research article a stoichiometric model of two producers and one...

August 28, 2009 19:49 Journal of Biological Dynamics Report

Journal of Biological DynamicsVol. 00, No. 00, Month 200x, 1–31

RESEARCH ARTICLE

A STOICHIOMETRICMODEL OF TWO PRODUCERS ANDONE CONSUMER

LAURENCE HAO-RAN LINa∗⋄, BRUCE PECKHAMa∗∗, HARLAN STECHa†,

and JOHN PASTORb‡

aDepartment of Mathematics and Statistics, University of Minnesota, Duluth MN 55812;

Dept. Fax: 218-726-8399;bDepartment of Biology, University of Minnesota, Duluth MN 55812; Dept. Fax: 218-726-8142;

∗Email: [email protected], Contact Phone: 218-349-3782∗∗Email: [email protected], Office Phone: 218-726-6188

†Email: [email protected], Office Phone: 218-726-8272‡Email: [email protected], Office Phone: 218-726-7001

(v1.0 released March 2009)

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 one-producer, one-consumer population model without stoichiometry. The generalization involvestwo independent steps: 1) introducing stoichiometry into the system, and 2) adding a secondproducer which competes with the first. Both generalizations introduce additional equilibria andbifurcations 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 enrich-ment. The model is closed for nutrient but open for carbon. The primary parameters we varyare the growth rates of both producers. Secondary variable parameters are the total nutrient inthe system, and the producer nutrient uptake rates. Depending on the parameters, the possibleequilibria are: no-life, one-producer, coexistence of both producers, the consumer coexisting witheither producer, and the consumer coexisting with both producers. Limit cycles exist in the lat-ter three coexistence combinations. Bifurcation diagrams along with corresponding representativetime series summarize the behaviors observed for this model.

Keywords: Stoichiometry, Producer-Consumer Model, Second Producer, Coexistence

1. Introduction

A population is the collection of inter-breeding organisms of a particular species.Mathematical population models follow the size of interacting populationsover time. Such models are fundamental in biology and ecology. A producer-consumer (prey-predator) population model studies the populations of twospecies which interact in a specific way. A variety of producer-consumer modelshas been developed and studied over the last century, usually with a singlecurrency, such as biomass, for each population. More recently, stoichiometry,which can be thought as tracking food quality as well as quantity, has been

⋄Corresponding author. Email: [email protected], [email protected]

ISSN: 1748-670X print/ISSN 1748-6718 onlinec© 200x Taylor & FrancisDOI: 10.1080/1748670YYxxxxxxxxhttp://www.informaworld.com


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introduced into population models [1, 2, 4, 5, 8–13, 19–21, 24, 25].

Every organism has its own ratios of nutrients in its body, and those ratios arecalled “stoichiometric ratios.” Stoichiometric ratios tend to be relatively constantfor most animals, but have been observed to have a wider range of valuesfor plants [19]. In this paper a single stoichiometric ratio, N/C, representedby an unnamed nutrient (N) and carbon (C), is tracked. Food that has a lowstoichiometric ratio is considered low quality, and contributes less to consumergrowth than high ratio food. Stoichiometric models, especially those with aclosed nutrient cycle, tend to exhibit a “paradox of enrichment”, a phenomenonin which the population of the consumer declines in response to an increase inthe population of the producer [4, 8, 10, 21, 24].

In 1963, Rosenzweig and MacArthur [18] presented a geometric analysisof producer growth and consumer response. It has become one of the classicproducer-consumer populationmodels with a single currency (i.e., no stoichiom-etry). Formulas (1) are the traditional realization of the Rosenzweig-MacArthurmodel, using logistic producer growth andMichaelis-Menten consumer responsewith food saturation.

dPdt

= rP − λP 2 − f(P )C

dCdt

= γf(P )C − dC

f(P ) = αPh+P

(1)

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 is the per-capita death rate of the consumer.f(P ) is a Holling Type II or Michaelis-Menten function that describes consumerfood 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 (1) is the producer growth rate r. For low rthe model predicts a producer monoculture system because there is not enoughfood to maintain the consumer. Increasing r sufficiently to pass a transcriticalbifurcation value allows both producer and consumer to stably coexist. Furtherincrease in r results in a destabilization of the coexistence equilibrium in a Hopfbifurcation, accompanied by the birth of an attracting limit cycle. The amplitudeof the cycle grows with continuing increase in r, but no further bifurcations areobserved [23].

We build our model starting from a basic Lotka-Volterra model for two com-peting producers. We introduce a consumer which interacts with the producersin a similar way to the Rosenzweig-MacArthur model. We keep track of bothcarbon (as a surrogate for biomass) and a nutrient (which can be thought of as


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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 thetwo producers and the consumer. The effect of stoichiometry is introduced bylowering the rate of biomass conversion from the producers to the consumerwhen the nutrient to carbon ratio of the food (producers) is below the levelneeded for consumer biomass creation. Details are in Section 2.

Although there are many parameters in our model, we focus on varying onlytwo “primary” parameters: the growth rates of the two producers. We also varythe total nutrient available in the system, and the nutrient uptake rates for theproducer(s). We perform a bifurcation study using a combination of analysis andnumerics. Our most significant contribution is identifying parameter combina-tions which allow the coexistence of all three species and the bifurcations whichoccur as one passes from inside to outside the parameter region of coexistence.Coexistence is lost as one species goes extinct in some version of a transcriticalbifurcation.

We have already listed several related stoichiometric producer-consumermodels. See [7] for a more detailed model of competing producers, but with noconsumer. See [6, 15, 22] for models with two producers and one consumer, butwith no stoichiometry. See [11] for a one-producer, two-consumer model withstoichiometry. The book of Andersen [1] includes a two-producer, one-consumermodel with stoichiometry, but it differs from our model in ways which we willdetail later.

The paper is organized as follows. In section 2 we provide the details of themodel construction. In section 3 we analyze the “reduced” model, having onlyoneproducer. This lays the groundwork for the techniques used in section 4 toanalyze the full model. The analysis of the reduced model provides a basis ofcomparison for the two-producer model. Discussion is in section 5, and conclu-sions in section 6.

2. 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 ameasure 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:P1, P2, C , and four nutrient variables: N1, N2, NC , M , where M denotes thenutrient concentration in the sediment. Variables of the model are summarized intable 1. 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 fiveby assuming a fixed amount of total nutrient, and a fixed stoichiometric ratioq = NC/C for the consumer. Figure 1 illustrates the flow of carbon and cycle ofnutrient in the model.

The equations for our model are given below in equation (2).


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:

dP1

dt= (b1 − λ11P1 − λ12P2)+P1 − dp1

P1 − P1

P1+P2

α(P1+P2)h+(P1+P2)

C

dP2

dt= (b2 − λ21P1 − λ22P2)+P2 − dp2

P2 − P2

P1+P2

α(P1+P2)h+(P1+P2)

C

dCdt

= min(γ, 1q

N1+N2

P1+P2

)α(P1+P2)

h+(P1+P2)C − dcC

dN1

dt= (NT − qC − N1 − N2)β1(b1 − λ11P1 − λ12P2)+P1 − P1

P1+P2

α(P1+P2)h+(P1+P2)

CN1

P1

− dp1N1

dN2

dt= (NT − qC − N1 − N2)β2(b2 − λ21P1 − λ22P2)+P2 − P2

P1+P2

α(P1+P2)h+(P1+P2)

CN2

P2

− dp2N2

(2)

We now describe the assumptions which lead to the equations (2).

Carbon flow: P1, P2, C .

The biomass (carbon) of producers is assumed to follow logistic growth. Notethat we have chosen to make the producer growth independent of the nutrient.This independence assumption simplifies the analysis and allows us to focuson the role of stoichiometry in the conversion of biomass from producer toconsumer. Lotka-Volterra competition is assumed between the two producers.The product of the competition coefficients λ12λ21 is assumed to be always lessthan the product of the self-limitation coefficients λ11λ22. Otherwise, the twoproducers could not coexist even in the absence of the consumer [3, 14]. Becausenutrient uptake depends on producer growth, a reduction in birth rate is notthe same as an increase in the death rate. Consequently, the producer deathterms (dPi

) are separated from the (logistic) growth terms (bi − λii − λij) in ourmodel. This is the reason we choose a different letter (bi) for maximum birthrate compared to r in Rosenzweig and MacArthur’s equation (1) for maximumgrowth rate. Rosenzweig and MacArghur’s growth rate is the birth rate minusthe death rate. This separation of birth terms from death terms is also why weuse the “positive part” function, ()+, in the first two equations of (2). Producerrespiration, not explicitly modeled, is assumed to be a fixed fraction of the carbonfixation through the photosynthesis. That is, “growth” is interpreted as “netgrowth”. We assume that litter decomposition and nutrient mineralization areinstantaneous. Since the amount of carbon in the sediment does not affect any ofthe other variables, there is no need to include a carbon sediment compartmentin the model. Mortality results in an exit of carbon, but not nutrient, from thesystem. See figure 1.

The predation of both producers (consumer response) is assumed to followa Michaelis-Menten function. Both producers are food for the consumer, whichforages nonpreferentially: the probabilities of producer P1 and P2 being eaten areassumed to be P1

P1+P2

and P2

P1+P2

, respectively. That is, the producers are consumedin proportion to their relative abundance.

