arXiv:1204.2822v1 [q-bio.PE] 12 Apr 2012 coexistence qiiru ofiuain sfudfrhg niheta well. as enrichment Keywords: high for found mea is enrichment configurations intermediate equilibrium (tran for immobilization only web low coexistence food At 3- of stable . m percentage of the the case of of the dimensionalities comparison For numerical system. a IGP present the we subpopulations in resource coexistence multiple robust from to a predictions extended support the is how module IGP explore the We when resources. and (resource predators immobilization of varying sp and with enrichment module predato increasing IGP an the of and effect by predator top rates the different by immobilization at with grazed module are in subpopulations subdivided the is resource in The resource. for a pr two analyze via of We indirectly consists model relationships. The IGP. IGP and in resources heterogeneous levels diversity trophic of 3 higher increase simplest of an the how persistence for is even question fundamental behaviour One their explored. webs food IGP o natural of restructuring in understanding importance widespread for principal are is (IGP) behaviour resilience intraguild with webs Food Abstract Coex Stable on Community Prey of Differentiation of Effect The fonvr odwb Pm n atn 1978; Lawton, and (Pimm webs food omnivory of Introduction 1. rpitsbitdt Elsevier to submitted Preprint a ilgsh ntl egln,Afe-eee Institute Alfred-Wegener Helgoland, Anstalt Biologische nsieo h rvlneadimportance and prevalence the of spite In mi address: Email nrgidpeain moiiain lentv eore Multiple resource, Alternative Immobilization, predation, Intraguild [email protected] .Sceioa .G .Loe,M orm,K .Wiltshire H. K. Boersma, L¨oder, M. J. G. M. Shchekinova, E. he-pce odWbModel Food–Web Three-Species E hhknv,M .J ¨dr .Besa .H Wiltshire H. K. Boersma, L¨oder, M. J. G. M. Shchekinova, (E. egln Germany Helgoland o oa n aieRsac,Krrmnd 0,D-27498 201, Kurpromenade Research, Marine and Polar for omnt.Ee nsml ytm plethora a the systems in simple species in Even three only community. for poorly even remain date understood, to dynamics population communities their natural in 2005) al., et Vadeboncoeur dtr ietyculdvaIPrlto and relation IGP via coupled directly edators wiea ihrtsalresto stable of set large a rates high at nwhile clgclcmuiis nsieo the of spite In communities. ecological f s ecnie w oes nIGP an models: two consider We rs. ihsal oxsec o different for coexistence stable with s odsic uppltos Individuals subpopulations. distinct to ftelws rpi ee mat the impacts level trophic lowest the of lil uppltoso h resource the of subpopulations ultiple h ai -pce oe r altered are model 3-species basic the rnfr aeo tbecoexistence stable a on rate transfer) seissseshsntbe fully been not has systems -species eivsiaewihparameters which investigate We . -pce odwbmdlwt a with model web food 3-species aias hi dpainand adaptation Their . fr ae u oe rdcsa predicts model our rates sfer) ce unvr eeaiethe examine We turnover. ecies eoretat,Stable traits, resource a ) sec na in istence etme ,2018 9, September of nonlinear effects such as flexible be- for the communal resource (Diehl and Feißel, 2001). haviour (Leibold et al., 2005), intraspecific interac- Yet empirical data suggest a robust persistence tions between competing consumers and resources of IGP systems in both terrestrial (Brodeur et al., (Holt et al., 1994), inhomogeneity of the environ- 2000; Arim and Marquet, 2004) and aquatic com- ment (Amarasekare, 2007; Janssen et al., 2007) and munities (Polis et al., 1989; Mylius et al., 2001; adaptive (Krivan, 1996; Krivan and Diehl, Borer et al., 2003; Denno and Fagan, 2003). 2005) precludes easy theoretical treatment and in- Theoretical models that are focused on the terpretation. aspects of stability and coexistence of species One example of a non-trivial omnivory in 3-level systems with the IGP (Polis and Holt, is a system with (Polis et al., 1992; Holt and Polis, 1997; Abrams et al., 1994a,b, 1989; Finke and Denno, 2002; Borer et al., 2003). 2010), as a rule, largely reduce the complex- Intraguild predation assumes that the same or- ity of interactions observed in realistic systems ganism is both competitor and predator to an- (Thomson et al., 2007). Such oversimplifications other member of the food web. The IGP mod- can influence the as well as els encompass a rich dynamical behaviour in- critically impact species persistence. Even though cluding coexistence (Polis and Holt, 1992) and the simplest model of the IGP encompasses only alternative stable states (Holt and Huxel, 2007; three species (Polis and Holt, 1992; Holt and Polis, Daugherty et al., 2007). Simple mathematical 1997; Diehl and Feißel, 2000, 2001) a number of models (Polis and Holt, 1992; Diehl and Feißel, empirical studies deal with larger food webs that 2000; Namba et al., 2008) have been evoked in at- involve more than three species potentially en- tempt to explain the persistence of IGP interac- gaged in IGP interactions (Rosenheim et al., 1993; tions in natural habitats. However predictions from Woodward et al., 2005). the mathematical theory of 3-species IGP systems Spatiotemporal heterogeneity of the environ- state that a high resource pro- ment often is invoked as one of the explana- motes the exclusion of intermediate trophic levels tory mechanisms for the coexistence between mul- and thus destabilizes interactions (Diehl and Feißel, tiple species competing for the same resources 2001). What is puzzling that various empirical (Hutchinson, 1961). It has been observed that such studies of omnivory document however coexistence, a spatiotemporal heterogeneity can affect the di- but not exclusion, over the entire range of nat- versity in prey populations (Amarasekare, 2006). ural resource productivities (Mylius et al., 2001; Indeed an inhomogeneity in prey items that share Borer et al., 2003). On the basis of experimental common resource and predators is critical in deter- observations a theoretical 3-species omnivory model mining the responses of ecological community. For (Stoecker and Evans, 1985; Holt and Polis, 1997; systems with multiple prey composition various co- Diehl and Feißel, 2001) predicts the coexistence existence patterns can be found depending on the only at superior competitive abilities of the IG prey levels of resource (Leibold, 1996). It is 2 not clear yet how the diversity in a prey community tems the IG prey has a mutualistic or at least will affect the behaviour in the IGP systems. facilitative relationship with the IG predator The effect of a structure on the IGP is (Crowley and Cox, 2011). Including such facilita- discussed in various recent models (Amarasekare, tion in ecological theory will fundamentally change 2006, 2007; Janssen et al., 2007). For example a many basic predictions and will enable a better un- stable coexistence of the intraguild prey due to derstanding of functioning of many natural commu- inhomogeneity of a habitat can be supported by nities (Bruno et al., 2003). creating temporal refuges for prey and reducing Especially in the IGP systems an emphasis the encounter rates among preys and predators should be given to the elucidation of the effects of (Janssen et al., 2007). In addition the stability of facilitation on community composition and stabil- the IGP can be enhanced by an inclusion of addi- ity (Crowley and Cox, 2011). Contrary to the com- tional factors such as behaviourally mediated effects petitive exclusion principle in systems with com- (Janssen et al., 2007). petitors for a single resource stability stems from To include the effect of an increasing diversity of (Hosack et al., 2009). Hereby one resource and IG predators on population dynamics consumer can in some way alter the habitat to ben- recently the 3-species IGP model (Holt and Polis, efit the other. Recently such an interaction was 1997) was modified by Holt and Huxel (2007). observed in experiments with a microzooplankton The authors extended the basic 3-species omnivory food web community (L¨oder et al., xxxx). The