Journal of Biological Dynamics ENRICHMENT IN A STOICHIOMETRIC MODEL OF TWO PRODUCERS AND ONE CONSUMER. Journal: Journal of Biological Dynamics

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1 ENRICHMENT IN A STOICHIOMETRIC MODEL OF TWO PRODUCERS AND ONE CONSUMER Journal: Manuscript ID: TJBD Manuscript Type: Original Article Date Submitted by the Author: 0-Oct-00 Complete List of Authors: Lin, Laurence; Virginia Polytechnic Institute and State University, Biological Sciences Peckham, Bruce; University of Minnesota Duluth, Mathematics and Statistics Stech, Harlan; University of Minnesota Duluth, Mathematics and Statistics Pastor, John; University of Minnesota Duluth, Biology Department Keywords: population dynamics, producer-consumer, predator-prey, stoichiometry, bifurcation, enrichment

2 October 0, 00 : Report Page of Vol. 00,No.00,Month00x, RESEARCH ARTICLE ENRICHMENT IN A STOICHIOMETRIC MODEL OF TWO PRODUCERS AND ONE CONSUMER LAURENCEHAO-RANLIN a, BRUCEPECKHAM a,harlan STECH a, andjohn PASTOR b a Departmentof MathematicsandStatistics,Universityof Minnesota, DuluthMN ; Dept. Fax: --; b DepartmentofBiology,Universityof Minnesota, DuluthMN ;Dept.Fax:--; linx0@d.umn.edu,ContactPhone: -- bpeckham@d.umn.edu,OfficePhone: -- hstech@d.umn.edu,officephone: -- jpastor@d.umn.edu,officephone: --00 (v.0 released June 00) In this paper, we consider a stoichiometric population model of two producers and one consumer. It is a generalization of the Rosenzweig-MacArthur population growth model, which is a oneproducer, one-consumer population model without stoichiometry. The generalization involves two independent steps: ) introducing stoichiometry into the system, and ) adding a second producer which competes with the first. Both generalizations introduce additional equilibria and bifurcations to those of the Rosenzweig-MacArthur model without stoichiometry. The primary focus of this paper is to study the attractors and bifurcations of the two-producer, one-consumer model with stoichiometry. We focus on bifurcations which are a result of enrichment.the model is closed for nutrient but open for carbon. The primary parameters we vary are the growth rates of both producers. Secondary variable parameters are the total nutrient in the system, and the producer nutrient uptake rates. Depending on the parameters, the possible equilibria are: no-life, one-producer, coexistence of both producers, the consumer coexisting with either producer, and the consumer coexisting with both producers. We observe limit cycles in the latter three coexistence combinations. Bifurcation diagrams along with corresponding representative time series summarize the behaviors observed for this model. Keywords: Population model, producer-consumer, predator-prey, stoichiometry, bifurcation, enrichment. Introduction A population is the collection of inter-breeding organisms of a particular species. Mathematical population models follow the size of interacting populations over time. Such models are fundamental in biology and ecology. A producer-consumer (prey-predator) population model studies the populations of two species which interact in a specific way. A variety of producer-consumer models has been developed and studied over the last century, usually with a single currency, such as biomass, for each population. More recently, stoichiometry, which can be thought as tracking food quality as well as quantity, has been introduced into population Correspondingauthor. linx0@d.umn.edu,laurenl@vt.edu ISSN: -0X print/issn - online c 00x Taylor& Francis DOI: 0.00/0YYxxxxxxxx

3 October 0, 00 : Report Page of models[,,,,0,,,]. The book of Andersen [], in particular, includes a two-producer, one-consumer model with stoichiometry, but it differs from our model in ways which we will detail later. See [,, ] for models with two producers and one consumer, but without stoichiometry. See [] for a more mechanistic model of competing producers, but without a consumer and without stoichiometry. Every organism has its own ratios of nutrients in its body, and those ratios are called stoichiometric ratios. Stoichiometric ratios tend to be relatively constant for most animals, but have been observed to have a wider range of values for plants []. In this paper we track carbon and one additional unnamed nutrient. Food that has a low stoichiometricratio nutrient to carbon is considered low quality, and contributes less to consumer growth than high ratio food. Stoichiometric models, especially those with a closed nutrient cycle, tend to exhibit a paradox of energy enrichment, a phenomenon in which the population of the consumer declines in response to an increase in the population of the producer [, 0,,, ]. In, Rosenzweig and MacArthur [] presented a geometric analysis of producer growth and consumer response. It has become one of the classic producer-consumer population models with a single currency(i.e., no stoichiometry). Formulas () are the traditional realization of the Rosenzweig-MacArthur model, using logistic producer growth and Michaelis-Menten consumer response with food saturation. dp dt = rp λp f(p)c dc dt f(p) = αp = γf(p)c dc h+p r is the maximal per-capita growth rate of producer in the absence of predation. λ is the producer self-limitation coefficient. γ is the efficiency of turning predated food into consumer biomass. d istheper-capitadeathrateoftheconsumer. f(p) is a Holling Type II or Michaelis-Menten function that describes consumer food saturation. α is the maximal producer death rate per consumer encounter due to predation. h is the half-saturation constant for the Michaelis-Menten function. A standard parameter to vary in () is the producer growth rate r. For low r the model predicts a producer monoculture system because there is not enough food to maintain the consumer. Increasing r sufficiently to pass a transcritical bifurcation value allows both producer and consumer to stably coexist. Further increase in r results in a destabilization of the coexistence equilibrium in a Hopf bifurcation, accompanied by the birth of an attracting limit cycle. The amplitude of the cycle grows with continuing increase in r, but no further bifurcations are observed[]. ()