The consumer gains biomass through predation. Consumer death is assumed tobe proportional to its population size. The biomass conversion efficiency is mod-elled as the minimum of γ and N1+N2

q(P1+P2), the ratio of the aggregate food source’s

stoichiometry, N1+N2

P1+P2

, to the consumer’s stoichiometry, q. When the food sourcecontains sufficient nutrient to support the maximum conversion efficiency γ, thenthe rate γ is achieved. Otherwise, stoichiometric limitation reduces the conver-sion efficiency to N1+N2

q(P1+P2), reflecting the maximum rate at which nutrient can be


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supplied to build structural consumer biomass. Implicit in the use of this termis the assumption that the consumer assimilates a mixture of the two producerstogether, rather that eating and assimilating one producer at a time. This assump-tion is similar to the foraging and biomass conversion assumptions in Andersen[1].Note that if the efficiency minimum function is replaced with the constant γ,

then the carbon equations decouple from the nutrient equations, and the systemreverts to a non-stoichiometric one.

Nutrient cycling: N1, N2.

The nutrient is assumed to cycle through the four compartments, N1, N2,NC , and M , as in figure 1. We eliminate NC by assuming that the consumernutrient to carbon ratio is kept constant q. So NC = qC . Additionally, weeliminate M by assuming a closed nutrient system with total nutrient NT , soM = NT −N1−N2−qC . This leaves us with the two nutrient variablesN1 and N2.

The producer gains nutrient via uptake from the mineralized nutrient, andloses nutrient due to predation and mortality. The nutrient uptake rate in thismodel is proportional to the mineralized nutrient M = NT − N1 − N2 − qC . Incontrast to most other stoichiometric models (but see [7]), nutrient uptake is alsohere assumed to be proportional to producer growth. Of course, proportionaluptake would be required of producers that maintain a constant N : C ratio.The nutrient uptake coefficients β1 and β2 are assumed to be constant. Nutrientis transferred from producer to consumer via predation, and returned to thenutrient pool when either a producer or the consumer dies.

To simplify somewhat the equilibrium analysis of formulas (2), we introduceproducer ratios Qi = Ni

Pi, i = 1, 2. Then calculating the derivatives of Q1 and Q2,

we get the equivalent “stoichiometric form” of the system:

dP1

dt= ((b1 − λ11P1 − λ12P2)

+ − dp1− αC

h+(P1+P2))P1

dP2

dt= ((b2 − λ21P1 − λ22P2)

+ − dp2− αC

h+(P1+P2))P2

dCdt

= (min(γ, 1q

P1Q1+P2Q2

P1+P2

) α(P1+P2)h+(P1+P2)

− dc)C

dQ1

dt= ((NT − qC − Q1P1 − Q2P2)β1 − Q1)(b1 − λ11P1 − λ22P2)

+

dQ2

dt= ((NT − qC − Q1P1 − Q2P2)β2 − Q2)(b2 − λ21P1 − λ22P2)

+

(3)

Note that dCdt

is not defined if both P1 and P2 are zero. It can, however, be

continuously extended to P1 = P2 = 0 by dCdt

= −dCC , because of the α(P1+P2)h+P1+P2

factor, assuming h > 0.

For our numerical investigations, we fix the following parameters, represent-ing two nearly identical producer species (differing only in maximal per-capitagrowth rates b1 and b2) and a nutrient-rich (relatively high q) consumer.


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Self-limiting coefficients: λ11 = λ22 = 0.5Interference coefficients: λ12 = λ21 = 0.2Producer per-capita natural death rates: dP1

= dP2= 0.05

Consumer per-capita death rate: dc = 0.17Consumer stoichiometric ratio: q = 0.05Maximal biomass conversation efficiency: γ = 0.1Maximal and half-saturation coefficients in the predation function, f: α = 2.75and h = 0.75,Nutrient uptake constants β1 = β2 = 0.3.

Our primary goal is to determine the effects of system enrichment withrespect to carbon and nutrients. Increasing bi is assumed to correspond to carbonenrichment via producer growth; increasing NT is assumed to enrich the systemwith nutrients. Consequently, the parameters, b1, b2, and NT are varied in ournumerical experiments. We also experiment with different values of β1, thefirst producer’s nutrient uptake rate. Rate constants are dependent on the unitsselected to measure biomass and nutrient densities. Rescalings and alternateparameter selection might be necessary to match specific systems. For example,if we rescale both time and the phase variables by a factor of three (t = 3t, P1 =

3P1, P2 = 3P2, C = 3C,N1 = 3N1, N2 = 3N2) then our variables and parametersare consistent with an algae-daphnia system with rescaled time measured indays, and rescaled phase variables measured in mg/liter [1]. Specifically, therescalings would result in replacing parameters b1, b2, dP1

, dP2, α, h, dC , NT , β1, β2

with b1/3, b2/3, dP1/3, dP2

/3, α/3, h/3, dC /3, NT /3, 3β1, 3β2, respectively. The λij’sand γ would remain unchanged. Theoretically, our model could apply to eitheraquatic systems or simple terrestrial systems (without significant root structures).See section 5.3 where we review the limitations of the model.

The papers which match our model assumptions most closely – but for oneproducer instead of two – are Loladze, Kuang, and Elser (LKE) [10], Kuang,Huisman and Elser (KHE) [8], and Wang, Kuang, and Loladze (WKL) [24]. TheLKE model does not include a nutrient pool. The KHE and WKL models doinclude a nutrient pool. All three models are closed for nutrient, and have pre-dation and stoichiometric limitation for conversion from producer to consumerwhich is similar to ours. All three models have nutrient uptake assumptionswhich differ from ours. Their models all include stoichiometric limitation forproducer 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 [12] investigate a model which, like ours, hastwo prodcers, one consumer and stoichiometry. Their model differs from oursin several facets, most significantly because they assume the two producers arein different patches, and that available nutrient for uptake (not total nutrient) re-mains constant. Andersen [1] includes amodel which also has two producers, oneconsumer and stoichiometry, but it also differs in many ways from our model.Most significantly, the Anderson study considers open nutrient systems, whileours is closed. The Anderson model does not assume any competition or self lim-itation between producers, and it makes different assumptions regarding nutri-ent uptake by the producers. Further, our interests are in examining enrichment-induced changes in system dynamics – a topic not considered in [1].


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3. One-Producer, One-Consumer Model

In order to more easily understand the model (3), we first consider the specialcase where there is a single producer in the system. We set P2 = 0 and ignoreQ2. For notational convenience we drop the “11” subscript from λ11, and the “1”subscripts from P1, Q1, b1, β1 and dP1

for the rest of this section to obtain:

dPdt

= (b − λP − dP − αCh+P

)P

dCdt

= (min(γ, Qq) αP

h+P− dC)C

dQdt

= ((NT − qC − QP )β − Q)(b − λP )

(4)

The positive part function from (3) is not necessary in (4) as a result of thefollowing Proposition.

Proposition 3.1: The following inequalities describe an invariant region of model (4):0 ≤ P ≤ b

λ, 0 ≤ C ≤ N

q, 0 ≤ Q ≤ NT β, and (NT − qC − QP ) ≥ 0.

Proof : The proof, which is standard for related models in the literature, followsby showing that all solutions which start on the boundary of the region will stayon the boundary or go to the interior of the region. See [9] for details.�

3.1. Equilibria

We calculate the system equilibria, with the variables listed in the order: (P,C,Q).

The no-life equilibrium (O): (0, 0, NT β). Neither producer nor consumer exist,but the (irrelevant) producer stoichiometric ratio is not necessarily zero.

The monoculture equilibrium (P): ( b−dP

λ, 0, NT λβ

λ+(b−dP )β ). In the absence of con-

sumers, the population of producers follows the logistic growth and approaches

its carrying capacity b−dP

λover time.

The coexistence equilibria (PC):

The coexistence equilibria are computed for two cases: that of high foodquality (case H), and that of low food quality (case L). We first compute case

H equilibrium candidates by replacing min(γ, Qq) with γ in eq. (4). Similarly, we

compute case L equilibrium candidates by replacing min(γ, Qq) with Q

q. Later we

will perform a “consistency check”: Qq

≥ γ for case H solutions, and Qq

≤ γ for

case L solutions. Candidate equilibria which satisfy the consistency check will becalled mathematically realized equilibria. Mathematically realized equilibria withnonnegative coordinates are said to be ecologically realizable.

For case H, there is a unique candidate equilibrium, denoted (PC)H . In terms of

the parameters, it is ( dChαγ−dC

, hγ(b−dP )αγ−dC

− γdCλh2

(αγ−dC )2 ,β(NT (αγ−dC)−hqγ(b−dP )+

λγqdC h2

αγ−dC)

αγ−dC(hβ−1) ).


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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 byincreased consumption by the consumer.

For case L, the candidate coexistence equilibrium set, denoted collectively by(PC)L, is more complicated. By setting the right hand sides of (4) to zero, solvingin the third equation for Q, plugging the result into the second equation and solv-ing for C , and substituting these two solutions into the first equation, it can beshown that the equilibria with nonzero P and C must satisfy the following cubicin P .

P 3c3 + P 2c2 + Pc1 + c0 = 0, (5)

wherec3 = −λc2 = b − dP + dC − hλc1 = (b − dP + dC)h − NT α

q+ dC

β

c0 = hdC

β

Thus there can be up to three candidate (PC)L equilibria. Along with theprevious candidate (PC)H equilibrium, there might exist up to four coexistenceequilibria. For the parameter sets we chose, however, we never observed morethan three equilibria which simultaneously satisfied the consistency check andwere therefore mathematically realized. Fewer still are ecologically realizablesince all equilibrium coordinates must be nonnegative.