ex- model to the so called ”partial IGP” model in which perimental system included two predators: a tintin- ”partial” overlap among competitors for a single re- nid species Favella ehrenbergii and a heterotrophic source exists and both predators have exclusive re- dinoflagellate species Gyrodinium dominans. They sources to exploit. It was shown (Holt and Huxel, are both on a phototrophic dinoflagellate 2007) that an alternative resource enhances the Scrippsiella trochoidea. The authors showed that tolerance of the IG prey against attacks from IG the IG predator F. ehrenbergii can precondition a predators. Independently of a competitive status substantial part of the common resource S. tro- of the IG prey in exploitation for a shared re- choidea during its feeding procedure by immobiliz- source it can persists by utilizing an alternative re- ing the common prey without ingestion. Such pre- source. An extended formulation of the IGP model conditioned individuals can be captured more easily with trophic supplementation has been proposed by by the IG prey G. dominans than the mobile indi- Daugherty et al. (2007). The authors investigated viduals of the same resource species. This mutual- three forms of a supplementary feeding outside of istic interaction leads to higher growth rates of the the basic IGP module and postulated a higher po- IG prey in the presence of the IG predator. The au- tential for persistence of the IG prey due to its ef- thors characterized their experimental observations ficient exploitation of external resources. as a facilitative IGP relationship with a commen- There is growing evidence that in many sys- salistic pattern. Our motivation for this modeling 3 study was to investigate if such commensalistic pat- paper is a resource turnover mechanism. This terns can create loopholes for a stable coexistence mechanism describes mutual interactions between of all species in the investigated system. Of our species from distinct resource subpopulations. The major interest was if in the IGP system an immo- interaction term depends exclusively on the re- bilization (L¨oder et al., xxxx) or the partitioning source subpopulation densities. The rate of of prey populations into distinct groups of individ- turnover is constant. If no turnover or immobi- uals offers opportunities for competition avoidance lization of individuals from one group to another among both consumer species. occurs then the basic IGP model with a single pop- We reformulated the 3-species IGP model pro- ulation of resource is recovered. We discuss the posed in (Polis and Holt, 1992; Holt and Polis, influence of immobilization and transfer of species 1997; Diehl and Feißel, 2000, 2001) to include mul- on the coexistence patterns in a system with dif- tiple subpopulations of prey. Furthermore, we ex- ferent subpopulations of the resource and compare plored the effect of diversification of the resource the results with the basic 3-species IGP. available to higher level consumers on the species This paper is organized as follows: in the first persistence by numerical simulations of an extended section we introduce a general 3-species IGP model IGP module. Specifically, we investigate how the with a new type of interaction that links the re- addition of new links to a focal IGP module en- source pools to the top consumers. In the following hances stability of population dynamics by reducing sections two distinct IGP formulations with n = 2 the competitive interactions of predators for their resource subpopulations are discussed. Both mod- shared resource. els are derived from the basic IGP module by in- In order to explain the results of the experimen- cluding additional links: (i) the immobilization by tal findings of L¨oeder et al. we investigated the the predator and (ii) the resource turnover. In the influence of multiple traits of the resource commu- Results section we numerically investigate stability nity on a stable coexistence in the 3-species model of equilibrium densities for various trophic configu- with different types of resource. For this purpose rations. Data from numerical analysis are presented we adapted and reformulated the original model for the IGP model with the immobilization and for by Holt and Polis (1997) and added a new type of the model with the resource turnover. At last we interaction. This link specifies the immobilization discuss results for a general IGP model with the re- mechanism that depends on the densities of mobile source turnover mechanism and n > 2 subpopula- and immobile resource items and the top predator tions of the resource. After sketching the main con- which creates the immobile resource fraction during clusions we review the model predictions and com- feeding. The immobilization term is used to model pare their relevance to the immobilization experi- the interactions between the IG predator and the ment (L¨oder et al., xxxx). Furthermore we discuss resource. possible alternative reformulations of the model. In Another type of interaction considered in this the Appendix explicit forms for the steady states for 4 n two simple analytical cases and multidimensional among different subpopulations {Sk}k=2 following system are specified. As a part of a linear stabil- the links in Fig. 1a. Another special case of the IGP ity routine the Jacobian matrices for two types of with two distinct populations of resource Sm and Si formulations are given. Finally, we carry over to is presented in Fig. 1b. Shown is a schematic view a higher dimensional formulation and describe the of trophic interactions including intraguild preda- parameters choice and the equilibrium densities. tion and two populations Si and Sm of immobilized and mobile resources respectively. The IG prey G 2. General model competes with the IG predator F for both resource types and is also an additional resource for the IG We introduce an omnivory model with an IGP predator. The size of the population S increases unit derived from a simple non-spatial Lotka- i due to immobilization of individuals from the pop- Volterra system with the linear functional responses ulation S by the IG predator F . adapted from Holt and Polis (1997). The origi- m nal model consists of populations of two predators (IG predator and IG prey) and a common resource. We begin with an overview of a general IGP Here, we include new features such as a resource model and all the important trophic links and pa- differentiation mechanism which affects palatability rameters that are used to define it. Later we focus of a fraction of resource for the predators. Specif- specifically on two different formulations of the gen- ically, the entire resource population is subdivided eral IGP model. into distinct groups under the assumption that the groups differ from each other by the quality and fit- ness of the individuals. They are consumed by the The general model for a food web with an in- predators at different group-specific grazing rates. homogeneous resource is derived from the Lotka- The differentiation of the resource could be due to Volterra omnivory model (Diehl and Feißel, 2000, damage by the predator or initial inhomogeneous 2001) with the interaction term that accounts for distribution of the resource quality. Afterwards, we the transitions among different pools. The Lotka- generalize our model to the case of the multiple re- Volterra omnivory model consists of n + 2 equa- sources. tions. It is used as an approximation for the food The food web model for a multiple number web community with the IGP and n ≥ 2 mutually n of prey subpopulations {Sk}k=1 is sketched in interacting subpopulations of the resources. In the 1 Fig. 1 a. The top predator F and the intermediate absence of predation a basal population S1 devel- predator G are engaged in the IGP and share a com- ops according to logistic growth (Diehl and Feißel, mon resource S1. The resource pools are not inde- 2000). The set of equations for the population den- pendent because there is an exchange of individuals sities are written as follows:

1Here and everywhere in the text the numerical subscripts denote species at the same . 5 Shared 1st resource : factors limiting the growth of the populations in (1). dS1 −1 = [r(1 − S1K ) − aG − fF ]S1 dt A key assumption of the model is that there is −z1(S1,S2,...,S , G, F ), n only one-directional movement between the basal

Shared kth resource (k =2 ...n) : resource S1 and its fractions {Sk}k6=1. The local in- teractions among individuals from alternative pools dSk = wk(S1,S2 ...,Sn, G, F ) n dt are embedded via functional terms {wk}k=2 pro-

−[bkG + fkF + mk]Sk, vided in Table 2 for each type of the IGP formu- lation. These terms account for transitions among Intermediate predator (IG prey): the resource items {Sk}k6=1. The general omnivory dG = z2(S1,S2,...,S , G, F ) model (1) can be reduced to three types of IGP dt n formulations: system with immobilization and sys- −(gF + mg)G, tems with the resource turnover for n = 2 sub- Top predator (IG predator): populations and for n > 2 pools. For each of dF the formulations specific expressions of functional = z3(S1,S2,...,S , G, F ) dt n 1 2 3 ′ forms z ,z ,z and {wk} are provided in the Ta- +(g gG − mf )F, (1) ble 2. The term z1 is responsible for the exchange

The parameters of the model and main popula- of individuals among subpopulations {Sk} due to tions are described in details in Table 1. Here r is the species turnover or the immobilization mecha- the maximum specific growth rate of the resource nism. The transfer of individuals from the popula- n population S1, K is the carrying capacity of the tion S1 to {Sk}k=2 happens instantaneously at con- n resource defined as enrichment factor in the pre- stant rates {ck}k=2 correspondingly. Analogously vious models (Diehl and Feißel, 2000, 2001). The {qk,j }k6=j are defined as instantaneous migration n n subpopulations {Sk}k=2 are derived from the basal rates among subpopulations {Sk}k=2. The terms resource S1 via immobilization or via individual-to- z2 and z3 are used to evaluate the total predation individual turnover. Species from S1 and {Sk}k6=1 of the IG prey and the IG predator on the resource. are consumed by the IG predator at potentially dif- To achieve a stable persistence of all species the n ferent rates f and {fk}k=2 and by the IG prey IG prey should benefit more from an alternative n at rates a and {bk}k=2 respectively. The differ- resource than the IG predator. For this reason, n entiation among subpopulations {Sk}k=2 is pre- whereas the attack rates of the IG predator are served by a choice of distinct predation pressures equal for different resource pools, the IG prey es- n n ′ n ′ n {bk}k=2, {fk}k=2, feeding rates {fk}k=2, {bk}k=2 tablishes a higher predation pressure on subpopu- n and mortality coefficients {mk}k=2. The density- lations {Sk}k6=1 than on the basal pool S1. The independent mortality rates for S1, G and F are numerical values for the attack rates are chosen m1,mg and mf correspondingly. They are used as to be close to the experimentally observed values 6 (L¨oder et al., xxxx). of densities of each subpopulation. In the following sections we present an explicit Holt and Huxel (2007) used an extended IGP formulation of the model with immobilization and module with alternative resources that are defined of the model with the resource turnover for n = 2 independently. They evolve according to their own subpopulations. intrinsic growth rates. As opposed to the formu- lation given by Holt and Huxel (2007) and to a model with trophic supplements (Daugherty et al., 2007) here we do not consider external alternative resources. In our model with immobilization the F a F b population density in every resource pool varies due to immobilization by the IG predator and consump- G G tion by the predators. Similarly in the formulation with the resource turnover the transfer mechanism S S S S between resource subpopulations plays a role in ex- 1 2 3 4 K S m Si change among the distinct resource pools. Alter- native pools grow due to the influx of species from the basal resource or the other pools. Therefore Figure 1: (a) General structure of the food web model with the sizes of subpopulations are controlled mainly by two predators and multiple resource subpopulations; (b) the a number of direct encounters with the IG preda- structure of the food web with the presence of the immo- tor (immobilization) or by a species turnover from bilization mechanism by the IG predator. The resource is one resource subpopulation to another. In addition, subdivided into populations of mobile Sm and immobilized Si individuals. The links represent: (solid) food resources the individuals in the different pools of the basal for the top predator, (dashed) food resources for the inter- resource are distinguished by group-specific preda- mediate predator, (dot-dashed) transitions between different tion pressures that establish a top-down regulation resource pools.

7 Table 1: The variables and parameters for the general model (1).

Definition

General model Populations

Sk of resource in the kth pool, G population size of IG prey F population size of IG predator Parameters r maximum specific growth rate of the resource S1 K resource carrying capacity or enrichment a attack rate of predator G on S1 subpopulation f attack rate of predator F on S1 g attack rate of predator F on G ck per capita effect of species S1 on Sk qk,j per capita effect of species Sk to Sj bk attack rate of predator G on Sk species fk attack rate of predator F on Sk species mk mortality rate of species from Sk subpopulation mg mortality rate of G mf mortality rate of F g′ converting efficiency of food resource G into F ′ fj growth rate of F from resource Sj ′ bj growth rate of G from resource Sj

8 Table 2: The variables and parameters for the models (2) and (3).

Definition

Model with immobilization Populations

Sm,Si population sizes of mobile (immobilized) resource Parameters r maximum specific growth rate of population Sm K resource carrying capacity a,b attack rates of predator G on mobile (immobilized) population f attack rate of predator F on mobile and immobilized populations im immobilization rate a′,f ′ conversion efficiency factors

Model with a resource turnover Populations

S1,S2 population sizes of resources Parameters r maximum specific growth rate of subpopulation S1 K resource carrying capacity a,b attack rates of predator G on subpopulations S1,S2 f attack rate of predator F on subpopulations S1 and S2 tr transfer rate or per capita effect of S1 on S2

9 Table 3: Description of the functional forms used in the system (1).

Description Model equations

′ System with immobilization: S1 = Sm,S2 = Si,z1 = imSmF, w2 = z1,z2 = a (aSm + bSi)G, ′ z3 = f f(Sm + Si)F

′ ′ System with resource turnover: z1 = trS1S2, w2 = z1, z2 = a (aS1 + bS2)G, z3 = f f(S1 + S2)F (n = 2 resource subpopulations)

n n ′ n ′ System with the resource turnover: z1 = S1 k=2 ckSk, z2 = G j=1 bj Sj ,z3 = F j=1 fj Sj , P n P P (n> 2 resource subpopulations) wk = Sk(ckS1 + j=2 qk,j Sj ), k =2 ...n P