4 October 0, 00 : Report Page of We build our model starting from a basic Lotka-Volterra model for two competing producers. We introduce a consumer which interacts with the producers in a similar way to the Rosenzweig-MacArthur model. We keep track of both carbon (as a surrogate for biomass) and a nutrient (which can be thought of as nitrogen or phosphorus). Our model is open for carbon, but closed for nutrient. Like most nutrient models, we track nutrient in the sediment as well as in the two producers and the consumer. The effect of stoichiometry is introduced by lowering the rate of biomass conversion from the producers to the consumer when the nutrient to carbon ratio of the food (producers) is below the level needed for consumer biomass creation. Details are in Section. Although there are many parameters in our model, we focus on varying two primary parameters: the growth rates of the two producers. Secondarily, we vary the total nutrient available in the system, and the nutrient uptake rates for the producer(s). We perform a bifurcation study using a combination of analysis and numerics. Our most significant contribution is identifying parameter combinations which allow the coexistence of all three species and the bifurcations which occur as one passes from inside to outside the parameter region of coexistence. We observe the loss of coexistence as one species goes extinct in some version of a transcritical bifurcation. The paper is organized as follows. In section we provide the details of the model construction. In section we analyze the reduced model, having only one producer. This lays the groundwork for the techniques used in section to analyze the full model. The analysis of the reduced model also provides a basis of comparison for the two-producer model. Discussion is in section, and conclusions in section.. Model Construction This model contains two producer populations, a single consumer population, and a sediment compartment. We choose to keep track of carbon density as a measure of biomass quantity, and nutrient density (later changed to the nutrient-to-carbon ratio) as a measure of food quality. This leads to three carbon variables: P, P, C,andfournutrientvariables: N, N, N C, M,where M denotes the mineralized nutrient concentration in the sediment. Variables of the model are summarized in table. We describe below why a sediment class for the carbon is not necessary. The model we will actually study is reduced from these seven variables to five by assuming a fixed amount of total nutrient, and a fixed stoichiometric ratio q = N C /C for the consumer. Figure illustrates the flow of carbonandcycleof nutrientinthe model. The equationsfor our modelare givenbelowin equation().

5 October 0, 00 : Report Page of >< >: dp dt = (b λ P λ P ) + P d p P P P +P α(p +P ) h+(p +P ) C dp dt = (b λ P λ P ) + P d p P P P +P α(p +P ) h+(p +P ) C dc dt = min(γ, N +N ) α(p +P ) q P +P h+(p +P ) C dcc dn dt = (N T qc N N )β (b λ P λ P ) + P P P +P α(p +P ) h+(p +P ) C N P d p N dn dt = (N T qc N N )β (b λ P λ P ) + P P P +P α(p +P ) h+(p +P ) C N P d p N We now describe the assumptions which lead to the equations(). Carbonflow: P, P, C. Carbon is fixed in the producers due to photosynthesis. This producer growth is assumed to be logistic. Producer respiration, not explicitly modeled, is assumed to be a constant fraction of the carbon fixation. That is, the growth term (b i P i λ ii Pi λ ij P i P j ) is interpreted as net photosynthesis. We have chosen to make the producer growth independent of the nutrient. This independence assumption simplifies the analysis and allows us to focus on the role of stoichiometry in the conversion of biomass from producer to consumer. Lotka-Volterra competition is assumed between the two producers. The product of the competition coefficients λ λ is assumed to be always less than the product of the self-limitation coefficients λ λ. Otherwise, the two producers couldnotcoexistevenintheabsenceoftheconsumer[,]. Becauseweassume nutrient uptake depends on producer growth, a reduction in growth rate is not the same as an increase in the death rate. Consequently, the producer death terms (d Pi P i ) are separatedfrom the growth terms in our model, and the growth terms are always assumed to be nonnegative. This is why we use the positive part function, () +, in thefirst twoequationsof(). Sincetheamountofcarboninthesedimentdoesnotaffectanyoftheothervariables, there is no need to include a carbon sediment compartment in the model. Mortalityresultsinanexitofcarbon,butnotnutrient,fromthesystem.SeeFig.. The predation of both producers (consumer response) is assumed to follow a Michaelis-Menten function. Both producers are food for the consumer, which foragesnonpreferentially:theprobabilitiesofproducer P and P beingeatenare P assumed to be P +P and P P +P,respectively.Thatis,theproducersareconsumed in proportion to their relative abundance. The consumer gains biomass through predation. Consumer death is assumed to be proportional to its population size. The biomass conversion efficiency is modelled as the minimum of γ and N+N, the ratio of the aggregate food q(p +P ) N source s stoichiometry, +N P +P, to the consumer s stoichiometry, q. If the food source contains sufficient nutrient to support the maximum conversion efficiency γ, then the rate γ is achieved. Otherwise, stoichiometric limitation reduces the conversion efficiency to N+N q(p +P ), reflecting the maximum rate at which nutrient can be supplied to build structural consumer biomass. Implicit in the use of this term is the assumption that the consumer assimilates a mixture of the two producers together, rather than eating and assimilating one producer at a time. ()