The stability of equilibria can be determined by computing the associatedeigenvalues and checking to see whether all have negative real parts. We do thisin our numerical experiments, but we do not provide Jacobian or eigenvalue for-mulas here.

3.2. Numerical Results

Bifurcations with varying b (maximal per-capita producer productivity) and NT (totalsystem nutrient) fixed

We provide representative bifurcation diagrams, varying b, in figure 2. The totalnutrient NT is fixed at 0.1. Figures 2a-c are traditional one-parameter bifurcationdiagrams, representing attractors with solid lines, and unstable equiilibriawith dashed lines. We plot only equilibria and limit cycle extrema which areecologically realizable.

Barely visible at this scale is (unlabelled) region O: 0 ≤ b ≤ dP = 0.05, wherethe growth rate of the producer is insufficient to overcome its natural death rate.The corresponding equilibrium components are plotted in grey. As b increasesthrough 0.05, a transcritical bifurcation leads to a (black) high nutrient producermonoculture, PH . (The nutrient level of the producer is in some sense irrelevantwithout a positive level of the consumer since in our model the stoichiometryis only effected through the conversion of biomass in predation. The producer

equilibrium, however, can still be classified as high nutrient since Qq

> γ.) A

second transcritical bifurcation at β ≈ 0.65 allows the transition from the PH


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producer monoculture to the (blue) coexistence equilibrium, (PC)H . A Hopfbifurcation at β ≈ 1.63 produces a stable periodic PC-limit cycle; maximum andminimum values along the periodic cycle (green) are displayed. The periodiclimit cycle is destroyed as a case L saddle-node develops on the cycle at b ≈ 2.6.We refer to this bifurcation as a “SNIC”, a saddle-node on an invariant circle. Theattracting case L coexistence equilibrium (PC)L born in the saddle-node bifur-cation replaces the periodic cycle as the attractor. A “nonsmooth saddle-node”bifurcation separates the two subregions of (PC)L. As we cross the boundaryfrom the left to the right at b ≈ 2.82, the case H saddle disappears with the caseL saddle, leaving the case L attracting equilibrium intact. The nonsmoothnessis caused by a crossing of the min function arguments in equation (4). It is alsoapparent in the “corner” formed where the blue and red dashed equilibriumlines meet. This is in contrast to the quadratic signature of a smooth saddle-nodebifurcation.

If the red consumer equilibrium curve in figure 2b were continued farther outin the parameter b, it would eventually intersect the b axis at b ≈ 14.14, resultingin a restabilization of the producer monoculture. This corresponds to enteringregion PL in figure 3(a). The restabilization of the producer monoculture is amanifestation of the “paradox of enrichment,” and appears to be a commonfeature of stoichiometric models with closed nutrient pools [8, 10, 24]. Recallthat the system is open for carbon, whose inputs increase with bi, but closed fornutrient. Therefore, increasing bi at first increases producer growth, resulting in

more food for the consumer. Once Qq

< γ, quality controls consumer growth,

and continued increase in bi enriches the producer(s) with carbon, without anycorresponding increase in system nutrients. Eventually, food quality is too lowto allow consumer growth to match consumer death, resulting in consumerextinction and reesablishment of a producer monoculture through a transcriticalbifurcation.

Figure 2d is an additional bifurcation diagram which is provided to betterunderstand which candidate equilibria are mathematically and/or ecologically

realized and which are not. We plot the ratio Qqat equilibria versus b. This is,

of course, merely a rescaled version of figure 2c, but along with the dashedgrey reference line at the value of γ = 0.1, it more directly relates to the minfunction in equation (4). The linetypes and colors are consistent with figures2a-c: blue for case H, red for case L, and green for the extremes along a periodicorbit. Solid lines still represent attractors and dashed lines represent unstableequilibria (saddles or sources). We have added to figure 2d dotted lines forcandidate equilibria which are not mathematically realized. Specifically, the caseH equilibrium is realized when it is above γ, and the case L equilibria are redwhen they are below γ. Note that in the PC cycle region, the full limit cycle is incase H (above γ) toward the left, but are part in case H and part in case L (belowγ) toward the right side of the region.

The nonsmooth saddle-node is more clear in figure 2d as the crossing of thecase H and case L equilibria, necessarily at γ. Figure 2d also suggests otherpossible bifurcation scenarios for alternate choices of our auxiliary parameters

(all parameters other than b). For example, if γ were increased above the Qqvalue

of the unrealized candidate saddle-node (about 0.2),the candidate saddle-nodewould become a realized saddle-node bifurcation. For γ below the Q

qvalue of the

realized saddle-node (about 0.004), neither case L candidate saddle-node would


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be realized.

For other auxiliary parameter choices, the red cubic case L curve could lose itsturning points, resulting in the loss of both candidate saddle-nodes. All of thesecases have been observed in our numerical experiments.

Table 2 summarizes the attractors in each region and the bifurcations betweenregions. We note that the sequence of bifurcations in figure 2, varying producergrowth rate b, is the same as the sequence displayed for the stoichiometricmodelsinvestigated in LKE, KHE, and WKL [8, 10, 24], where the authors directly varyproducer carrying capacity.

Bifurcations with varying b and NT

To better place the one-parameter bifurcation diagrams in of figure 2 in context,we show a two-parameter bifurcation diagram in b and NT in figure 3. The solidcurves in figure 3a indicate those bifurcations resulting in a change of attractor;the dashed curves indicate bifurcations which do not change the attractor. Thedotted black line corresponds to the NT = 0.1 one parameter cut of figure 2.We use color to distinguish bifurcation curve types: blue for transcritical, greenfor saddle-node, and red for Hopf. Other curves are in black. Correspondingtime series plots are shown in figure 4. In each time series, solutions of P , C ,and “magnified” Q are plotted against time. Table 2 summarizes the bifurcationsbetween the adjacent regions of (b,NT ) parameter space.

Most of the transitions between regions have been described above for theone-parameter cut. The geometry of the regions in the two-parameter planeadds additional insight to the behavior of the model. For example, if we wereto take a one-parameter cut with NT less than about 0.02??, then we wouldnever see the consumer persisting. Note that the PL region extends to the b axisbecause we chose for simplicity to make the producer growth independent ofits nutrient content (recall comments in section 2). If we had included a nutrientrequirement for producer growth, then the region of no growth would extendto a thin L-shaped region next to both axes, rather than just next to the NT axis.Note also the black dashed lines separating regions PH from PL and (PC)H from(PC)L. The respective regions are distinguished by the producer equilibriumlevel changing from high nutrient to low nutrient. The dashed green and blacklines together are significant because they indicate parameter combinations forwhich equilibria are located at the nonsmoothness boundary in the phase space.

For comparison, figure 3(b) shows a bifurcation diagram of the corre-

sponding model without stoichiometry (replacing min(γ, Qq) with γ). The non-

stoichiometric model is, of course, independent of NT since the model reverts toa single currency model.

4. The Two-Producer, One-Consumer Model

We now turn to an analysis of the full model (3), with both producers, one con-sumer, and stoichiometry. Much of the analysis parallels that performed in theprevious section for a single producer.


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Proposition 4.1: The following inequalilties describe a positively invariant region ofmodel (3): 0 ≤ P1 ≤ b1

λ11

, 0 ≤ P2 ≤ b2λ22

, 0 ≤ C ≤ NT

q, 0 ≤ Q1 ≤ NT β1, 0 ≤ Q2 ≤

NT β2, and (NT − qC − Q1P1 − Q2P2) ≥ 0.

Proof : The proof is similar to that of Proposition 3.1. See [9] for details.�

4.1. Equilibria

Equilibria with P2 = 0 and/or P1 = 0 are same as the equilibria in the single-producer-single-consumer model in the previous section, but with extra “trivial”coordinates. Since Qi is not ecologically relevant when Pi = 0, we denote thecorresponding mathematical equilibrium value as being “not applicable” (NA).Coordinates listed in the order: (P1, P2, C,Q1, Q2).

• The no-life equilibrium (O): (0, 0, 0, NA,NA)

• The monoculture equilibrium (P1): (b1−dP1

λ11

, 0, 0,NT β1λ11

λ11+(b1−dP1)β1

, NA)

• The monoculture equilibrium (P2): (0,b2−dP2

λ22

, 0, NA,NT β2λ22

λ22+(b2−dP2)β2

)

• The one-producer coexistence equilibrium ((P1C)H : ( dchγα−dc

, 0,γh(b1−dP1

)

αγ−dc− γdch2λ11

(αγ−dc)2,

β1(NT (αγ−dc)−hqγ(b1−dP1)−

dcqγh2λ11

αγ−dc)

αγ−dc(hβ1−1), NA).

• The one-producer coexistence equilibrium ((P2C)H : (0, dchγα−dc

,γh(b2−dP2

)

αγ−dc− γdch2λ22

(αγ−dc)2,

NA,β2(NT (αγ−dc)−hqγ(b2−dP2

)−dcqγh2λ22

αγ−dc)

αγ−dc(hβ2−1)).

The remaining coexistence equilibria involve both producers and are thereforedistinct from the equilibria in the single producer analysis. Note that for P1 andP2 to be positive equilibrium values, the growth terms (b1 − λ12P2 − λ11P1) and(b2−λ21P1−λ22P2) must both be positive, so we ignore the positive part notationin the four relevant terms in equation (3).