2.1. System with immobilization by predator where the state variables Sm and Si are the densi- The system with the immobilization illustrated ties of mobile and immobilized species. Note that in Fig. 1 b is derived from the equations (1) for two the feeding rates of the top predator F on both resource subpopulations by substituting the inter- populations Si and Sm are equal. By contrast, the action terms z1,z2,z3 and w2 from Table 1. After attack rate of the IG prey on immobilized subpopu- the substitution the set of equations for the IGP lation is higher than on mobile species. The relation model with immobilization yields: b>a holds in the presence and in the absence of Mobile resource: the predator F . This assumption is well justified

dSm 1 by the observations of an experiment with artificial = [r(1 − S K− ) − aG]S dt m m immobilization (L¨oder et al., xxxx). G. dominans −(f + im)FSm, demonstrate a strongly selective behaviour towards Immobilized resource: immobilized species when offered in a mixture with dS i = i FS − [bG + fF ]S , mobile cells of S. trochoidea. It was measured that dt m m i ingestion rates of the predator in the immobilized IG prey: prey treatment were by a factor of 20 greater than dG = [a′aS + a′bS − gF − m ]G, dt m i g those in the control treatment. IG predator: The stability of equilibrium densities and the dF = [f ′f(S + S )+ g′gG − m ]F, (2) dt i m f persistence zones of the system (2) with a non-zero 10 immobilization rate are discussed in Section 3.1. (Diehl and Feißel, 2000, 2001). By contrast, when- ever the turnover of species takes place and non- 2.2. System with the resource turnover zero densities are produced in the resource pool S2 2.2.1. General case of n =2 subpopulations the intraguild predation introduces a higher pres-

The model with the resource turnover is derived sure on the second subpopulation S2. This will from the general case (1) by substituting the func- potentially lead to a negative effect on the popu- tional forms from Table 2. It is written as follows: lation density in S2 and to higher levels of subpop-

1st resource: ulation S1. The result of this interaction is that the

dS1 −1 3-species coexistence is reached via the IGP com- = [r(1 − S1K ) − aG]S1 dt petition trade-off. −[fF + trS2]S1,

2nd resource: 3. Main results

dS2 = [t S1 − bG − fF ]S2, dt r We illustrate an emergent dynamical behaviour

IG prey: for the three formulations provided in Table 3 with stability diagrams. Due to high dimensionality of dG ′ ′ = [a aS1 + a bS2 − gF − m ]G, dt g the models (1)-(3) the analysis of an entire param-

IG predator: eter space is intractable. Only several illustrative examples for every formulation will be shown here. dF ′ ′ = [f f(S1 + S2)+ g gG − m ]F. (3) dt f 3.1. Model with immobilization All the parameters are chosen the same as for the system with immobilization (2). Note that the evo- In Fig. 2 the regions of stable positive equilib- lution equations are written as in (2) but immo- rium solutions versus immobilization and enrich- bilization term is replaced with the transfer term ment are shown. The parameter space is parti- that is dependent on the population densities. The tioned into several stability zones associated with transfer between the two subpopulations occurs the regions of coexistence, exclusion of both preda- each time whenever species from two different pools tors and exclusion of the IG prey at G = 0. The encounter each other. In the simplest case the num- boundaries defined for partitioning of the diagram ber of encounters is proportional to the population are found from the eigenvalue analysis (see Ap- densities of S1 and S2. pendix). As shown in Fig. 2 at low enrichment the If the density of second subpopulation is zero densities of both predators decay to zero and the and no differentiation in the resource takes place at summed of the resource reaches steady tr = 0 than the top predator F outcompetes the state at Sm + Si = K. The case of zero im- predator G due to a higher predation rate (f >a). mobilization has been already considered in previ- This outcome is predicted by the basic IGP model ous studies (Holt and Polis, 1997; Diehl and Feißel, 11 2000). At low immobilization and at high enrich- source subpopulation reach saturation threshold at ment only the top predator and resource are stable a higher enrichment (see Fig. 4). and positive, just as in the 3-species IGP model (Diehl and Feißel, 2000), whereas the coexistence between both predators and common resource is possible only in the regions of intermediate enrich- ment. A higher mortality rate for the predator G 0.3 a 0.25 results in its extinction in the region of low immo- S + S + G + F 0.2 i m m bilization in Fig. 2b where an extra resource can no i 0.15 longer support its persistence. Only the IG preda- 0.1 S + S + F S =0 i m tor and resource persist in this region of parame- 0.05 S i 0 ters. Situation is different for higher immobilization 1 1.5 2 2.5 3 K where a large region of coexistence for both preda- tors exists. The equilibrium densities shown on the 0.3 b diagrams are defined in the eq. (A.5) in Appendix. 0.25 Fig. 3 shows equilibrium densities of the four S + S + G +F 0.2 i m m components of the food web and their dependence i 0.15 on enrichment and immobilization rates. For a high 0.1 S S + S + F immobilization rate the resource population is dom- 0.05 i m inated by immobilized individuals. Meanwhile at 0 1 1.5 2 2.5 3 low immobilization mobile and immobilized popu- K lations increase along the gradient of enrichment an adverse pattern occurs at high immobilization.

The growth rate of the IG predator is noticeably Figure 2: Regions of stable coexistence for the immobi- reduced at im =0.2 due to an increase of the com- lization model (2). Stability diagrams are partitioned into petitive trade-off with the IG prey. The dependence the regions of stable (unstable) coexistence and alterna- tive states with exclusion of one of the predators. Let- of the population densities on the enrichment of re- ters stand for persistence of different trophic configurations: source is shown in Fig. 4 for the immobilization (S) =exclusion of both predators, (Si+Sm+F ) =coexistence im = 0.3. Meanwhile as predicted from the stan- of top predator and resource, (Si+Sm+F +G) =3-species co- existence. Parameters are: r = 0.4,f = 0.12, a = 0.025, b = dard IGP model (Diehl and Feißel, 2000) the IG ′ ′ ′ 0.1, a = 0.8,g = 0.025,f = 0.2,g = 0.5, mf = 0.04. Two prey is excluded at high enrichment in the model plots for:(a) mg = 0.02, (b) mg = 0.06 are shown. with immobilization at im = 0.3 the IG prey ben- efits from immobilized resource and its persistence is increased at a broader range of carrying capac- ities. The density of the mobile (immobilized) re- 12 Figure 3: Equilibrium densities along immobiliza- tion and enrichment gradients. Shown are densities of: (a) mobile resource, (b) immobilized resource,(c) predator G,(d) Figure 5: Regions of stable coexistence for the immobiliza- predator F . Parameters are used as in Fig. 2 tion model (2). The level curves for the grazing rate a de- fine the boundaries of the stability regions. The colorbar shows the values of a. Parameters are: r = 0.5,f = 0.1, b = ′ ′ ′ 0.1, a = 0.8,g = 0.07,f = 0.2,g = 0.5, mg = 0.02, mf =