6 October 0, 00 : Report Page of This assumption is similar to the foraging and biomass conversion assumptions in Andersen[], but differsfrom thoseof[]. Note that if the efficiency minimum function is replaced with the constant γ, then the carbon equations decouple from the nutrient equations, and the system reverts to a non-stoichiometric one. Nutrientcycling: N, N. The nutrient is assumed to cycle through the four compartments, N, N, N C, and M, as in Fig.. We eliminate N C by assuming that the consumer nutrient to carbon ratio is constant q. So N C = qc. Additionally,we eliminate M by assuming a closednutrient systemwith totalnutrient N T, so M = N T N N qc. This leavesuswith thetwo nutrientvariables N and N. The producer gains nutrient via uptake from the mineralized nutrient, and loses nutrient due to predation and mortality. The nutrient uptake rate in this model is proportional to the mineralized nutrient M = N T N N qc. In contrast to most other stoichiometric models, nutrient uptake is assumed, as in [], to be proportional to producer growth. Of course, proportional uptake would be required of producers that maintain a constant N : C ratio. The nutrient uptake coefficients β and β are assumed to be constant. Nutrient is transferred from producer to consumer via predation, and returned to the nutrient pool when either a producer or the consumer dies. When the producer is nutrient-rich with nutrient to carbon ratio greater than γ the consumer does not use all the producer nutrient it consumes. The excess is assumed to be returned to the nutrient pool via excretion. We assume that litter decomposition and nutrient mineralization are instantaneous, so the nutrient is available(in compartment M) for uptake as soonasitleavesthe producersor theconsumer. To simplify somewhat the equilibrium analysis of formulas (), we introduce producer ratios Q i = Ni P i,i =,. Then calculating the derivatives of Q and Q, we get the equivalent stoichiometric form of the system: dp dt = ((b λ P λ P ) + d p dp dt = ((b λ P λ P ) + d p dc dt = (min(γ, q P Q +P Q P +P ) α(p+p) h+(p +P ) d c)c αc h+(p )P +P ) αc h+(p )P +P ) dq dt = ((N T qc Q P Q P )β Q )(b λ P λ P ) + dq dt = ((N T qc Q P Q P )β Q )(b λ P λ P ) + Note that dc dt is not defined if both P and P are zero. Assuming h > 0, it can, however, be continuously extended to P = P = 0 by dc dt = d C C, because of the α(p+p) h+p +P factor. For our numerical investigations, we fix the following parameters, representing two nearly identical producer species (differing only in maximal per-capita ()

7 October 0, 00 : Report Page of growth rates b and b ) andanutrient-rich(relativelyhigh q)consumer. Maximalproducerphotosytheticgrowth rate: b i [0,0] Totalnutrientdensity: N T [0,0.], default: 0. Self-limitingcoefficients: λ = λ = 0. Interferencecoefficients: λ = λ = 0. Producer per-capitanaturaldeathrates: d P = d P = 0.0 Consumerper-capitadeathrate: d c = 0. Consumer stoichiometric ratio: q = 0.0 Maximal biomass conversation efficiency: γ = 0. Maximal predation rate: α =. Predation half-saturation coefficients: h = 0., Nutrientuptakeconstants β {0.,0.,0.}, default: 0.; β = 0.. Our primary goal is to determine the effects of system enrichment with respecttocarbonandnutrients.increasing b i isassumedtocorrespondtocarbon enrichment;increasing N T isassumedtoenrichthesystemwithnutrients.consequently, the parameters, b, b, and N T are varied in our numerical experiments. We also experiment with different values of β, the first producer s nutrient uptake rate. Rate constants are dependent on the time scale and the units selected to measure biomass and nutrient densities. Rescalings and alternate parameter selection would likely be necessary to match specific systems. For example, if we rescale both time and the phase variables by a factor of three ( t = t,p = P,P = P,C = C,N = Ñ,N = Ñ) then our variables and parameters are consistent with an algae-daphnia system with rescaled time measured in days, and rescaled phase variables measured in mg/liter []. Specifically, the rescalings would result in replacing parameters b,b,d P,d P,α,h,d C,N T,β,β with b /,b /,d P /,d P /,α/,h/,d C /,N T /,β,β, respectively.the λ ij s and γ would remain unchanged. Theoretically, our model could apply to either aquatic systems or simple terrestrial systems. See section. where we review the limitations of the model. The papers which match our model assumptions most closely but for one producer instead of two are Loladze, Kuang, and Elser (LKE) [], Kuang, Huisman and Elser (KHE) [0], and Wang, Kuang, and Loladze (WKL) []. The LKE model does not include a nutrient pool. The KHE and WKL models do include a nutrient pool. All three models are closed for nutrient, and have stoichiometric limitation for conversion from producer to consumer that is similar to ours. All three models have nutrient uptake assumptions that differ from ours. Additionally, their models all include stoichiometric limitation for producer growth. All three studies examine system behavior under enrichment, but the enrichment is effected by directly increasing producer carrying capacity, rather than increasing producer growth rates. Miller, Kuang, Fagan and Elser [] investigate a model which, like ours, has two producers, one consumer and stoichiometry. Their model differs from ours in several facets, most significantly because they assume the two producers are in different patches, and that available nutrient for uptake (rather than total nutrient) remains constant. Andersen [] includes a model which also has two producers, one consumer and stoichiometry, but it also differs in many ways from our model. Most significantly, the Anderson study considers open nutrient systems, while ours is closed. The Anderson model does not assume any compe-

8 October 0, 00 : Report Page of tition or self limitation between producers, and it makes different assumptions regarding nutrient uptake by the producers. Further, our interests are in examining enrichment-induced changes in system dynamics a topic not considered in [].. One-Producer, One-Consumer Model In order to more easily understand the model (), we first consider the special casewhere there isasingleproducer inthe system.we set P = 0and ignore Q. Fornotationalconveniencewedropthesubscriptsfrom λ, P,Q,b,β and d P for therest ofthissectiontoobtain: dp dt = (b λp d P αc h+p )P dc dt = (min(γ, Q q ) αp h+p d C)C dq dt = ((N T qc QP)β Q)(b λp) The positive part function from () is not necessary in () as a result of the following Proposition. Proposition.: The following inequalities describe a positively invariant region of model(): 0 P b λ, 0 C N q, 0 Q N Tβ, and (N T qc QP) 0. Proof: The proof, which is standard for related models in the literature, follows by showing that all solutionswhich start on the boundary of the region will stay on the boundaryor go tothe interiorofthe region.see[]for details... Equilibria We calculate the system equilibria, with the variables listed in the order: (P, C, Q). The no-life equilibrium (O): (0,0,N T β). Neither producer nor consumer exist, but the(irrelevant) producer stoichiometric ratio is not necessarily zero. The monoculture equilibrium (P): ( b dp λ, 0, N T λβ λ+(b d P )β). In the absence of consumers, the population of producers follows the logistic growth and approaches itscarryingcapacity b dp λ over time. The coexistence equilibria(pc): The coexistence equilibria are computed for two cases: that of high food quality (case H), and that of low food quality (case L). We first compute case H equilibrium candidates by replacing min(γ, Q q ) with γ in eq. (). Similarly, we compute case L equilibrium candidatesby replacing min(γ, Q q ) with Q q. Later we will perform a consistency check : Q q γ for case H solutions, and Q q γ for case L solutions. Candidate equilibria which satisfy the consistency check will be ()