• The producer equilibriumwithout consumer (P1P2):

We set C = 0 and assume that all other variables are positive. The equilib-rium solution is computed to be (P1, P2, C,Q1, Q2) =

(λ22B1−λ12B2

Sprod,

λ11B2−λ21B1

Sprod, 0,

NT β1Sprod

B1(β1λ22−β2λ21)+B2(β2λ11−β1λ12)+S,

NT β2Sprod

B1(β1λ22−β2λ21)+B2(β2λ11−β1λ12)+S) ,

where B1 = b1 − dP1, B2 = b2 − dP2

, and Sprod = λ11λ22 − λ12λ21

In the absence of the consumer, our model reduces to a Lotka-Volterracompetition model. Our simplifying assumption that the producer growth beindependent of the stoichiometry renders Q1 and Q2 irrelevant. It is clear fromthe first two equations in (3) that in order to have P1 and P2 persist at positivelevels, we must have b1 > dP1

and b2 > dP2. Be analyzing nullclines in the

P1-P2 plane, it can be shown that to avoid competitive exclusion, we must also

requireb1−dP1

λ12

>b2−dP2

λ22

andb2−dP1

λ21

>b1−dP1

λ11

. for fixed values of λij with λ11 andλ22 “sufficiently larger” than λ12 and λ12, these four inequalities determinea wedge-shaped region in the b1-b2 plane. Look ahead to wedges formed bythe green lines (mostly dashed) in Figures 5a and c. In words, the birth rates


12

for the two species must be “sufficiently close” to allow for coexistence of P1

and P2. Again looking ahead to Figures 5a and c, we see that the coexistenceregions: (P1P2C)H , P1P2C-cycle, and (P1P2C)L, are a subset of the coexistencewedge for just P1 and P2 without C .

• “Interior” coexistence equilibria (all three species positive):As we did for the one-producermodel, we first calculate all possible candidate

equilibria. There remain two cases: case H, where min(γ, Q1P1+Q2P2

q(P1+P2)) = γ, and

case L, where min(γ, Q1P1+Q2P2

q(P1+P2)) = Q1P1+Q2P2

q(P1+P2).

• Case H: high food quality coexistence equilibrium (P1P2C)H :

Assuming all variables are positive, the equilibrium solution is

(P1P1P2C , P2P1P2C , CP1P2C ,β1(NT −qCP1P2C)

1+β1P1P1P2C+β2P2P1P2C,

β2(NT −qCP1P2C )1+β1P1P1P2C+β2P2P1P2C

),

where B1 = b1 − dP1, B2 = b2 − dP2

,Ssum = λ11 + λ22 − λ12 − λ21,Sprod = λ11λ22 − λ12λ21,

P1P1P2C = B1−B2

Ssum− dch(λ22−λ12)

Ssum(αγ−dc),

P2P1P2C = B2−B1

Ssum− dch(λ11−λ21)

Ssum(αγ−dc),

and CP1P2C = γh(B1(λ22−λ21)−B2(λ11−λ12))Ssum(αγ−dc)

+ γdch2Sprod

Ssum(αγ−d−c)2

Note that the easiest way to guarantee all components of the all-species-coexistence equilibrium are positive is to require both Ssum and Sprod

to be positive. Both terms reflect the dominance of self-limitations overcompetition-limitation.

• Case L: low food quality coexistence equilibria (P1P2C)L :

In this case we assume the min(γ, N1+N2

q(P1+P2)) = N1+N2

q(P1+P2)= Q1P1+Q2P2

q(P1+P2). The

procedure for finding (P1C)L was described in the last section. Similar to theone-producer, one-consumer model, there are up to three equilibria in thetwo-producer, one-consumermodel. The P1 equilibriumvalues for (P1P2C)Lcan be shown to be the roots of a cubic polynomial. The formula takes up apage in Mathematica [9], so we do not provide it here. The remaining fourcomponents can then be determined in terms of P1.

As in the one-producer reducedmodel, stability of equilibria can be determinedby computing the associated eigenvalues and checking to see whether all havenegative real parts. We do this in our numerical experiments, but we do not pro-vide Jacobian or eigenvalue formulas here.

4.2. Numerical results

Figure 5 contains several bifurcation diagrams summarizing the dynamicsand corresponding transitions observed for our set of auxiliary parameters (allparameters except b1, and b2). Representative time series are plotted in figure 6.Because of the symmetry by interchanging P1 with P2, we provide time seriesfor only one of any pair of “conjugate” regions. Figure 8 provides the results ofadditional experiments with varying producer P1’s nutrient uptake rate β1. Wefirst briefly identify the bifurcation diagrams in figures 5 and 8, and then describeeach diagram in more detail. Table 3 summarizes the bifurcations between the


13

adjacent regions.

Figures 5a-c are different views of the same bifurcation diagram in the (b1, b2)parameter plane. NT is still fixed at 0.1. Figure 5a gives an overview for a largerange of parameters, and includes the paradox of enrichment regions where theconsumer cannot exist for high b1 and/or b2: (P1)L, (P2)L, and (P1P2)L. Figure5b is an enlargement of figure 5a, showing more detail, but including only bifur-cations which change the attractor. Figure 5c is the same as figure 5b, but withseveral additional bifurcation curves which do not involve the attractors. Figure5d is a bifurcation diagram for the analogous two-producer-one, consumer

model without stoichiometry (replace min(γ, Q1P1+Q2P2

q(P1+P2)) with γ). Comparing this

diagramwith figure 5b helps to emphasize the role of stoichiometry in our model.

Figure 8a is a two-parameter bifurcation diagram for the one-producer modelshowing β vs. b. It is analogous to figure 3a, with β replacing NT . Figures 8b andc are bifurcation diagrams in the same (b1, b2) space as 5a-c, but with P1’s uptakecoefficient β1 perturbed from its original value of 0.3. In figure b, β1 is 0.2, and infigure c, β1 is 0.1.

We now provide a more detailed description of the bifurcation diagrams.We begin with the two-parameter diagrams in figure 5. Bifurcation types arecolor coded: blue for transcritical of equilibria, green for saddle-node, red forHopf, and pink for transcritical of limit cycles. Other curves are in black. Solidlines indicate the most significant bifurcations – those that result in a changein the attractor. All other bifurcations are indicated by dashed lines. Regionsare labelled by the associated attractor. (We observe no regions of bistability forour choice of auxiliary parameters, so this labelling scheme is unique. ¿Fromnumerical work with similar models, however, we expect that certain parametersets could allow bistability, with, for example, a coexistence equilibrium and aP1P2 equilibrium both being stable.)

Figure 5a is an overview of the full parameter space. We observe no bifurca-tions outside this region of the (b1, b2) parameter plane. Figure 5b is “minimal” inthe sense that we provide only the bifurcations that are ecologically meaningful– those that involve a change of attractor in the first quadrant. Figure 5c showsthe same region as figure 5b, but with additional nonattractor bifurcations alsodisplayed. Region labels correspond to the region’s attractor. Subscripts H and Lcorrespond to equilibria that are high or low nutrient, respectively. The subscriptdepends on whether P1, P2, or the combination of both have a stoichiometryrelative to the consumer stoichiometry q that is above or below the predation

conversion factor of γ. This ratio (P1Q1+P2Q2

P1+P2

/q) makes sense even without theexistence of any consumer. There are three regions that allow for coexistence ofall three species: (P1P2C)H , P1P2C-cycle, and (P1P2C)L. All other regions havebehavior that can be described by a “reduced” model.

We denote by O the region (0 ≤ b1 ≤ dP1) and (0 ≤ b2 ≤ dP2

). The region is notlabelled in the figures 5a-d; it is too small to see clearly. We do not provide timeseries for region O either; the consumer and both producers go extinct for this setof parameters.

For low values of b2 – less than 0.05 – and following any of figures 5a, b,or c for increasing b1, we see a sequence identical to that described for the


14

one-producer model in section 3. This is because the growth rate for producerP2 is not sufficient to allow its survival. So the model reduces to a one-producermodel. A transritical bifurcation at b1 = 0.05 allows P1 to survive; a secondtranscritical bifurcation at b1 ≈ 0.52 allows the consumer to survive; a Hopfbifurcation (b1 ≈ 1.61) destabilizes the (P1C)H equilibrium and gives rise to aperiodic limit cycle; the limit cycle is destroyed in a SNIC bifurcation and theattractor becomes the low nutrient equilibrium (P1C)L which has been born inthe saddle-node bifurcation; finally, the consumer goes extinct in a transcriticalbifurcation (b1 ≈ 14.2). A vertical path for 0 ≤ b1 ≤ 0.05 results in a symmetricsequence of bifurcations but with P1 = 0. A path along the diagonal results ina similar sequence but with both producers existing. The most interesting newfeature is the transition from coexistence close to the diagonal to the extinction ofP1 as we move toward the b2 axis, and the analogous extinction of P2 as we movetoward the b1 axis. All these transitions occur as some form of a transcriticalbifurcation – either the crossing of an equilibrium (across gf or cde in figure c) ora limit cycle (across fc) from an “interior” attractor to a P2 = 0 attractor.