2 0.04

How sensitive is a stable coexistence to small 1.5 mobile prey S m immobilized prey S variations of the attack rates of the intermediate i 1 predator G predator G? Will our predictions be still valid? To predator F examine the system behaviour for different attack 0.5 Biomass density rates of G the coexistence zones are exemplified for different values of a in Fig. 5. Colorcode is assigned 0 1 1.5 2 2.5 3 K according to grazing rate a. Overall the stability di- agram exhibits similar pattern as in Fig. 2. Specif- ically, the region of stable coexistence enlarges for higher immobilization. As it seems reasonable the number of stable solutions and the 3-species per- Figure 4: Equilibrium biomass densities versus enrichment manence zone in Fig. 5 gradually broadens with the for fixed immobilization rate im = 0.3. increase of predation pressure from predator G. Si- multaneously fewer exclusion steady states for the predator G are discovered. 13 3.2. Model with a resource turnover ified on the diagram in Fig. 6 d the IG prey levels remain positive. Since the IG prey has an advan- 3.2.1. Case of n =2 subpopulations tage as a competitor for the shared resource only In this section the equilibrium solutions and sta- the IG predator gets excluded from the system. bility of the equilibria are discussed for the sys- The stability behaviour of the system (3) is tem (2) with the mechanism of species turnover. highly sensitive to the alternations of attack rates of In Fig. 6 the regions of stable (unstable) equilibria G and the productivity of resource r. Changes of are plotted versus the enrichment and the transfer these parameters produce different emergent pat- rate. The results are contrasted on the stability di- terns as shown in Fig. 7. The location of states agrams in Figs. 6 a − d for different predation rates of stable (unstable) permanence and the exclusion of the IG predators. Four different states are lo- zone of G is still comparable to the patterns shown calized in the parameter space that corresponds to in Fig. 6, however the region of 3-species coexis- stable (unstable) persistence and exclusion of the tence gets visibly reduced. The reduction is more IG prey (IG predator). We investigate how the dy- evident on the plots Fig. 7 a,c and d. At higher namics in the extended IGP system responded to transfer rates the coexistence of all 3-species is no variation of enrichment levels. For each case shown longer observed and only the population of IG prey in Fig. 6 an increase of enrichment is accompanied and resource persist. Due to low productivity the with a series of bifurcations in the system mani- densities of the basal resource S1 are quickly de- fested by an invasion of higher trophic levels simi- pleted and the IG predator is driven to extinction. lar to predictions from the linear theory On the contrary, conditions become more profitable (Oksansen et al., 1981). For instance, at low en- for the IG prey that is released from the IGP pres- richment both regimes 4 and 1 are stable. Further sure and simultaneously obtains more benefits by increase of K at fixed tr =0.05 results in a chain of predation on the extra resource S2. bifurcations from a stable regime 4 to an unstable At low transfer rates (Fig. 7c and d) the IG 2 and subsequently to a stable coexistence regime predator is excluded independently on carrying ca- 1. A further increase of carrying capacity favours pacity of the resource. As expected, with increase an exclusion of G and shifts the population densi- of the attack rate of G the population of the IG ties towards the of the IG predator. An predator is driven to extinction due competition interesting feature is that at low transfer rate only with IG prey. However, situation becomes more a coexistence of the IG predator and the resource favourable for the IG predator at higher values of is found. The second subpopulation S2 is extin- the transfer coefficient tr. For high enrichment and guished fast due to predation and low transfer rate. intermediate transfer the IG prey is excluded from The steady states found for low enrichment are sim- the system. At a fixed enrichment several alter- ilar to the case of a single prey population without nating states are found along the gradient of tr transfer mechanism at tr = 0 and S2 = 0. As typ- (see Fig. 7 d). For example, at K > 2.5 the be- 14 haviour of the food web is very sensitive even to a enrichment K and limiting value tr. The colorcode small alternations of tr. Indeed, the system passes is assigned according to the percentage of stable co- through distinct steady states just within a small existence solutions found for 300 food webs. In all increment of transfer rate. The exclusion of the IG the replicas of the simulated system the steady state predator is observed at tr < 0.05, the coexistence densities for G, F and S1 are fixed (see (A.9) in the is found at tr 0.06 and the exclusion of the IG prey Appendix). Thus only the variations among pos- is achieved at tr 0.07. Finally at a higher transfer sible equilibrium densities {Sk}k6=1 are examined. values (tr > 0.14) both predators enter the system The constraints for the parameters of high dimen- and persistence is reached. sional system (1) are given in eqs. (A.10)-(A.12) in After presenting the results for the systems (2) the Appendix. and (3) we proceed to a more complex situation The stability diagrams in Fig. A.5 show some with n > 2 of distinct subpopulations of the re- similarities to the regions of coexistence in Figs. 6 source. and 7 found for the n = 2 subpopulation model (3). The size of the stability zone expands with the 3.2.2. Case of n> 2 prey subpopulations increase of the transfer coefficient. At low transfer For a multipopulation model the choice of pa- rates no stable persistence is found, but different al- rameters including predation rates can be enor- ternative traits. The percentage of stable food webs mously large. As a consequence more freedom is with 3-species is substantially lower for a large sys- provided for choosing equilibrium densities that can tem with n = 7 subpopulations than for n = 2, 3. fit the model (1). Since it is impossible to inves- This reduction in stability is independent on the tigate the entire range of biologically plausible pa- number of simulated food webs and a choice of main rameters we make a particular choice of parameters parameters of the system. It is possible that an in- that allow an easier comparison of the case n> 2 in crease of food web connectivity in this case impacts (1) with the model (2). The details of the procedure negatively the system (1) stability. Another feature are provided in Appendix. is that for n = 7 the percentage of stable equilib- In this section we show the results of the nu- ria at a fixed enrichment value decreases for large merical simulation for the model with n > 2 prey values of tr unlike in previous cases in Fig. 8 a-c. subpopulations. The system (1) for the case n> 2 The results of the numerical simulation demon- is integrated numerically. For the calculation of the strate that for n = 2 subpopulation up to 95% stability diagrams at different fixed values of enrich- of stable systems are found at a higher transfer ment and transfer rate we perform 300 simulations. rate and an intermediate enrichment. Second, for a The results of the simulations for n = 2, 3, 4 and 7 larger food web with n = 7 subpopulations a higher subpopulations are illustrated in Fig. 8. The per- percentage of stable steady states (up to 45%) are centage of stable 3-species coexistence is calculated identified at low transfer rate and at high enrich- for every point in the parameter space with fixed ment. 15 We compare the results of simulation for four a b cases (n =2, 3, 4, 7) at fixed enrichment K =1.341 0.2 0.2 1 1