9 October 0, 00 : Report Page of called mathematically realized equilibria. Mathematically realized equilibria with nonnegative coordinates are said to be ecologically realizable. ForcaseH,thereisauniquecandidateequilibrium,denoted (PC) H.Intermsof the parameters, it is ( dch αγ d C, hγ(b dp ) αγ d C γdcλh (αγ d C), λγqd β(nt (αγ dc) hqγ(b dp )+ C h ) αγ d C αγ d C(hβ ) ). Note that the producer equilibrium level is independent of the growth rate b; increasing b results in increased producer growth, but the growth is balanced by increased consumption by the consumer. For case L, the candidate coexistence equilibrium set, denoted collectively by (PC) L,ismorecomplicated.Bysettingtherighthandsidesof()tozero,solving inthethirdequationfor Q,pluggingtheresultintothesecondequationandsolving for C, and substituting these two solutions into the first equation, it can be shown that the equilibria with nonzero P and C must satisfy the following cubic in P. where c = λ c = b d P + d C hλ c = (b d P + d C )h NT α q + dc β c 0 = hdc β P c + P c + Pc + c 0 = 0, () Thus there can be up to three candidate (PC) L equilibria. Along with the previous candidate (PC) H equilibrium, there might exist up to four coexistence equilibria. For the parameter sets we chose, however, we never observed more than three equilibria which simultaneously satisfied the consistency check and were therefore mathematically realized. Fewer still are ecologically realizable since all equilibrium coordinates must be nonnegative. The stability of equilibria can be determined by computing the associated eigenvalues and checking to see whether all have negative real parts. We do this in our numerical experiments, but we do not provide Jacobian or eigenvalue formulas here... Numerical Results Bifurcations with varying b(maximalper-capita producerproductivity) and N T (total system nutrient) fixed We provide representative bifurcation diagrams, varying b, in Fig.. The total nutrient N T is fixed at 0.. Figures (a)-(c) are traditionalone-parameter bifurcation diagrams, representing attractors with solid lines, and unstable equilibria with dashed lines. We plot only those equilibria and limit cycle extrema which are ecologically realizable. Barely visible at this scale is (unlabelled) region O: 0 b d P = 0.0, where the growth rate of the producer is insufficient to overcome its natural death rate. The corresponding equilibrium components are plotted in brown. As b increases through 0.0, a transcritical bifurcation leads to a (green) high nutrient producer

10 October 0, 00 : Report Page of monoculture, P H. (The nutrient level of the producer is in some sense irrelevant without a positive level of the consumer since in our model the stoichiometry is only effected through the conversion of biomass in predation. The producer equilibrium, however, can still be classified as high nutrient since Q q > γ.) A second transcritical bifurcation at β 0. allows the transition from the P H producer monoculture to the (blue) coexistence equilibrium, (PC) H. A Hopf bifurcation at β. produces a stable periodic P C-limit cycle; maximum and minimum values along the periodic cycle (purple) are displayed. The periodic limit cycle is destroyed as a case L saddle-node develops on the cycle at b.. Werefertothisbifurcationasa SNIC,asaddle-nodeonaninvariantcircle.The attractingcaselcoexistenceequilibrium (PC) L (red)borninthesaddle-nodebifurcation replaces the periodic cycle as the attractor. A nonsmooth saddle-node bifurcation separates the two subregions of (PC) L. As we cross the boundary from the left to the right at b., the case H saddle disappears with the case L saddle, leaving the case L attracting equilibrium intact. The nonsmoothness is caused by a crossing of the min function arguments in equation (). It is also apparent in the corner formed where the blue and red dashed equilibrium lines meet. This is in contrast to the quadratic signature of a smooth saddle-node bifurcation. If the red consumer equilibrium curve in Fig. (b) were continued farther out in theparameter b, itwouldbeseentoeventuallyintersectthe b axisat b., resulting in a restabilization of the producer monoculture. This corresponds to entering region P L in Fig. (a). The restabilization of the producer monoculture is a manifestation of the paradox of energy enrichment, and appears to be a common feature of stoichiometric models with closed nutrient pools [0,, ]. Recall that the system is open for carbon, whose inputs increase with b i, but closed for nutrient. Therefore, increasing b i at first increases producer growth, resulting in more food for the consumer. Once Q q < γ, food quality controls consumer growth, and continued increase in b i enriches the producer(s) with carbon, without any corresponding increase in system nutrients. Eventually, food quality is too low to allow consumer growth to match consumer death, resulting in consumer extinction and reesablishment of a producer monoculture through a transcritical bifurcation. Figure (d) is an additional bifurcation diagram which is provided to better understand which candidate equilibria are mathematically and/or ecologically realized and which are not. We plot the ratio Q q at equilibria versus b. This is, of course, merely a rescaled version of Fig. (c), but along with the dash-dot reference line at the value of γ = 0., it more directly relates to the min function in equation (). The linetypes and colors are consistent with Fig. (a)-(c): blue for case H, red for case L, and purple for the extremes along a periodic orbit. Solid lines still represent attractors and dashed lines represent unstable equilibria(saddles or sources). We have added to Fig. (d) dotted lines for candidate equilibria which are not mathematically realized. Specifically, the case H equilibrium is realized when it is above γ, and the case L equilibria are realized when they are below γ. Note that in the PC cycle region, the full limitcycle is in case H (above γ) toward the left, but are part in case H and part in case L (below γ) toward the right side of the region. The nonsmooth saddle-node is more clear in Fig. (d) as the crossing of the case H and case L equilibria, necessarily at γ. Figure (d) also suggests other