Figures 5a and c include the blue dashed transcritical wedge boundaries.Coexistence of both producers without the consumer is possible only on theinterior of this wedge. We discuss this reduction of the region of coexistence bythe introduction of the consumer later in section 5.1. Note that these lines aresolid when they bound regions without the consumer: (P1P2)H and (P1P2)L.Figure c includes several more dashed bifurcation lines: vertical ones whichcorrespond to bifurcations in the P2 = 0 “side hyperplane” and horizontal oneswhich correspond to bifurcations in the P1 = 0 “side hyperplane.” The extensionof the horizontal line at b2 = 0.05 above b1 = 0.05 is not displayed. This does notinvolve any attractor, and in fact is only an ecologically significant bifurcationin the absence of both P1 and C . The symmetric part of b1 = 0.05 is also not dis-played. All horizontal and vertical bifurcation lines actually extend indefinitely,but we have terminated them both to keep figure 5c from further clutter, andto emphasize the connection between the side hyperplane bifurcations and thecorresponding interior bifurcations. For example, the P2 = 0 side hyperplanenonsmooth saddle-node (line kda) is displayed only up to the point a, where itconnects with the interior nonsmooth saddle-node (line aa′).

Other curves indicate equilibrium bifurcations not involving an attractor: atranscritical of the unstable (P1P2C)H equilibrium from the “interior” of thephase space (defined by P1, P2 and C all being positive) to P2 = 0 as we pass fromregion {ff ′b′b} to region {fbc}. A similar transition occurs from region {b′ba′a} toregion {abc}. Curve ca is a transcritical of different unstable equilibrium from theinterior to P2 = 0. Recall that there are up to three interior equilibria: the originalequilibrium which underwent a Hopf bifurcation across ff ′, and two additionalequilibria born in the SNIC saddle-node bifurcation across cbb′c′. A count ofinterior equilibria yields one in region {ff ′b′b}, none in region {bcf}, three inregion {bb′a′a}, two in region {abc}, and one in the rest of region (P1P2C)L.Region {bcf} is especially noteworthy because there exists an interior limit cycle,but no interior equilibrium.

With the exception of the vertical and horizontal curves which are not fullyextended, figure 5c appears to include all ecologically relevant bifurcations forour choice of parameter values.

Figure 5d applies to the nonstoichiometric model, with min(γ, Q1P1+Q2P2

q(P1+P2))


15

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 thetypical Rosenzweig-MacArthur bifurcation sequence which in which continuedenrichment results in a limit cycle which seems to persist for arbitrarily largeenrichment. This assumes only that enrichment results in a path of positive slopeheading away from the origin.

Figure 8a shows the two-parameter bifurcation diagram in β and b1 forthe single producer model of section 3. By comparing to the two-parameterbifurcation diagram of figure 3a, it can be seen that increasing the producer nutri-ent uptake rate is roughly analogous to increasing the total nutrient in the system.

Figures 8b and c are analogous to figure 5b, but for unequal values of theproducer uptake coefficients. The symmetry of figure 5b is broken in figures 8band c. Figure 8b is topologically equivalent to figure figure 5b, but figure 8c isno longer equivalent. Figure 8a helps exlain parts of figures 8b and c as follows.The horizontal one-parameter cut at β = 0.2 (respectively β = 0.1) in figure8a corresponds to a horizontal one-parameter cut in figure 8b (respectively c)at b2 − 0. Similarly, a one-parameter cut at β = 0.3 in figure 8a corresponds tothe vertical one-parameter cut in figures 8b and c at b1 = 0. We address furtherthe ecological implications of the mismatch in producer nutrient uptake rates insection 5.1.

Figure 7 is the two-producer analogue to the one-parameter bifurcationdiagram in figure 2 for the single producer case. Phase variables P1, P2, and Care plotted against b1 in a-c. The ratio of the consumers food stoichiometry toits structural ratio is plotted in d. We have fixed NT = 0.1, and b2 = 2. Linecolors and types match those of figure 2: blue for case H, red for case L, solidfor attractors, dashed for unstable, dotted for candidate equilibria which areunrealized. If b2 were less than 0.05, this one-parameter “cut” would matchfigure 2 even more closely. We have chosen a larger value of b2 for contrast. Thecut can be placed in context by looking at the dotted black lines illustrating theb2 = 2 cut in the (b1, b2) plane in the two-parameter diagrams of figures 5a, b or c.

We will walk through the cut from low to high b1 and identify attractors first.Then we will describe the extra curves corresonding to nonattracting equilibriaand candidate equilibria (in figure d). For low b1 values, P2 and C coexist on anattracting periodic limit cycle in region P2C cycle. The solid purple lines denotethe extrema of this cycle. This is consistent with single producer analysis: thepoint (b,NT ) = (2.0, 0.1) is in the PC-cycle region of figure 3a. Note that thephase values of the extrema of the limit cycle do not change as b1 is increasedbecause P1 is not yet enriched enough to survive. There is a transcritical crossingof periodic limit cycles at b1 ≈ 1.38. Producer N1 now survives along with P2

and C on an interior limit cycle (solid green extrema). The interior limit cycle isdestroyed in a SNIC bifurcation at b1 ≈ 2.405. The saddle-node is best illustratedin figure b where the solid red curve shows the attracting equilibrium (P1P2C)L.At b1 ≈ 4.28, producer P1 “outgrows” producer P2, and P2 goes extinct in atranscritical bifurcation. Not shown in any of the figure 7 diagrams, but apparentin the black dotted b2 = 2 cut displayed in figure 5a, the consumer is lost in theparadox of enrichment transcritical bifuration at b1 ≈ 14.2.

Dashed lines correspond to unstable equilibria, either saddles or sources. Two


16

of the dashed line segments correspond to interior equilibria: dashed blue inregion Cyclea, and dashed red in the first subregion of (P1P2C)L. The dashedblue equilibrium is born in a P1 = 0 transcritical bifurcation (see figure 7a), anddies in a P2 = 0 transcritical bifurcation (see figure 7b at b1 ≈ 1.62). The reddashed equilibrium is born in the saddle-node bifurcation already mentionedabove at b1 ≈ 2.405, and dies in a P2 = 0 transcritical bifurcation at b1 ≈ 2.6. Bothbifurcations are best viewed in figure 7b.

Two additional side hyperplane equilibria are plotted: black dashed for (P1)H ,and brown for (P2C)H . The (P1)H monoculture is included mainly for figure7a, to show that all coexistence P1 values are below the P1 monoculture value.The brown dashed (P2C)H side hyperplane equilibrium is included only tobetter see the transcritical bifurcation between regions Cycleb′ and Cyclea. Seefigure 7b or c. We chose not to display the analogous connecting P2 = 0 orbitat the transcritical bifurcation between regions Cyclea and Cycleb because itscontinuation would unnecessarily complicate the diagrams.

As we did in figure 2d, we show extra dotted curves denoting “candidate”equilibria in figure 7d. The only difference in interpretation of the dotted linesin the two figures is that in the former, the dotted candidates are mathematicallyunrealized because they don’t satisfy the “consistency check”, while in the latter,portions of the dotted lines are ecologically unrealizable because at least one com-ponent is negative. The part the dotted red curve that is above the γ value of 0.01and the part of the dotted blue curve that is below γ are candidates which areunrealized for both reasons.

5. Discussion

5.1. Ecological Implications

All the bifurcations in the preceding sections are theoretically relevant to eco-logical systems, but we emphasize in this section three new behaviors of thismodel that have interesting ecological implications and which are experimentallytestable:

(1) in a one-producer-consumer system, increasing producer growth rate (b)produces a different sequence of bifurcations and behaviors when the to-tal amount of nutrient (NT ) is low compared to when it is moderate orhigh (Fig. 3a);

(2) the presence of the consumer decreases the region of coexistence of thetwo producers in the (b1, b2) parameter plane (Fig. 5c); and

(3) differential response of producer growth rates (bi) to enrichment may pro-duce different sequences of bifurcations and species extinctions (Fig. 5c).

We have assumed a reasonable set of parameter values for aquatic systems(with rescaling, as indicated at the end of section 2, Model Construction), anddifferent aquatic systems or terrestrial systems might be modeled by otherparameter selections. However, because the equilibria, their stability, and bi-furcation behaviors of the model depend critically on parameter values, it isimportant in any experimental test to first determine accurate values of theparameters, especially producer growth rates bi, producer nutrient uptake ratesβi, and the reductions in growth rates due to self limitation and interspeciescompetition (λij). Once parameter values have been established for the particular


17

species chosen, then bifurcation diagrams would need to be recomputed todetermine specific values of parameters where critical transitions are predictedto happen.

To conform to the assumptions of the model, the experimental system mustbe closed to nutrient fluxes from the outside environment but open to carbonfluxes via producer uptake in photosynthesis and outputs via respiration. Acommonly used system which meets these assumptions is a simple aquariumwith an initial nutrient pool that remains fixed (but which may be dynamicallyallocated amongst the different species) and which is open to carbon fluxes andcontains producer(s) such as algae and a consumer such as Daphnia or similarzooplankton. In contrast, a chemostat is open to both carbon and nutrients andwould not be an appropriate experimental environment.

The general strategy would be to experimentally vary parameter values to seeif the experimental system can be forced across bifurcations, thereby causinga large and observable qualitative change of behavior, such as extinction ofone or more species, initiation or quenching of limit cycles, etc. The simplestparameter to experimentally vary is the total amount of nutrient, NT , whichcan be done by simply fertilizing the system. Species growth rates, bi, can bevaried in two ways: first by judicious choice of different species, secondly bydetermining whether b1 and b2 respond differently to light enrichment because ofdifferent photosynthetic response curves for the two producers. If the latter canbe experimentally established and calibrated to light enrichment, then the twobi values could be simultaneously changed by shading or light augmentation.Finally, nutrient uptake rates, βi, can be varied by choice of producer speciesused in the experiment.