r 0.15 0.15 and variable transfer coefficient in Fig. 9. The t 3 0.1 3 0.1 4 yields are derived for 1000 simulations of food webs. 0.05 4 2 0.05 The estimations of the number of steady states 1 2 3 1 2 3 c d show that for food webs with n ≤ 4 a higher per- 0.2 1 0.2 r t 0.15 3 0.15 1 centage of solutions with a stable coexistence are 0.1 0.1 0.05 2 0.05 identified than for food web with n = 7 pools. In- 4 4 0.6 0.8 1 1 2 3 deed, the yield for n = 2 reaches almost 95% mean- K K while the percentage of stable food webs found for n = 7 saturates at 23% for large tr. The non- Figure 6: Regions of stable (unstable) equilibria are marked monotonic variations of the yields in Fig. 9 reveal according to species composition: 1(2) - stable (unstable) co- existence of resource and both predators G and F ; 3 - exclu- a highly sensitive behaviour of the IGP model (1) sion of IG prey; 4 - exclusion of IG predator. The predation to a change in transfer rate in all cases. For n =7 rates are: (a) f = 0.1, a = 0.0155 ;(b) f = 0.01, a = 0.0155; the percentage of stable food webs reaches 41% at (c) f = 0.2, a = 0.0155; (d) f = 0.1, a = 0.065. The re- ′ a low transfer rate. It decreases substantially for maining parameters are: r = 0.5, b = 0.1, a = 0.8,g = ′ ′ 0.07,f = 0.2,g = 0.5, mg = 0.02, mf = 0.04 higher values of the transfer rate. For n ≤ 4 there is an overall incline from 60% at tr ∼ 0.1 to 95% at t ∼ 0.3 of stable configurations. a r 0.2 1 0.2 b 0.15 0.15 r t 1 Two types of stable equilibrium solutions are il- 0.1 2 0.1 lustrated in Fig. 10. Both solutions are obtained 0.05 3 0.05 4 4 inside the stable coexistence region as indicated in 1 2 3 1 2 3

Fig. 9. The system (1) is simulated with n = 5 0.2 c 0.2 d 0.15 3 0.15 number of subpopulations and initial conditions as r 1 t 0.1 0.1 3 defined in the Appendix. For the steady state in 0.051 4 0.05 Fig. 10 a and the oscillatory state in Fig. 10 b most 4 1 2 3 1 2 3 of resource subpopulations are unstable and their K K densities rapidly decline to zero after some initial transient. Nevertheless, coexistence in the system Figure 7: Regions of stable (unstable) equilibria are marked according to stable trophic configurations: 1(2) - stable (un- is typically supported by one or two resource pools stable) coexistence; 3 - exclusion of IG prey; (4) exclusion with non-zero densities. of IG predator. The predation rates are: (a) f = 0.1, a = 0.0155 ;(b) f = 0.01, a = 0.0155; (c) f = 0.15, a = 0.065; (d) f = 0.1, a = 0.065. The remaining parameters are: ′ ′ ′ r = 0.3, b = 0.02, a = 0.8,g = 0.07,f = 0.2,g = 0.5, mg =

0.02, mf = 0.04

16 2 10 0.3 0.3 prey S (a) a b 1 80 prey S 0.2 0.2 0 3

r 10 predator G t predator F 0.1 0.1

60 Biomass density n=2 n=3 0 500 1000 1500 t 0.5 1 1.5 2 0.5 1 1.5 2 150 40 c predator G (b) d 100 predator F 0.2 0.2 prey S

r 4 t 20 50 0.1 0.1

Biomass density 0 n=4 n=7 0 500 1000 1500 0 t 0.5 1 1.5 2 0.5 1 1.5 2 K K

Figure 10: Two stable solutions for the system (1) with Figure 8: The percentage of 3-species stable coexistence n = 5 subpopulations. The equilibrium densities and in- found for the model (1). The stability region is presented teraction rates are described in Appendix. Parameters are: versus enrichment and transfer rate. Simulations for the r = 0.7,f = 0.1, b = 0.1, a′ = 0.8,g = 0.07,f ′ = 0.2,g′ = a b c four cases are given: ( ) n = 2, ( ) n = 3, ( ) n = 4, 0.5, mg = 0.02, mf = 0.04,K = 1.8, tr = 0.2 (d) n = 7. Parameters are: r = 1,f = 0.1, b = 0.1, a′ = ′ ′ 0.8,g = 0.07,f = 0.2,g = 0.5, mg = 0.02, mf = 0.04 4. Discussion

There is growing evidence from theoretical and

100 empirical studies that creating additional trophic n=2 links have a stabilizing effect on food webs (Moore, 80 n=3 n=4 2005; Ives and Carpenter, 2007). Generalized mod- n=7 60 els reveal that the stability of food webs can