11 October 0, 00 : Report 0 Page 0 of possible bifurcation scenarios for alternate choices of our auxiliary parameters (all parametersother than b). For example,if γ were increasedabovethe Q q value of the unrealized candidate saddle-node at (b, Q/q) (., 0.)), the candidate saddle-node would become a realized saddle-node bifurcation. For γ below the Q q value of the realized saddle-node (about 0.00), neither case L candidate saddle-node would be realized. For other auxiliary parameter choices, the red cubic case L curve could lose its turning points, resulting in the loss of both candidate saddle-nodes. All of these cases have been observed in our numerical experiments. Table summarizes the attractors in each region and the bifurcations between regions. We note that the sequence of bifurcations in Fig., varying producer growth rate b, is the same as the sequence displayed for the stoichiometric models investigated in LKE, KHE, and WKL [0,, ], where the authors directly vary producer carrying capacity. Bifurcations with varying band N T To better place the one-parameter bifurcation diagrams in Fig. in context, we show a two-parameter bifurcation diagram in b and N T in Fig.. The solid curves in Fig. (a) indicate those bifurcations resulting in a change of attractor; the dashed curves indicate bifurcations which do not change the attractor. The dotted black line corresponds to the N T = 0. one parameter cut of Fig.. We use color to distinguish bifurcation curve types: blue for transcritical, green for saddle-node, and red for Hopf. Other curves corresponding to nonsmooth bifurcations are in black. Corresponding time series plots are shown in Fig.. In each time series, solutions of P, C, and magnified Q are plotted against time.tablesummarizesthebifurcationsbetweentheadjacentregionsof (b,n T ) parameter space. Most of the transitions between regions have been described above for the one-parameter cut. The geometry of the regions in the two-parameter plane adds additional insight to the behavior of the model. For example, if we were to take a one-parameter cut with N T less than about 0.0, then we would never see the consumer persisting. Note that the P L region extends to the b axis because we chose for simplicity to make the producer growth independent of its nutrient content(recall comments in section ). If we had included a nutrient requirement for producer growth, then the region of no growth would extend to a thin L-shaped region next to both axes, rather than just next to the N T axis. Note also the black dashed lines separating regions P H from P L and (PC) H from (PC) L. The respective regions are distinguished by the producer equilibrium level changing from high nutrient to low nutrient. The dashed green and black lines together are significant because they indicate parameter combinations for which equilibria are located at the nonsmoothness boundary in the phase space. For comparison, Fig. (b) shows a bifurcation diagram of the corresponding model without stoichiometry (replacing min(γ, Q q ) with γ). The nonstoichiometric model is, of course, independent of N T since the model reverts to a single currency model.

12 October 0, 00 : Report Page of The Two-Producer, One-Consumer Model We now turn to an analysis of the full model (), with both producers, one consumer, and stoichiometry. Much of the analysis parallels that performed in the previous section for a single producer. Proposition.: The following inequalities describe a positively invariant region of model (): 0 P b λ, 0 P b λ, 0 C NT q, 0 Q N T β, 0 Q N T β, and (N T qc Q P Q P ) 0. Proof: Theproof issimilarto thatofproposition..see[]for details... Equilibria Equilibria with P = 0 and/or P = 0 are the same as the equilibria in the single-producer-single-consumer model in the previous section, but with extra trivial coordinates.since Q i isnotecologicallyrelevantwhen P i = 0,wedenote the corresponding mathematical equilibrium value as being not applicable (NA). Coordinatesare listedinthe order: (P,P,C,Q,Q ). Theno-life equilibrium(o): (0, 0,0, NA,NA) Themonocultureequilibrium(P ): ( b d P N, 0, 0, T β λ, NA) λ λ +(b d P )β Themonocultureequilibrium(P ): (0, b d P N, 0, NA, T β λ ) λ λ +(b d P )β Theone-producercoexistenceequilibrium((P C) H : ( dch, 0, γh(b d P ) γdch λ γα d c αγ d c (αγ d c), β (N T (αγ d c) hqγ(b d P ) dcqγh λ ) αγ dc αγ d c(hβ ), NA). Theone-producercoexistenceequilibrium((P C) H : (0, NA, β (N T (αγ d c) hqγ(b d P ) dcqγh λ ) αγ dc αγ d c(hβ ) ). d ch, γh(b d P ) γdch λ γα d c αγ d c (αγ d c), The remaining coexistence equilibria involve both producers and are therefore distinct from the equilibria in the single producer analysis. Note that for P and P to be positive equilibrium values, the growth terms (b λ P λ P ) and (b λ P λ P )mustbothbepositive,soweignorethepositivepartnotation in the four relevant terms in equation(). Theproducer equilibriumwithoutconsumer(p P ): We set C = 0 and assume that all other variables are positive. The equilibriumsolutioniscomputedtobe (P,P,C,Q,Q ) = ( λ B λ B, λ B λ B, 0, S prod S prod N T β S prod ), B (β λ β λ )+B (β λ β λ )+S N T β S prod B (β λ β λ )+B (β λ β λ )+S, where B = b d P, B = b d P, and S prod = λ λ λ λ In the absence of the consumer, our model reduces to a Lotka-Volterra competition model. Our simplifying assumption that the producer growth be independent of the stoichiometry renders Q and Q irrelevant. It is clear from the first two equations in () that in order to have P and P persist at positive levels, we must have b > d P and b > d P. By analyzing nullclines