Assuming that these preliminary experimental details can be addressed, thefollowing experiments could test the new behaviors of this model:

(1) Different bifurcation sequence with increased producer growth rates under lowcompared with high nutrient levels. In Fig. 3a, increasing producer growth bresults in a different bifurcation sequence at different nutrient levels, NT .At low nutrient levels (0.2 < NT < 0.04) as b increases, there is first atranscritical bifurcation where the producer survives without a consumer,a second transcritical bifurcation where the producer and consumer coex-ist at a stable fixed point equilibrium, and finally at very high producergrowth rates the loss of the consumer at another transcritical bifurcationbecause of very low food quality. This sequence can be tested by increas-ing b and examining where either species or both exist. In the coexistenceregion, increase in b should correspond with a decrease in producer N/C ,but sustained high consumer N/C . When the difference in the ratios isgreat enough, then the consumer should go extinct. Repeating the exper-iment at moderate to high nutrient levels (NT > 0.08) should result in abifurcation sequence similar to that displayed in figure 2, including thebirth of a limit cycle via a Hopf bifurcation and the subsequent loss of thelimit cycle at higher values of b.

(2) The presence of the consumer decreases the region of coexistence of the two pro-ducers. In our model, the two producers can coexist without the consumerbecause we have assumed weak competition (λii > λij), a common find-ing of non-stoichiometric Lotka-Volterra models [3, 14]. The region of co-existence of the two producers in the (b1, b2) parameter plane (Fig. 5c) is


18

delineated by the wedge shape within the dashed blue lines. However,introducing a consumer restricts the region of coexistence further to thewine glass shaped region within the wedge shaped region (Fig. 5c). Thisis in contrast to the findings of Grover [? ] and Pastor and Cohen [15]where consumers facilitate coexistence of two producers who competestrongly for a single resource. Whether the consumer decreases the regionof coexistence of two producers can be tested by first establishing a mi-crocosm containing two producers which coexist and which have growthrates which put them within the wedge shaped region of coexistence inthe (b1, b2) parameter plane but outside the wine glass region of coexis-tence of two producers and a consumer. Then introduce a suitable con-sumer who preys on both species. The producer with the slower growthrate should then go extinct.

(3) differential response of producer growth rates (bi) to enrichment may producedifferent sequences of bifurcations and species extinctions (Fig. 5c).Two different producers might respond differently to various sources of

carbon enrichment. Therefore, each source of enrichment corresponds toa specific path through the (b1, b2) parameter space in figure 5(c). For ex-ample, suppose the system is known to be at the red dot in region Cycleb.If a specific source of enrichment causes b1 and b2 to increase at the samerate, we will follow a path of slope one from our starting point. This willresult in a straight-forward bifurcation sequence: loss of the interior limitcycle in a SNIC bifurcation, and eventual loss of the consumer in a tran-scritical bifurcation (as in figure 5(a)). On the other hand, another sourceof enrichment might increase b1 much more than b2. This will result ina path in figure 5(c) that has a slope less than one. This could result ina complicated bifurcation sequence in which, for example, we pass fromthe interior limit cycle to a P1C-cycle (region {cfnm}), to a (P1C)L equi-librium (region {cmkd}), back to coexistencewith a (P1P2C)L equilibrium(region {acd}), out of coexistence again by crossing curve de and return-ing to (P1C)L, and finally losing the consumer as it crosses from region(P1C)L to region (P1)L. Many other paths are, of course, also possible.This emphasizes the necessity referred to above of recomputing the ana-logue of the bifurcation diagram in figure 5 using the identified param-eters for specific species, as well as determining the producer birth rateresponse to various forms of enrichment. The asymmetries apparent infigure 8(c), caused by differing producer nutrient uptake rates, coupledwith differing responses of producer growth rates to enrichment, couldcreate further possibilities.Such experiments would require a careful determination of and precise

control over parameter values and probably also virtuoso experimentaltechnique. But if they could be done, then the large and qualitative differ-ences in the sequences of bifurcations, coexistence, and extinctions shouldprove a strong confirmation of the model presented here.

5.2. Topology versus geometry

The use of bifurcation theory to analyzemodels has the great advantage of group-ing behavior of systems into broad equvalence classes. The mathematical defini-tion of topological equivalence of systems of differential equations, however, may betoo broad to capture all of the ecological significance of a model. For example, wehave observed the following geometrically different, but topologically equivalent


19

behavior for “interior” limit cycles. For certain parameter combinations, the limitcycle, starting from small initial conditions for both producers and consumer, ex-hibits an increase first in one producer, then the second producer, and finally theconsumer. As happens in producer-consumer models with a single producer, thegrowth in consumer leads to a reduction in producer, and then consumer, beforereturning to the small initial conditions. For different parameter combinations,we observe small initial conditions leading to first the growth of one producer,then the growth of the consumer, and finally the growth of the second producer.These two distinct ecological scenarios are not distinguished by our bifurcationclassification.

5.3. Model limitations

There are, of course, many limitations of our model due to both its simplifyingassumptions and its generality. We have mentioned in section 2 several of them.

(1) The producers’ growth does not depend on the total nutrient, NT . This isa simplifying assumption. That we still get a bifurcation sequence (figure2) similar to LKE [10], KHE [8] andWKL [24], emphasizes the importanceof the role of stoichiometry in the conversion efficiency of the consumer,as opposed to the producer growth.

(2) Another limitation occurs when this model is applied to a terrestrial sys-tem. In this case, the model could be enhanced by considering the under-ground components - roots and decomposers, each having different stoi-chiometry than the photosynthetic component which we have modelledhere. It would be interesting to see whether adding this new complicationleads to any new behaviors.

(3) From a technical standpoint, the use of the minimum function leads to anonsmoothmodel. The model could be made smooth in a variety of ways,such as using the Kooijman approach of using synthesizing units ??. Theadvantage of the smooth system, however, is counteracted by the addedcomplexity of the model. Our “caseH” and “case L” coexistence equilibriawould be analytically intractable in a smoothed version. The only conclu-sions that would be affected are the fine details of the bifurcation dia-grams, since nonsmoothness tends to collapse portions of such diagrams.The gross bifurcation behavior, however, can be expected to persist.

(4) Finally, the fact that our model is closed to nutrients is an assumptionthat would require care to reproduce experimentally, but also one thatcontributes to the simplicity of the model.

5.4. Other behaviors of our model

We remind the reader that our bifurcation analysis is dependent on the auxiliaryparameters chosen for the study. We do not claim that we have provided acomplete description of all possible behaviors, even in the one-producer reducedmodel. There are many parameters in equation (4) which we have not variedand which could lead to different behavior. For example, similar models wehave studied have the first saddle-node taking place off the limit cycle (noSNIC), resulting in a coexisting attracting limit cycle and attracting equilibrium.A later homoclinic destroys the limit cycle and returns the equilibrium to theonly attractor. Other choices of the parameters in our model could lead to otherscenarios.


20

As we indicated in the Ecological Implications subsection above, our bifurca-tion analysis can be viewed as a paradigm for studying many specific systems.Identification of species specific parameters, and recomputation of bifurcation di-agrams such as those in figure 5 would be the first step in such a study.

6. Conclusions

The model developed and studied in this paper is a simple Lotka-Volterra modelof two competing producers and one consumer, but with stoichiometry. The pro-ducers grow logistically. Stoichiometric limitation is modeled only by restrictingthe conversion efficiency from producer to consumer. The stoichiometric effectis thus focussed on this mechanism. The model’s nutrient uptake term, beingproportional to both available nutrient and to producer growth, differs from thenutrient uptake in other models, but does not appear to produce new dynamicalbehavior.

The main goal in our analysis has been to determine the response of thesystem to increased carbon (b1 and b2) and nutrient (NT ) enrichment. Consistentwith many stoichiometric models, the overall effect of including stoichiometricrestrictions is to prevent enrichment from leading to large amplitude producer-consumer oscillations; especially with a fixed nutrient pool, there is not enoughnutrient to support large populations. More specifically, in both the reduced,single producer model, and the full two-producer model, the response of thesystem to enrichment (b, or b1 and b2, respectively) leads to roughly the samebifurcation sequence as observed in previous stoichiometric studies with asingle producer [10, 24] where carrying capacity was directly varied. This is trueeven though the latter models include additional stoichiometric restrictions onproducer growth. By envisioning one-parameter cuts – with fixed total nutrientNT – through figure 3a one can see that the same qualitative bifurcation scenariomight be expected as long as NT is sufficiently high (above approximately 0.05).The specific cut exhibited in figure 2 is one example. A comparison of figure3 with figure 8a shows that increasing nutrient uptake effects the system in amanner similar to increasing the total nutrient.

In the two-producer model, the large scale bifurcation scenario due to carbonenrichment (heading radially out from the origin in and of figures 5a,b,c) is sim-ilar to the single-producer reduced model: no life, producer without consumer,coexistence, cycle coexistence, cycle replaced by new (low nutrient) equilibrium,and finally loss of consumer due to low producer stoichiometry. In general, asone moves away from the diagonal in the producer growth space (b1, b2), theproducer with the lower growth rate eventually goes extinct in a transcriticalbifurcation involving either an equilibrium or a periodic cycle. The details of thecoexistence of one versus both producers, illustrated in figures 5a,b,c, is new tothis study. This is possibly the most interesting result in the paper. Analysis ofthe figures reveals some nonintuitive possibilities. It can be seen from the figuresthat there are, for example, straight line paths with positive slope (correspondingto enrichment) that go from coexistence of all three species, to the loss of oneproducer, and back to coexistence. Features like the existence of interior periodiccycles without interior equilibria are also of dynamical interest.