40 be enhanced when species at higher trophic lev-

Percentage els graze upon multiple prey species (Gross et al., 20 of stable equilibria 2009). In particular, for low dimensional food 0 0.1 0.15 0.2 0.25 0.3 webs it is demonstrated that an addition of alter- t r native food resources can stabilize the interactions (Holt and Huxel, 2007) and open up a possibility for feedbacks on population dynamics due to ap- Figure 9: The percentage of 3-species stable coexistence for parent competition. The predictions of our model the model (1). Nsim = 1000 simulations are performed for the cases: n = 2, 3, 4, 7. Parameters are as in Fig. 8. Enrich- confirm the main conclusions given in a theoretical ment: K = 1.341. study of an extended IGP model (Holt and Huxel, 2007). In the alternative formulations used here the IG prey has the access to an extra resource be- 17 yond the shared resource for which both predators ate immobilization. A similar type of experiments compete. This extra resource is a more attractive performed for various initial species densities could resource item for the IG prey and is thus attacked furnish a justification of this hypothesis. at higher rates by the IG prey whereas the attack The above results demonstrate that a persis- rate of the IG predator stays the same. Moreover tence of IG predator, IG prey and resource is the IG predator indirectly stimulated the growth of achieved even at a low value of immobilization rate. the IG prey population by providing this extra re- Moreover, a significantly higher percentage of ob- source. Our predictions tested by the application served stable configurations is found when the im- of a stability analysis are robust in the sense that mobilization and transfer links in (1) and (2) are they are independent of the form of the interac- strengthened. Because our model is an oversim- tion term that is responsible for the availability of plification of the experimental behaviour the parti- an additional resource. We demonstrate that for tioning of the parameter space according to stable different formulations of the basic IGP model with versus unstable coexistence could be used as an ap- the embedded interactions a stable 3-species coex- proximation of the population dynamics found in a istence is ultimately reached whenever a moderate real experimental situation. Firstly, the conditions strength of the omnivorous links is used. for long term stable coexistence found by numerical However, the problem to relate the experimental simulations are not so easy to examine experimen- findings (L¨oder et al., xxxx) to the theoretical pre- tally because of technical and temporal restrictions. dictions of our model (2) still remains open. Since Experimental samples in ref. (L¨oder et al., xxxx) the experiment is aimed to observe a short term are taken during 3 days of incubation due to a de- populations development it is not easy to find a di- of the prey population. Secondly, due to the rect correspondence between empirical population existence of stable limit cycles as predicted by our dynamics and theoretically predicted behaviour. In linear stability analysis (see Fig. 10) the oscillatory the 3 day batch culture experiment of L¨oder et al. solutions go through a period of very low densities with all 3 species present both predators G. dom- and might be driven to extinction in the presence inans and F. ehrenbergii displayed positive growth of random fluctuations of the environment. while the prey population S. trochoidea displayed si- We point out that our numerical simulations of multaneously a sharp decline to almost zero. How the extended 3-species IGP module (1) do not ex- can this behaviour be classified according to our plore the entire swath of parameters and configura- theoretical model? Could it be a part of an oscil- tions for the steady states. Rather our analysis fo- latory cycle or an unstable state? It is not easy cuses on explaining conceptual features of the IGP to answer these questions, however, we can make a model with diverse prey populations. guess that the short term evolution observed in the Earlier studies focussing on dynam- experiment recasts as a part of an oscillatory cycle ics of complex ecological communities for a periodic equilibrium state found at intermedi- (DeWitt and Langerhans, 2003) demonstrated the 18 importance of multiple prey traits in mitigating locities of resource species. Another question is if predator selection pressures and altering predator- the growth rate of the IG predator will be affected induced behavioural shifts in natural environments by the inclusion of the time of resource capture. (Lima and Dill, 1990; Trussell et al., 2002). Our How will the inclusion of the time lag change the model can also be adapted to food web communi- predictions of our immobilization model? These ex- ties in which differentiation among prey individuals, tensions of a general IGP model will be a topic for namely, variation in individual traits such as fitness our future investigations. and mobility, is a result of heterogeneities in their Finally, we point out that it is of potential inter- natural habitat and/or adaptation of the species to est for biological control and conservation manage- the local conditions of the habitat. We show that ment to understand functioning of omnivory and an existing diversity of resource items traits can sig- IGP systems in relation to global changes of the nificantly alter the emergent community patterns. environment. Since IGP food webs are widespread Adding new subpopulations of resources with dis- in natural communities their adaptation and re- tinct traits that are more vulnerable to an attack silience behaviour is principal for understanding the from the IG predator facilitates the coexistence restructuring of natural communities. of both IGP-related predators which compete for the common food resource. Thus, the presence of 5. Conclusions an alternative resource indirectly induces shifts in exploitative competition. We have used three formulations of a general It is important to note that a general math- IGP model to explore the effects of increasing di- ematical model with density- dependent interac- versity in the prey population on higher trophic lev- tions and immobilization do not render a unique els. The reformulated IGP model alters the results theoretical description for the results of the exper- from the basic IGP theory (Polis and Holt, 1992; iment (L¨oder et al., xxxx). Our model predictions Holt and Polis, 1997; Diehl and Feißel, 2000). We can be tested against alternative formulations. In- show that an increase of a number of trophic inter- deed, the main features of the experimental system actions in the system via differentiation of resource can be examined by the inclusion of predation rates can stabilize the population dynamics of the IGP that are dependent on the mobility of the resource module. This conclusion holds for the densities of species. Since slow and immobile individuals can the IG prey that level up even when the IG preda- also be found among mobile species one can use an tor is a superior competitor for the common basal inhomogeneous distribution of velocities of the re- resource. source species in a theoretical model. To guarantee The results of our numerical simulations can be more benefit for an intermediate consumer in catch- summarized as follows. ing a certain type of individual distinct predation First, we show that for the system with the rates should be assigned according to different ve- immobilization term up to three regions of stable 19 trophic configurations are observed along the en- crease of the attack rates of the IG prey depresses richment gradient. Meanwhile at low enrichment the population of the IG predator until it is com- both IG prey and IG predator are excluded, at high pletely excluded. enrichment the presence of only small concentra- Numerical simulations of food web (1) with tion of immobilized cells is sufficient to facilitate n = 2, 3, 4 and 7 distinct pools demonstrate that the coexistence of the competitors in the IGP rela- the high dimensional food webs overall manifest far tionship. Moreover the percentage of all admissible less stable behaviour than the food webs with only trophic configurations for the 3-species persistence two distinct subpopulations. An interesting feature inclines substantially for higher immobilization. is that the percentage of stable states for n = 7 Second, given that immobilization is high substantially decreases from 40% to 23% with an enough it prompts the exchange between pools of apparent increase of the value of transfer rate. By mobile and immobilized resource and facilitates fast contrast, for food webs with n ≤ 4 an increase in decline of the mobile population and a growth of im- transfer rate leads to the growth of the percentage mobilized subpopulation. Meanwhile the exchange of stable coexistence solutions from about 60% to between the basal mobile resource and the preda- 95%. tors gets weaker due to the low density of mobile species the immobilized individuals become a ma- Appendix A. jor food resource for the predators. Because the IG prey is a superior competitor for immobilized re- source a robust coexistence of both predators will In the Appendix we review the steady state so- be easily supported. In addition, along an increas- lutions for the Lotka-Volterra models (2) and (3) ing gradient of immobilization the relative abun- and provide Jacobian matrices to examine their lo- dance of IG prey becomes higher than the abun- cal stability for the coexistence of both predators dance of IG predator. and the resource. Restructuring of the basic IGP module by First, equilibrium solutions are derived for the adding individual-to-individual turnover facilitates 3-species model (2) with zero immobilization (im = the coexistence and stabilizes the otherwise unsta- 0) and zero initial size of immobilized population ble system. Moreover a strengthening of the inter- (Si = 0). Second, the steady states are given for action link leads to a significantly broader range of the model (2) with immobilization (im 6= 0). At enrichment values at which stable coexistence could last, the equilibrium solutions are presented for the be found. At low transfer rate two types of equilib- system with the resource turnover (3). For every ria are observed: (i) if the IG predator is a superior case various trophic configurations are considered: competitor for the resources than at low enrichment (i) exclusion of both predators, (ii) exclusion of IG both predators are excluded and at high enrichment prey or IG predator and (iii) the 3-species coexis- only the IG predator stays in the system; (ii) an in- tence. 20 Appendix A.1. Steady state solutions for 3-species Also the Jacobian matrix for the 3-species coex- model istence is provided in the explicit form.

The equilibrium solution of (2) for the 3-species For the model (2) with immobilization (im 6= 0) coexistence without immobilization is stated as fol- we define a set of equilibrium densities to satisfy lows: the equalities below:

−1 amf mgf Q1 = [r(1 − SmK ) − aG Seq = r − +  gg′ g  1 −(f + im)F ]Sm, r fa f ′ − × − − a′ , K g  g′  Q2 = imFSm − Si(bG + fF ), (A.4) ′ ′ −1 Geq = (mf − ff Seq )(g g) , ′ ′ Q3 = (a aSm + a bSi − gF − mg)G, ′ −1 Feq = (a aSeq − mg)g , (A.1) ′ ′ Q4 = (f f(Si + Sm)+ gg G − mf )F.

