1 Supporting Information 2 3 Michael Sieber and Frank M. Hilker. Prey, predators, parasites: intraguild predation or 4 simpler community modules in disguise? 5 6 Appendix S1. Details of the coordinate transformation 7 8 Here we show how to change the state variables of a differential equation model. The idea is to 9 subsume two populations (e.g. total prey) and to keep track of the distribution of the two prey 10 populations in another new variable (e.g., ratio of prey B to prey A, A > 0). That is, we transform 11 the model according to the map 12 A A B N B B / A i . P P P 13 14 15 With the assumption of indiscriminate predation (see main text), the transformed system (6-8) is 16 then obtained as follows. 17 1. The differential equation for the predators P is easiest to determine, as we keep this state 18 variable. We simply have to subsume A+B=N in the original Eq. (3) to give Eq. (8) in the 19 main text. 20 2. The differential equation for the total prey population N=A+B is obtained by 21 dN dA dB . dt dt dt 22 23 24 The predation terms can be subsumed by using the identity N=A+B. The growth term 25 applies only to the susceptible prey A= N /(1+i), and the virulence only to the infected prey 26 B=N i /(1+i). This gives Eq. (6). 27 3. Finally, the differential equation for the ratio i of prey species (e.g. infecteds to susceptibles) 28 requires a bit more care. The reason is that we have to use the quotient rule in order to get 29 the time derivative. That is 30 ' ' B B A BA i , A2 A ' 1 ' 2 3 where the prime denotes differentiation with respect to time. We can replace B' and A' by 4 the right-hand sides of Eqs. (2) and (1), respectively. Then we are left with substituting A 5 and B as before. Simplifying terms eventually gives Eq. (7). 6 7 Appendix S2. Basins of attraction in the case of multistability 8 9 In the case of alternative attractors, the initial conditions determine which one of the attractors will 10 be eventually approached. This raises the question how the basins of attraction of the two attractors 11 are arranged in phase space. Here, the basin of attraction of an attractor denotes the set of all initial 12 values for which the corresponding solution asymptotically approaches this attractor. Since we 13 consider a three-dimensional model, a high resolution scan of the phase space would require an 14 immense amount of computation time. Thus, we restrict the scan to a plane with B = 0.5 fixed in 15 order to give an impression of the basins of attraction. 16 1 2 Figure S1: Basins of attraction for the exploitative competition (EC) model (Armstrong & 3 McGehee, 1980). Black indicates initial values, for which the corresponding solution 4 approaches the chaotic attractor. White indicates initial values, for which the periodic 5 attractor is approached. For this diagram, solutions starting in 46,522 different points on the 6 fixed plane B = 0.5 have been investigated for their asymptotic behavior. Model equations are 7 (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. 8 9 The result of the scan is shown in Fig. S1. The parameter set is chosen such that one attractor is 10 periodic and the other one is chaotic. Even though only a two-dimensional transect has been 11 explored and the resolution of the scan is not high enough to reproduce all details of the basin 12 boundaries, the intricate structure of the basins of attraction is nevertheless visible. 13 The basins of attraction seem to be made up of interleaved circular regions on the plane. This gives 14 a hint of the three-dimensional structure of the basins, which might look like nested tubes. 15 16 17 Appendix S3. Transformation of IGP into a food chain 1 Here we show that an IGP module can be transformed into a food chain when the IG predator and 2 prey are similar from the basal prey's point of view. We start with the following IGP model studied 3 by Tanabe & Namba (2005), where A, B, and P are the basal prey, IG prey and IG predator, 4 respectively: 5 dA r (1 A) A dt logistic growth 6 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 7 8 Note that the two predators have equal preferences for and attack rates of the basal prey. This 9 satisfies the condition of indiscriminate predation and yields a special case of the original Tanabe- 10 Namba model ( a12 a13 and a 21 a31 in their formulation). Introducing the total predator 11 population C=P+B and the ratio i=P/B of IG predator to IG prey leads to 12 dA r (1 A) A dt logistic growth 13 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 14 15 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 16 i is a bit more complicated. For 1 (perfect conversion efficiency), the “top-predator” i is linear 17 in Eq. (S6) and nonlinear in Eq. (S5), yielding again a linear nullcline structure. This assumption 18 would apply to the case of P and B being diseased and healthy predators, respectively. 19 In this model, coexistence of all three species is possible on a stable equilibrium. For some 20 parameter ranges, we found numerical evidence of bistability (stable coexistence vs. extinction of 21 one species and oscillations in the remainder two populations). However, we could not find chaotic 22 dynamics that has been reported in Tanabe & Namba (2005) and is also known to occur in tri- 1 trophic food chains with nonlinear predators (Hastings & Powell, 1991). This may be due to the 2 fact that we are considering the special case of non-discriminating predators. 3 4 References 5 6 Armstrong, R. A. & McGehee, R. (1980) Competitive exclusion. The American Naturalist, 115, 7 151-170. 8 9 Hastings, A. & Powell, T. (1991) Chaos in a three-species food chain. Ecology 72, 896-903. 10 11 Tanabe, K. & Namba, T. (2005) Omnivory creates chaos in simple food webs models. Ecology, 86, 12 3411-3414.