To summarize, paying attention to stoichiometry in population models candrastically change predicted population behavior. This study adds to the under-


21

standing of the role of stoichiometry, especially in the restrictions on coexistenceof a consumer which preys on more than one producer.

Acknowledgements

We thank University of Minnesota for support our project funding.

7. Appendix

7.1. Numerical Methods

The bifurcation diagrams in this paper were computed using a variety of analyticand numerical methods.

• Figure 2. Most curves were plotted using Mathematica along with the explicitexpressions for the equilibria in section 3. The exceptions were the curves forthe “case L” coexistence equilibria (red) and the curves representing the peri-odic limit cycles extrema (green). For the case L coexistence equilibria, Math-ematica was used (FindRoots) to detemine (P1C)L as a solution to the cubicequation (5). The other components were then calculated using the previouslydetermined P1 value. For the limit cycle extrema (green), Mathematica wasused to compute numerical solutions (NDSolve) from time 1 through time 2000.The numerical solutionswere then processed from time 1000 through time 2000to determine the extrema corresponding to each solution component. Steps ofsize 0.1 in b1 were used.

• Figure 3. The transcritical bifurcation curves (blue) were computed by setting acomponent of the equilibrium to zero. The Hopf curve (red) was computedby setting the real part of the eigenvalue of the equilibrium point to zero.The saddle-node curve (green) was computed using equation (5): they corre-sponded to extrema the cubic in P1. The dashed lines – boundaries betweenlow and high nutrient cases – were computed by solving for equilibria whichalso satisfied the stoichiometric condition that Q/q = α.

• Figure 5. Bifurcation curves were mostly computed in a manner similar tothe techniques described for figure 3. There were two exceptions. The “inte-rior” Hopf curve was determined using Mathematica (ContourPlot) to solvefor equilibria with eigenvalues that had zero real part. The curves representingtranscritical bifurcations between side hyperplane cycles and interior cycles(cyan), forming part of the boundary of the (P1P2C)L cycle region in figuresb,c,d, were computed differently. To approximate these curves, numerical so-lutions to equation (3) were computed using Mathematica from time 0 to 6000.Then the difference, accurate to 0.000001, between maximum and minimum ofP1 (P2, respectively) was computed from the last 1000 time steps of the solu-tions. Newton’s method was then used to converge to a value of b2, given avalue of b1, which satisfied (to within 0.001) found to have a difference of zero.

• Figure 7. Methods were analogous to those used to compute figure 2.• Figure 8. Methods for figure a were analogous to methods for figure 3. Methods

for figures b and c were analogous to methods for figure 5.

7.2. Mathematical bifurcation notes

The two-parameter bifurcation diagrams in this paper naturally raise the ques-tion of identification of codimension-two bifurcation points wherever more than


22

one codimension-one bifurcation curve cross or come together. Many of thecodimension-two bifurcations in this paper are nonstandard because of the preva-lence of the transcritical bifurcations. These are nongeneric in bifurcation theoryexcept in the presence of invariant planes. Others are nonstandard because theyinvolve nonsmooth changes. We identify all bifurcation points in figures 3a and5c where codimension-one bifurcation curves either cross or come together. “Triv-ial” crossings, which correspond to the crossing of two codimension-one curvesthat to not interact in the same region of phase space, are denoted as bifurcationone × bifurcation two.

• Figure 3a:(1) Point (a) is a nonsmooth × transcritical point. A smooth approximation

should eliminate this point.(2) Point (b) is also a nonsmooth × transcritical point. A smooth approxima-

tion should eliminate this point as well.(3) Point (c) is a nonsmooth saddle-node/Hopf point. Based on bifurcation

diagrams computed for previous research [16, 17], we expect a smoothapproximation to resolve the bifurcation diagram into saddle-node curvewith a codimension-two cusp, and a codimension-two Takens-Bogdanovpoint. The nonsmoothness appears to collapse these two codimension-twopoints together.

• Figure 5c:(1) Points (o) and (g) are double transcritical bifurcation points.(2) Point (f) is a transcritical/Hopf point.(3) Point (b) is a SNIC × transcritical point.(4) Point (c) is a SNIC/transcritical point. This should have the same local

unfolding as the saddle-node/transcritical point identified in ??.(5) Point (a) is a nonsmooth saddle-node/transcritical point. Smooth approxi-

mation should resolve into a smooth saddle-node/transcritical bifurcationlike point (c).

(6) Point (d) is a nonsmooth saddle-node × transcritical point.

Analysis of the double transcritical point and the transcritical/Hopf pointshould be especially tractable. This would allow us to more precisely identifythe corresponding points in our bifurcation diagrams.


23

Table 1. Table of VarialesVariables Definitions Variables Definitions

P1 Population Density of N1 Nutrient of Producer P1

Producer P1 (Carbon)P2 Population Density of N2 Nutrient of Producer P2

Producer P2 (Carbon)C Population Density of NC Nutrient of Consumer C

Consumer C (Carbon)

Q1 = N1

P1

Stoichiometric Ratio of P1 Q2 = N2

P2

Stoichiometric Ratio of P2

M Mineralized Nutrient

Table 2. Bifurcations between the adjacent regions in figures 2 and 3: one producer, one consumer

Regions Type of Bifurcation Involved Attractors Non-Attractor Transitions

O → PH Transcritical O → PPH → (PC)H Transcritical P → (PC)H(PC)H → PC-cycle Hopf (PC)H

→ PC-cyclePC-Cycle → (PC)L {cde} SNIC PC-cycle An unstable (PC)L

→ (PC)L is also created.(PC)L {cde} → (PC)L {bcef} Non-Smooth The unstable (PC)H and unstable (PC)L

Saddle-node are eliminated.(PC)L {bcef} → PL Transcritical (PC)L → PL

Table 3. Bifurcations between the adjacent regions in figures 5(b) and 5(c): two producers, one consumer

Regions Type of Bifurcation Attractors Non-Attractor Transitions

O → (P1P2)H Transcritical O → (P1P2)O → P1H Transcritical O → (P1)P1H → (P1P2)H Transcritical (P1) → (P1P2)(P1P2)H → (P1P2C)H Transcritical (P1P2) → (P1P2C)H(P1)H → (P1C)H Transcritical (P1) → (P1C)H(P1C)H → (P1P2C)H Transcritical (P1C)H → (P1P2C)H(P1P2C)H → Cyclea Hopf (P1P2C)H

→ Interior cycle(P1C)H → P1C-Cycle Hopf (P1C)H

→ P1C-cycleP1C-cycle → Cycleb Transcritical P1C-cycle

→ Interior cycleCycleb → Cyclea Transcritical (P1C)H → (P1P2C)HCyclea → (P1P2C)L {abb’a’)} SNIC Interior cycle Unstable (P1P2C)L is created.

→ Stable (P1P2C)LCycleb → (P1P2C)L {abc} SNIC Interior cycle Unstable (P1P2C)L is created.

→ (P1P2C)LP1C − Cycle → (P1C)L {cdkm} SNIC P1C plane cycle Unstable (P1C)L is created.

→ (P1C)L(P1C)L ({cdkm} → (P1P2C)L {acd} Transcritical (P1C)L → (P1P2C)L(P1P2C)L {abb’a’} → (P1P2C)L Non-Smooth Unstable (P1P2C)H and

Saddle-node unstable (P1P2C)Lare eliminated.

(P1P2C)L {abb’a’} → (P1P2C)L {abc} Transcritical (P1P2C)H → (P1C)H(P1P2C)L {abc} → (P1P2C)L {acd} Transcritical (P1P2C)L → (P1C)L(P1P2C)L {acd} → (P1P2C)L Non-Smooth Unstable (P1C)H and

Saddle-node unstable (P1C)Lare eliminated.

(P1C)L {cdkm} → (P1C)L {edk} Non-Smooth Unstable (P1C)H andSaddle-node unstable (P1C)L

are eliminated.(P1C)L {cdkm} → (P1P2C)L {acd} Transcritical (P1C)L → (P1P2C)L(P1C)L {jedk}→ (P1P2C)L {edaa’d’e’} Transcritical (P1C)L → (P1P2C)L(P1P2C)L → (P1P2)L Transcritical (P1P2C)L → (P1P2)(P1C)L → (P1)L Transcritical (P1C)L → (P1)(P1)L → (P1P2)L Transcritical P1 → (P1P2)L)


24

(a) Flow of Carbon (b) Nutrient Cycling

Figure 1. Two-Producer-One-Consumer Flow Charts: Carbon Flow and Nutrient Cycling

HPCLLPC−CycleHPCLHPH

1 2 3 4b

0

2

4

6

8

P

(a) Producer P

HPCLLPC−CycleHPCLHPH

1 2 3 4b

0.0

0.5

1.0

1.5

2.0

2.5

C

(b) Consumer C

HPCLH PC−Cycle HPCLLPH

1 2 3 4b

0.005

0.010

0.015

0.020

0.025

0.030

Q

(c) Producer stoichiometry Q

a b

PC−CycleHPCLH HPCLLPH

0 1 2 3 4b0.0

0.1

0.2

0.3

0.4

0.5

0.6

Q

q

(d) Ratio of producer stoichiometry to consumer sto-

ichiometry: QqBifurcation Diagram

Figure 2. One-parameter bifurcation diagrams for the reduced One-Producer-One-Consumer Model of equa-tion (4). (a),(b),(c) Traditional one-parameter bifurcation diagrams of the respective phase variable versus theparameter b. Attracting behaviour is illustrated by solid curves; unstable equilibria are illustrated by dashedlines. (d) Producer stoichiometry to consumer stoichiometry ratio; dotted blue and red curves denote unreal-ized candidate equilibria. See text for more explanation.