′ where g,g > 0. The necessary condition for the The system (A.5) has four alternative solutions : (i) coexistence requires that the right hand side is pos- exclusion of both predators at S = K; (ii) exclu- itive in (A.1). F F sion of the IG predator (Sm, G ) at F = 0; (iii) The expressions for the equilibrium densities for the coexistence of resource and the IG predator the survival of the IG prey and the resource with G G G (Sm,Si , F ) at G = 0; (iv) the 3-species coex- e e e e exclusion of the IG predator at Feq = 0 yield: istence (Sm,Si , G , F ).

m r SF In the absence of F the immobilization mecha- SF = g , GF = 1 − eq . (A.2) eq aa′ eq a K  nism is not active and the model (2) reduces to the

The condition for persistence of the IG prey and system without immobilization (Diehl and Feißel, ′ the resource reads: a aK >mg. 2000) where the equilibrium solutions written as At zero density of the intermediate consumer (A.2). Upon exclusion of the IG prey in (A.5) one

(Geq = 0) one yields the steady states of the re- obtains expression for the equilibrium densities of G G resource and the IG predator: source Seq and the IG predator Feq :

G G mf G im G G mf G r Seq Sm = ′ , Si = Sm, S = , F = 1 − . (A.3) f (im + f) f eq ff ′ eq f  K  r SG F G = 1 − m . (A.5) The densities are positive if and only if the condi- f + im  K  ′ G tion ff K>mf holds true. Note that the size of mobile population Sm is pro- G portional to the size of immobilized population Si . Appendix A.2. Steady state solutions and linear As is expected the population of immobilized preys stability analysis is impacted positively by the increase of immo- Appendix A.2.1. Model with immobilization bilization. Although the predation pressures are Here we describe alternative steady state solu- equal for both resource subpopulations the immo- G tions and discuss their local stability derivation. bile population Si extinguishes faster than the mo- 21 G bile population Sm. Indeed, with the increase of of the predators at S1 + S2 = K; (ii) the exclusion F F F predation rate f the following approximations hold: of the IG predator (S1 ,S2 , G ) at F =0; (iii) the G G G G G Sm ∼ 1/(im + f) and Si ∼ 1/f(im + f). exclusion of the IG prey (S1 ,S2 , F ) at G = 0; e e e e The equilibrium densities for the 3-species coex- (iv) the coexistence of 3-species (S1 ,S2, G , F ). istence are derived from (A.5) by setting the right The solution for the coexistence of the resource hand side to zero. and the IG prey in the absence of the IG predator At last, to evaluate the stability of the equilib- is expressed as follows: e e e e rium solution Sm,Si , G and F one solves for the F trmg S1 = K 1 − , eigenvalues of the stability matrix :  ra′b  ′ F F mg − a aS1 F tr F S2 = , G = S1 . (A.7) e e a′b b D1 0 −aSm −(f + im)Sm   e e e e Note that at t = 0 the IG prey is excluded and imF D2 −bSi imSm − fSi r    ′ e ′ e e   aa G ba G D3 −gG  steady state density for the resource approach the    ′ e ′ e ′ e   ff F ff F gg F D4  carrying capacity limit K. The positive solution of   (A.7) exists if the parameters satisfy the inequality: The matrix diagonal is written in terms of the ′ ra b>trmg. equilibrium densities Se ,Se, Ge and F e as follows: m i The steady state for the resource and the IG r D = − Se , −bGe − fF e, 0, 0 . (A.6) predator in the absence of IG prey yields:  K m  G trmf e e e e S1 = K 1 − , The solution (Sm,Si , G , F ) is globally  f ′fr  G mf asymptotically stable in the phase space S2 = (1+ Kt ) − K, r f ′f (Svirezhev and Logofet, 1983) if the condition for G Ktr mf F = r − tr . (A.8) stability is satisfied. For the stable coexistence it rf  f ′f  is necessary that the real parts of all four eigen- A nontrivial solution for the 3-species coexistence is e e e e values λi, (i = 1,... 4) of the stability matrix are found by solving for the equilibrium (S1,S2, G , F ) non-positive. To obtain the boundary for stability of the following system: regions in the parameter space the eigenvalues of 0 = r(1 − SeK−1) − aGe the above stability matrix are evaluated numeri- 1 e e cally at different parameters combinations. The −fF − trS2 , e e e resulting stability diagrams are presented in Fig. 2 0 = trS1 − bG − fF ,

′ e ′ e e and Fig. 5. 0 = a aS1 + a bS2 − gF − mg,

0 = f ′f(Se + Se)+ g′gGe − m . (A.9) Appendix A.3. Model with prey–to–prey interac- 1 2 f

tions and n =2 subpopulations Finally, the condition for the stable coexistence is As in the previous case the system 3 for n = 2 provided by solving for the eigenvalues of the sta- pools permits four steady states: (i) the exclusion bility matrix : 22 e e e e 3 −r/KS1 −trS1 −aS1 −fS1 ues for the grazing rates {fk}k> , are randomly   e e e trS2 0 −bS2 −fS2 assigned from the interval [0,f/n]. In the next    ′ e ′ e e  step an antisymmetric matrix Q of individual-to-  aa G ba G 0 −gG     ′ e ′ e ′ e  individual interactions is defined with the upper di-  ff F ff F gg F 0    The eigenvalues of the stability matrix are func- agonal coefficients {qjk}j2 are expressed from the equations as follows: Appendix A.4. Model with prey–to–prey interac-

tions and n> 2 subpopulations mk = ckS1 − qjkSj − bkG − fkF. (A.11) X Here we define initial conditions that are used The equation (A.11) for k = 2 is used to solve for the numerical simulation of the model (1) with for the attack rate f2. Finally, the attack rates n > 2 number of subpopulations. Unlike for the 1 1 {b′ }n− and the feeding rates {f ′ }n− are ran- system (3) with n = 2 subpopulations the equilib- k k=2 k k=2 domly assigned from the intervals [0,a′b/(n − 1)] rium densities can no longer be determined analyt- and [0,f ′f/(n − 1)] correspondingly. The choice ically. To simplify the search for the equilibria in of the feeding rates enables to reduce the preda- (1) we implement several assumptions. 1 tion pressure on populations {S }n− by a factor We chose the parameters and equilibrium den- k k=2 ′ of 1/(n − 1). The remaining grazing rates f1 and sities to fulfil several constraints provided below. ′ ′ ′ b1 are defined from the relations: f1 = f f and Initial densities are equal to the steady states (A.9) ′ ′ e b1 = a a. Finally, two constraints hold to solve for for n = 2 subpopulations, namely: S1 = S1,S2 = ′ ′ e e e the coefficients bn and fn: S2, G = G and F = F . For the sake of simplic- n−1 n n ity the values for the remaining densities {Sk}k=2 ′ ′ e ′ ′ e fj Sj = f fS2, bj Sj = a bS2. (A.12) are defined to be less than the equilibrium density Xj=2 Xj=2 e S2. Provided that the interaction rates {ck}k6=1 are ′ We want to emphasize that the attack rate b1 randomly assigned values not exceeding tr/(n − 1) ′ n should be higher than any of the rates {bj}j=2. the equilibrium density S can be found from the n Nevertheless since a>b relation holds the total constraint: predation pressure of G summed over alternative n e pools exceeds the predation exclusively on S1. This cj Sj = trS2. (A.10) Xj=1 far the positive population density of the IG prey is

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