13 October 0, 00 : Report Page of in the P -P plane, it can be shown that to avoid competitive exclusion, we must also require b dp λ > b dp λ and b dp λ > b dp λ. For fixed values of λ ij with λ and λ sufficiently larger than λ and λ, these four inequalities determine a wedge-shaped region in the b -b plane. Look ahead to wedges formed by the blue lines (mostly dashed) in Figures (a) and (c). In words, the growth rates for the two species must be sufficiently close to allow for coexistenceof P and P. AgainlookingaheadtoFig.(a)and(c),we seethat the coexistenceregions: (P P C) H, P P C-cycle,and (P P C) L, are a subsetof thecoexistencewedgefor just P and P without C. Interior coexistence equilibria(all three species positive): As we did for the one-producer model, we first calculate all possible candidate equilibria. There remain two cases: case H, where min(γ, QP+QP ) = γ, and q(p +P ) casel, where min(γ, QP+QP q(p ) = QP+QP +P ) CaseH:high food qualitycoexistenceequilibrium (P P C) H : q(p +P ). Assuming all variables are positive, the equilibrium solution is (P, P, C, β (N T qc ) +β P +β P, β (N T qc ) +β P +β P where B = b d P, B = b d P, S sum = λ + λ λ λ, S prod = λ λ λ λ, P = B B S sum dch(λ λ) S, sum(αγ d c) P = B B S sum dch(λ λ) S, sum(αγ d c) and C = γh(b(λ λ) B(λ λ)) S sum(αγ d c) + γdch S prod S sum(αγ d c) ), Note that the easiest way to guarantee all components of the all-speciescoexistence equilibrium are positive is to require both S sum and S prod to be positive. Both terms reflect the dominance of self-limitations over competition-limitation. CaseL:low foodqualitycoexistenceequilibria (P P C) L : N +N In this case we assume the min(γ, q(p ) = N+N +P ) q(p = QP+QP +P ) q(p. The +P ) procedureforfinding (P C) L wasdescribedinthelastsection.similartothe one-producer, one-consumer model, there are up to three equilibria in the two-producer,one-consumermodel.the P equilibriumvaluesfor (P P C) L can be shown to be the roots of a cubic polynomial []. The formula is too complicated to include here, but once a value for P is determined, the remainingfour componentscanthen bedeterminedintermsof P. As in the one-producer reduced model, stability of equilibria can be determined by computing the associated eigenvalues and checking to see whether all have negativereal parts. We do this in our numericalexperiments,but we do not provide Jacobian or eigenvalue formulas here... Numerical results Figure contains several bifurcation diagrams summarizing the dynamics and corresponding transitions observed for our set of auxiliary parameters (all

14 October 0, 00 : Report Page of parameters except b, and b ). Representative time series are plotted in Fig.. Because of the symmetry by interchanging P with P, we provide time series for only one of any pair of conjugate regions. Figure illustrates a specific one-parameter cut through Fig. (a), (b) or (c) at b =.0. Figure provides the resultsofadditionalexperimentswithvaryingproducer P snutrientuptake rate β. We first briefly describe the bifurcation diagrams in Fig. and, and then describe each diagram in more detail. Table summarizes the bifurcations between the adjacent regions. Figures (a)-(c) are different views of the same bifurcation diagram in the (b,b ) parameter plane. N T is still fixed at 0.. Figure (a) gives an overview for a large range of parameters, and includes the paradox of enrichment regions wheretheconsumercannotexistforhigh b and/or b : (P ) L, (P ) L,and (P P ) L. Figure (b) is an enlargement of Fig. (a), showing more detail, but including only bifurcations which change the attractor. Figure (c) is the same as Fig. (b), but with several additional bifurcation curves which do not involve the attractors. Figure (d) is a bifurcation diagram for the analogous two-producer, one-consumer model without stoichiometry (replace min(γ, QP+QP q(p ) with +P ) γ). Comparing this diagram with Fig. (b) helps to emphasize the impact of stoichiometry in our model. Figure (a) is a two-parameter bifurcation diagram for the one-producer model showing β vs. b.itisanalogoustofig.(a),with β replacing N T.Figures(b)and (c) are bifurcation diagrams in the same (b,b ) space as (a)-(c), but with P s uptake coefficient β perturbed from its original value of 0.. In Fig. (b), β is 0., and infig.(c), β is 0.. We now provide a more detailed description of the bifurcation diagrams. We begin with the two-parameter diagrams in Fig.. Bifurcation types are color coded: blue for transcritical of equilibria, green for saddle-node, red for Hopf, and cyan for transcritical of limit cycles. Solid lines indicate the most significant bifurcations those that result in a change in the attractor. All other bifurcations are indicated by dashed lines. Regions are labelled by the associated attractor. (We observe no regions of bistability for our choice of auxiliary parameters, so this labelling scheme is unique; from numerical work with similar models, however, we expect that certain parameter sets could allow bistability, with, forexample,acoexistenceequilibriumandap P equilibriumbothbeingstable.) Figure (a) is an overview of the full parameter space. We observe no bifurcations outside this region of the (b,b ) parameter plane. Figure (b) is minimal in the sense that we provide only the bifurcations that are ecologically meaningful those that involve a change of attractor in the first quadrant. Figure (c) shows the same region as Fig. (b), but with additional nonattractor bifurcations also displayed. Region labels correspond to the region s attractor. Subscripts H and L correspond to equilibria that are high or low nutrient, respectively. The subscript depends on whether P, P, or the combination of both have a stoichiometry relative to the consumer stoichiometry q that is above or below the predation conversion factor of γ. This ratio ( PQ+PQ P +P /q) makes sense even without the existence of any consumer. There are three regions that allow for coexistence of all three species: (P P C) H, P P C-cycle, and (P P C) L. All other regions have behavior that can be described by a reduced model.