25

PC−CycleHPCLHPH

HPCLL

PLa b

c

e

d

f

1 2 3 4 5b

0.1

0.2

0.3

0.4

0.5

NT

(a) Bifurcation Diagram With Stoichiometry

PC−CycleHPCLHPH

1 2 3 4 5b

0.1

0.2

0.3

0.4

0.5

NT

(b) Bifurcation Diagram Without Stoichiometry

Figure 3. Two-parameter bifurcation diagrams in b and NT for the One-Producer-One-Consumer Model. (a)

Equation (4). (b) Equation (4) without stoichiometry: min(γ,Qq

) is replaced by γ. See figure 4 for representative

time series corresponding to each region, and table 2 for bifurcations between adjacent regions. See text for moreexplanation.


26

P

Q/q

C

ý

50 100 150 200t

0.5

1.0

1.5

2.0

population�ratio

(a) Producer monoculture: PH

P

Q/q

C

ý

50 100 150 200t

0.5

1.0

1.5

2.0

population�ratio

(b) High nutrient coexistence: (PC)H

P

C

Q/q

ý

50 100 150 200t

1

2

3

4

5

population�ratio

(c) Periodic coexistence: PC-Cycle

5 � Q/q

C

P

5 � ý

10 20 30 40 50t

1

2

3

4

5

population�ratio

(d) Low nutrient coexistence: (PC)L

5 � Q/q

C

P

5 � ý

10 20 30 40 50t

1

2

3

4

5

population�ratio

(e) Low nutrient monoculture: PL

Figure 4. Representative time series corresponding to parameter regions in figure 3. The dashed line ismultipler × γ, which is used as a reference to indicate whether producer’s N:C ratio, Q, is relatively high orlow to the consumer. When Q is high, consumer’s conversion efficiency is γ. Otherwise, the conversion ef-

ficiency is Qq. Green is for the producer P ; red is for the consumer C; blue is for the producer’s (rescaled)

N : C ratio Q over q. The parameters of figures are listed below. (a) b = 0.3, NT = 0.3, and initial(P1, C, NT ) = (2, 1, 0.02); (b) b = 1.2, NT = 0.3, and initial (P1, C, NT ) = (2, 1, 0.02); (c) b = 3, NT = 0.3,and initial (P1, C, NT ) = (2, 1, 0.02); (d) b = 3, NT = 0.08, and initial (P1, C, NT ) = (2, 1, 0.02); (e) b = 3,NT = 0.02, and initial (P1, C, NT ) = (0.5, 0.3, 0.003).


27

HP1P2CLL

HP2CLL

HP1CLL

HP1P2LL

P1L

P2L

5 10 15 20b1

5

10

15

20

b2

(a) Parameter space overview

P1C−Cycle

P2C−Cycle

HP1CLL

HP1CLH

HP2CLH

HP2CLLHP1P2CLL

HP1P2CLH

CycleP1P2C−

P2HHP1P2LH

P1H

1 2 3 4 5b1

1

2

3

4

5

b2

(b) Enlargement near origin; attractor bifurcationsonly

a

b

cd

e

j

k

f

gmno

a‘

Cycleb

Cyclea

Cycleb‘

P1C−Cycle

P2C−Cycle

f‘

c‘

b‘

g‘

HP2CLL

HP2CLH

HP1P2CLL

HP1P2CLH

HP1CLL

HP1CLH

r

1 2 3 4 5b1

1

2

3

4

5

b2

(c) Enlargement near origin: including unstable equi-librium bifurcations (Details)

CycleP1P2C−

P1C−Cycle

P2C−Cycle

P1P2C

P2C

P1CP1P2P1

P2

1 2 3 4 5b1

1

2

3

4

5

b2

(d) Without stoichiometry: replace

min(γ, P1Q1+P2Q2

P1+P2

) with γ in equation (3)

Figure 5. Two-parameter bifurcation diagrams, b1 versus b2, for the Two-Producer-One-Consumer Model ofequation (3). Solid lines indicate a change of attractor; dashed lines indicate bifurcations involving equilibriawhich are not attracting. Dotted black lines correspond to the one-parameter cut of figure 7. Representative timeseries are displayed in figure 6. See a summary of the attractors in each region and bifurcations between regionsin table 3. See the text for more description.


28

P2

P15 � ý

5 � Q/q

C

100 200 300 400t

0.5

1.0

1.5

2.0

2.5

3.0

Populations�ratio

(a) Producer monoculture: (P1P2)H

5 � ý

P2

P1

5 � Q/q

C

100 200 300 400t

0.5

1.0

1.5

2.0

2.5

3.0

Populations�ratio

(b) High nutrient coexistence: (P1P2C)H

P1

P2

C5 � Q/q

50 100 150 200t

0.5

1.0

1.5

2.0

2.5

3.0

Populations�ratio

(c) Periodic coexistence: Cyclea

P1

P2

C

5 � Q/q

50 100 150 200t

1

2

3

4

Populations�ratio

(d) Periodic coexistence: Cycleb

5 � Q/q

5 � ý

C

P2

P1

50 100 150 200t

1

2

3

4

5

Populations�ratio

(e) Low nutrient coexistence: (P1P2C)L

100 � Q/q

P2

P1

C

100 � ý

50 100 150 200t

5

10

15

Populations�ratio

(f) Low nutrient monoculture: (P1P2)L

Figure 6. Representative time series corresponding to parameter regions in figure 5. The dashed line ismultipler × γ, which is used as a reference to indicate whether the combined producers’ N:C ratios, Q =P1Q1+P2Q2

P1+P2

, is relatively high or low to the consumer. When Q is high, consumer’s conversion efficiency is

γ. Otherwise, the conversion efficiency is Qq. Red is for the producer P1; brown is for the producer P2; blue is

for the consumer C; green is for the combined producers’ (rescaled) N : C ratios Q over q. The time series inP1H, (P1C)H , P1C − Cycle, (P1C)L , and P1L are similar to the ones in figure (4). The parameters of figuresare listed below. (a) b1 = 0.4, b2 = 0.3, NT = 0.1, and initial (P1, P2, C, Q1, Q2) = (2, 1, 2, 0.005, 0.005); (b)b1 = 1, b2 = 0.9, NT = 0.1, and initial (P1, P2, C, Q1, Q2) = (2, 1, 2, 0.005, 0.005); (c) b1 = 2, b2 = 1.9,NT = 0.1, and initial (P1, P2, C, Q1, Q2) = (2, 1, 2, 0.005, 0.005); (d) b1 = 2.3, b2 = 1.7, NT = 0.1, and initial(P1, P2, C, Q1, Q2) = (2, 1, 2, 0.005, 0.005); (e) b1 = 3.6, b2 = 3.5, NT = 0.1, and initial (P1, P2, C, Q1, Q2) =(2, 1, 2, 0.005, 0.005); (f) b1 = 11, b2 = 10.5, NT = 0.1, and initial (P1, P2, C, Q1, Q2) = (2, 1, 2, 0.005, 0.005);


29

HP1CLLHP1P2CLLP2C−Cycle Cycle

b‘ a b(ac)

P1H

P1L

1 2 3 4 5b1

0

2

4

6

8

10

P1

(a) Producer P1


HP2CLH

(ac)b‘ a b

1 2 3 4 5b1

0

1

2

3

4

5

6

P2

(b) Producer P2


(ac)

b‘ a b

HP2CLH

1 2 3 4 5b1

-1

0

1

2

3

4

C

(c) Consumer C

Cycle

b‘ aP2−C Plane HP1P2CLL HP1CLL

b

(ac)

HP2CLH

a

1 2 3 4 5b1

0.1

0.2

0.3

0.4

0.5

Q1 P1+Q2 P2

q HP1+P2L

(d) Ratio of combined producer stoichiometry to con-

sumer stoichiometry: Q1P1+Q2P2

q(P1+P2)

Figure 7. One-parameter bifurcation diagrams in b1 for the Two-Producer-One-Consumer Model of equation(3) with b2 = 2.0. Solid lines denote attractors; dashed lines denote unstable equilibria; and dotted lines denoteunrealized candidate equilibria. See text for more explanation.


30

PH HPCLH PC−Cycle HPCLL

PL

1 2 3 4 5b

0.1

0.2

0.3

0.4

0.5

Β

(a) β versus b bifurcation diagram for the One-producer-one-consumer model of equation (4)

HP1P2CLL

HP2CLL

P2C−Cycle

CycleP1P2C−

HP1CLLP1C−CycleHP1P2CLH

HP2CLH

HP1CLHHP1P2LHP1H

P2H

1 2 3 4 5b1

1

2

3

4

5

b2

(b) Same as figure 5b, except for β1 = .2

HP1P2CLL

HP2CLL

P2C−Cycle

HP1CLL

HP1CLH

HP1P2CLH

HP2CLH

HP1P2LHP1H

P2H P1C−Cycle

CycleP1P2C−

1 2 3 4 5b1

1

2

3

4

5

b2

(c) Same as figure 5c, except for β1 = .1

Figure 8. Bifurcation diagrams for varying producer nutrient uptake rate β (or β1). See text for further descrip-tion.


REFERENCES 31

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