC
  a AC

dt consumptio
n of
 aC A ,

(S4)
predation by C
A
(1   )  C  m P
m

Ci  B C ,
1
 i 
1

i


loss due to i
di   i

C i  (m P  m B ) i .



dt 
1

i

mortality
(S5)
mortality
(S6)
gain from C
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This is a food chain from A to C to i. While C is a linear predator on A, the biomass flow from C to
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i is a bit more complicated. For   1 (perfect conversion efficiency), the “top-predator” i is linear
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in Eq. (S6) and nonlinear in Eq. (S5), yielding again a linear nullcline structure. This assumption
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would apply to the case of P and B being diseased and healthy predators, respectively.
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In this model, coexistence of all three species is possible on a stable equilibrium. For some
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parameter ranges, we found numerical evidence of bistability (stable coexistence vs. extinction of
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one species and oscillations in the remainder two populations). However, we could not find chaotic
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dynamics that has been reported in Tanabe & Namba (2005) and is also known to occur in tri-
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trophic food chains with nonlinear predators (Hastings & Powell, 1991). This may be due to the
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fact that we are considering the special case of non-discriminating predators.
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References
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Armstrong, R. A. & McGehee, R. (1980) Competitive exclusion. The American Naturalist, 115,
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151-170.
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Hastings, A. & Powell, T. (1991) Chaos in a three-species food chain. Ecology 72, 896-903.
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Tanabe, K. & Namba, T. (2005) Omnivory creates chaos in simple food webs models. Ecology, 86,
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3411-3414.