15 October 0, 00 : Report Page of We denoteby O the region (0 b d P ) and (0 b d P ). The region isnot labelled in the Fig. (a)-(d); it is too small to see clearly. We do not provide time seriesforregionoeither;theconsumerandbothproducersgoextinctforthisset of parameters. For low values of b less than 0.0 and using any of Fig. (a), (b), or (c) for increasing b, we see a sequence identical to that described for the one-producer model in section. This is because the growth rate for producer P is not sufficient to allow its survival. So the model reduces to a one-producer model. A transritical bifurcation at b = 0.0 allows P to survive; a second transcritical bifurcation at b 0. allows the consumer to survive; a Hopf bifurcation (b.) destabilizes the (P C) H equilibrium and gives rise to a periodic limit cycle; the limit cycle is destroyed in a SNIC bifurcation and the attractor becomes the low nutrient equilibrium (P C) L which has been born in the saddle-node bifurcation at b.; finally, the consumer goes extinct in a transcritical bifurcation at b.. A vertical path for 0 b 0.0 results in a symmetric sequence of bifurcations but with P = 0. A path along the diagonal results in a similar sequence but with both producers existing. The most interesting new feature is the transition from coexistence close to the diagonal to the extinctionof P as we move toward the b axis, and the analogous extinction of P as we move toward the b axis. All these transitions occur as some form of a transcritical bifurcation either the crossing of an equilibrium (across curve segment gf or cde in Fig. (c)) or a limit cycle (across fc) from an interior attractorto a P = 0 attractor. Figure (c) includes the blue transcritical wedge boundaries part dashed and part solid. Coexistence of both producers without the consumer is not possible outside this wedge. We discuss the reduction of the region of interior coexistence by the introduction of the consumer later, in section.. Note that the wedge boundaries in Fig. (a) and (c) are solid when they bound regions without the consumer: (P P ) H and (P P ) L. Figure (c) includes several more dashed bifurcation lines: vertical ones which correspond to bifurcations in the P = 0 side hyperplane and horizontal ones which correspond to bifurcations in the P = 0 sidehyperplane. The extensionof the horizontalline at b = 0.0 to the rightof b = 0.0isnotdisplayed.Thisdoesnotinvolveanyattractor,andinfact is only an ecologically significant bifurcation in the absence of both P and C. The symmetricpart of b = 0.0 is also not displayed.all horizontaland vertical bifurcation lines actually extend indefinitely, but we have trimmed them both to keep Fig. (c) from further clutter, and to emphasize the connection between the side hyperplane bifurcations and the corresponding interior bifurcations. For example, the P = 0 side hyperplane nonsmooth saddle-node (line kda) is displayed only up to the point a, where it connects with the interior nonsmooth saddle-node(line aa ). Other dashed curves indicate equilibrium bifurcations not involving an attractor: a transcritical bifurcation of the unstable (P P C) H equilibrium from the interior of the phase space (P,P and C all positive) to P = 0 as we pass from region {ff b b} to region {fbc}. A similar transition occurs from region {b ba a} to region {abc}. Curve ca denotes a transcritical bifurcation of a different unstable equilibrim from the interior to P = 0. Recall that there are up to three interior equilibria: the original equilibrium which underwent a Hopf bifurcation across ff, and two additional equilibria born in the SNIC saddle-node bifurcation

16 October 0, 00 : Report Page of across cbb c. A count of interior equilibria yields one in region {ff b b}, none in region {bcf}, three in region {bb a a}, two in region {abc}, and one in the rest of region (P P C) L. Region {bcf} is especially noteworthy because there exists an interior limit cycle, but no interior equilibrium. That is, within this region of parameter space, interior coexistence is possible only in a nonequilibrium(cyclic) state. With the exception of the vertical and horizontal curves which are not fully extended, Fig. (c) appears to include all ecologically relevant bifurcations for our choice of parameter values. Figure (d) corresponds to the nonstoichiometric model, with min(γ, QP+QP q(p +P ) ) replaced by γ. Its bifurcations are a subset of the stoichiometric bifurcations, so we provide no further explanation here, except to point out that we see the typical Rosenzweig-MacArthur bifurcation sequence in which continued carbon enrichment manifested in any positive slope path away from the origin in the (b,b ) plane results in a limit cycle which seems to persist for arbitrarily large enrichment. The cycle is an interior one for (b,b ) values sufficiently close to the diagonal, and in one of the hyperplanes otherwise. Figure is the two-producer analogue to the one-parameter bifurcation diagram in Fig. for the single producer case. Phase variables P,P, and C are plottedagainst b in Fig.(a)-(c). The ratioofthe consumer sfood stoichiometry to its structural ratio is plotted in (d). We have fixed N T = 0., and b =. Line colors and types match those of Fig. : blue for case H, red for case L, solid for attractors, dashed for unstable, dotted for candidate equilibria which are unrealized. If b were less than 0.0, this one-parameter cut would match Fig. even more closely.we have chosenalarger valueof b for contrast.the cut can be placed in context by looking at the dotted black lines illustrating the b = cut inthe (b,b )planein thetwo-parameterdiagramsof Fig.(a), (b)or (c). We will walk through the cut from low to high b and identify attractors first. Then we will describe the extra curves corresonding to nonattracting equilibria and candidate equilibria (in Fig. (d)). For low b values, P and C coexist on an attracting periodic limit cycle in region P C cycle. The solid magenta lines denote the extrema of this cycle. This is consistent with single producer analysis: the point (b,n T ) = (.0,0.) is in the PC-cycle region of Fig. (a). Note that the phase values of the extrema of the limit cycle do not change as b is increased because P is notyet enrichedenoughto survive.there isatranscriticalcrossing of periodic limit cycles at b.. Producer N now survives along with P and C on an interior limit cycle (solid purple extrema). The interior limit cycle is destroyed in a SNIC bifurcation at b.0. The saddle-nodeis best illustrated infig.(b)wherethesolidredcurveshowstheattractingequilibrium (P P C) L. At b., producer P outgrows producer P, and P goes extinct in a transcritical bifurcation. Not shown in any of the Fig. diagrams, but apparent in the black dotted b = cut displayed in Fig. (a), the consumer is lost in the paradoxofenrichmenttranscriticalbifurationat b.. Dashed lines correspond to unstable equilibria, either saddles or sources. Two of the dashed line segments correspond to interior equilibria: dashed blue in region Cycle a, and dashed red in the first subregion of (P P C) L. The dashed

17 October 0, 00 : Report Page of blue equilibrium is born in a P = 0 transcritical bifurcation (see Fig. (a) at b.), anddiesinap = 0transcriticalbifurcation(seeFig.(b)at b.). The red dashed equilibrium is born in the saddle-node bifurcation already mentioned above at b.0, and dies in a P = 0 transcritical bifurcation at b.. Bothbifurcationsare bestviewedin Fig.(b). Two additionalside hyperplaneequilibriaare plotted:green dashedfor (P ) H, and pink for (P C) H. The (P ) H monoculture is included mainly for Fig. (a), to showthatallcoexistence P valuesarebelowthe P monoculturevalue.thepink dashed (P C) H side hyperplane equilibrium is included only to better see the transcriticalbifurcationbetween regions Cycle b and Cycle a. See Fig. (b) or (c). We chosenotto displaytheanalogousconnecting P = 0orbitatthetranscritical bifurcation between regions Cycle a and Cycle b because its continuation would further complicate the diagrams. As we did in Fig. (d), we show extra dotted curves denoting candidate equilibria in Fig. (d). The only difference in interpretation of the dotted lines in the two figures is that in the former, the dotted candidates are mathematically unrealized because they don t satisfy the consistency check, while in the latter, portions of the dotted lines are ecologically unrealizable because at least one componentisnegative.the part thedottedred curvethat isabovethe γ valueof 0. andthepartofthedottedbluecurvethatisbelow γ arecandidateswhichare unrealized for both reasons. Figure (a) shows the two-parameter bifurcation diagram in β and b for the single producer model of section. By comparing to the two-parameter bifurcation diagram of Fig. (a), it can be seen that increasing the producer nutrient uptake rate is roughly analogous to increasing the total nutrient in the system. Figures (b) and (c) are analogous to Fig. (b), but for unequal values of the producer uptake coefficients. The symmetry of Fig. (b) is broken in Fig. (b)and(c).figure(b)istopologicallyequivalenttofig.(b),butfig.(c)isno longer equivalent. Figure (a) helps exlain parts of Fig. (b) and (c) as follows. The horizontal one-parameter cut at β = 0. (respectively β = 0.) in Fig. (a) corresponds to a horizontal one-parameter cut in Fig. (b) (respectively (c)) at b = 0. Similarly, a one-parameter cut at β = 0. in Fig. (a) corresponds to the vertical one-parameter cut in Fig. (b) and (c) at b = 0. We address further the ecological implications of the mismatch in producer nutrient uptake rates in section... Discussion.. Ecological Implications All the bifurcations described in the preceding sections and associated figures are theoretically relevant to ecological systems, but we emphasize in this section three new behaviors of this model that have interesting ecological implications and which are in principle experimentally testable. () In a one-producer-consumer system, increasing producer growth rate (b) produces a different sequence of bifurcations and behaviors when the total amount of nutrient (N T ) is low compared to when it is moderate or high (Fig.(a)).

18 October 0, 00 : Report Page of () The presence of the consumer decreases the region of coexistence of the two producersinthe (b,b )parameterplane(fig.(c)). () Differentialresponseofproducergrowthrates(b i )toenrichmentmayproduce different sequences of bifurcations and species extinctions(fig. (c)). We have assumed a reasonable set of parameter values for aquatic systems (with rescaling, as indicated at the end of section, Model Construction), and different aquatic systems or simple terrestrial systems might be modeled by other parameter selections. However, because the equilibria, their stability, and bifurcation behaviors of the model depend critically on parameter values, it is important in any experimental test to first determine accurate values of the parameters. Standard experimental techniques could be used to establish, for example, producer growth rates b i, producer nutrient uptake rates β i, and the reductions in growth rates due to self limitation and interspecies competition (λ ij ). Once parameter values have been established for the particular species chosen, then bifurcation diagrams would need to be recomputed to determine specific values of parameters where critical transitions are predicted to happen. To conform to the assumptions of the model, the experimental system must be closed to nutrient fluxes from the outside environment but open to carbon fluxes via producer uptake in photosynthesis and outputs via respiration. A commonly used system which meets these assumptions is a simple aquarium with an initial nutrient pool that remains fixed (but which may be dynamically allocated amongst the different species) and which is open to carbon fluxes and contains producer(s) such as algae and a consumer such as Daphnia or similar zooplankton. In contrast, a chemostat is open to both carbon and nutrients and would not be an appropriate experimental environment. The general strategy would be to experimentally vary parameter values to see if the experimental system can be forced across bifurcations, thereby causing a large and observable qualitative change of behavior, such as extinction of one or more species, initiation or quenching of limit cycles, etc. Our simulations suggest that algae-daphnia experiments would need to be run for at least one year to determine periodic behavior and convergence. The simplest parameter to experimentally vary is the total amount of nutrient, N T, which can be done by simplyfertilizingthe system.speciesgrowth rates, b i, can be varied in two ways: first by judicious choice of different species, secondly by determining whether b and b respond differently to light enrichment because of different photosynthetic response curves for the two producers. (See [] for an example of altering growth and stoichiometry of algae by augmenting light levels.) If the response of b and b can be experimentally established and calibrated to light enrichment, then shading or light augmentation would correspond to a specific path through the (b,b ) plane. Finally, nutrient uptake rates, β i,canbe variedbychoiceofproducer speciesusedin the experiment. Assuming that these preliminary experimental details can be addressed, the following experiments could test the new behaviors of this model: () Different bifurcation sequences with increased producer growth rates under low compared with high nutrient levels. In Fig. (a), increasing producer growth b results in a different bifurcation sequence at different nutrient levels, N T. At low nutrient levels (such as N T 0.0 in Fig. (a)), as b increases,