research article a stoichiometric model of two …

August 22, 2008 9:35 Journal of Biological Dynamics Report

Journal of Biological DynamicsVol. 00, No. 00, Month 200x, 1–18

RESEARCH ARTICLE

A STOICHIOMETRIC MODEL 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 August 2008)

A stoichiometric population model of two producers and one consumer is a generalization of theRosenzweig-MacArthur population growth model, which is a single-producer-single-consumerpopulation model without stoichiometry. The generalization involves two steps: 1) addinga second producer which competes with the first, and 2) introducing stoichiometry into thesystem. Both generalizations introduce additional equilibria and bifurcations to the Rosenzweig-MacArthur model without stoichiometry.

The primary focus of this paper is to study the equilibria and bifurcations of the two-producer-one-consumer model with stoichiometry. The nutrient cycle in this model is closed. The primaryparameters are the growth rates of both producers, and the secondary parameter is the total nutri-ent in the system. Depending on the parameters, the possible equilibria are: no-life, one-producer,coexistence with both producers, the consumer coexisting with either producer, and the consumercoexisting with both producers. Limit cycles exist in the latter three coexistence combinations. Bi-furcation diagrams along with corresponding representative time series summarize the behaviorsobserved 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 beenintroduced into population models [1, 5, 6, 8, 12–15].

�Corresponding author. Email: [email protected]

ISSN: 1748-670X print/ISSN 1748-6718 onlinec© 200x Taylor & Francis

DOI: 10.1080/1748670YYxxxxxxxxhttp://www.informaworld.com


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Every organism has its own ratios of nutrients in its body, and those ratios arecalled “stoichiometric ratios.” Stoichiometric ratios are very specific for mostanimals, but plants have been observed to have a wide range of values [12].In this paper, a single stoichiometric ratio, N:C, represented by an unnamednutrient (N) and carbon (C) is tracked. Food that has a very low stoichiometricratio compared to the consumer’s ratio is not of as much benefit to the consumer.Stoichiometric models, especially those with a closed nutrient cycle, tend toexhibit a “paradox of enrichment”, a phenomenon in which the population of theconsumer declines in response to an increase in the population of the producer[4, 8, 14].

In 1963, Rosenzweig and MacArthur [11] suggested geometric forms forproducer growth and consumer response. It has become one of the classicproducer-consumer population models with a single currency, carbon. Formulas(1) are the equations for a version of the Rosenzweig-MacArthur model, usinglogistic producer growth and Michaelis-Menten consumer response with foodsaturation.

dPdt = αP − λP 2 − f(P )C

dCdt = ef(P )C − dC

f(P ) = βPχ+P

(1)

α is the natural growth rate of producers in the absence of predation.λ is the self limitation coefficient.e is the efficiency of turning predated food into consumers’ own biomass.d is the natural death rate of consumers.f(P ) is a Holling Type II or Michaelis-Menten function that describes consumerssaturation on food.β is the maximal death rate per encounter of producer due to predation.χ is the half-saturation constant for the Michaelis-Menton function.

A standard parameter to vary in (1) is the producer growth rate α. For low αthe ecosystem is a producer monoculture system because there is not enoughfood to maintain the consumer. Increasing α sufficiently to pass a transcriticalbifurcation value allows both producers and consumers to stably coexist. Furtherincrease in α results in a destablilization 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 α, but no further bifurcations areobserved.

In 2000, Loladze, Kuang and Elser (LKE) [8] introduced a producer-consumerpopulation model with stoichiometry, which has become a basis of compari-son for most subsequent stoichiometric models. Their model was based on theRosenzweig-MacArthur model and exhibits the “paradox of enrichment” [4, 14].


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The equations are:

dPdt = αP (1− P

min(K, (NT−θC)/q))− f(P )C

dCdt = e min(1, (NT−θC)/P

θ )f(P )C − dC(2)

α is the natural growth rate of producers in the absence of predation.K is the carrying capacity of producers.e is the maximal production efficiency.d is the natural death rate of consumers.NT is total amount of nutrient in the model.θ is a fixed stoichiometric ratio of consumers.q is the minimum stoichiometric ratio of producers.f(P ) is a function to describe consumers’ response to food. This is typically aMichaelis-Menton (Holling type II) fucntion, as in (1).

The total nutrient, NT is assumed to be fixed, consistent with a closed nutrientsystem. There are two stoichiometric terms, the producer’s carrying capac-ity, min(K, (NT − θC)/q), and the consumer’s biomass conversion efficiency,min(1, (NT−θC)/P

θ ). If the two stoichiometric “min” functions were each replacedwith the first arguments in the respective min functions, the system wouldrevert to the nonstoichiometric Rosenzweig-MacArthur model, equivalent toequation (1). (The equivalence is seen by replacing the first equation in (1)with dP

dt = αP (1 − PK ) − f(P )C. Varying K with this replacement results in

the same qualitative bifurcation sequence as just described above for varyingα in (1)). With the stoichiometric terms, however, varying K results in newbifurcations. The typical sequence mimics the bifurcations described above forthe Rosenzweig-MacArthur model through the Hopf bifurcation, but continuedincrease in K now results in the destruction of the limit cycle due to a thesaddle-node birth of equilibria on the limit cycle. The stable node born in thebifurcation replaces the limit cycle as the attractor. Continued increase in Kexhibits the paradox of enrichment. The consumer values of the equilibriumdecrease until a transcritical bifurcation value after which the consumer can nolonger exist. The nutrient is distributed in the large producer population, that isof in sufficient quantity for the consumer to survive.

A natural way to generalize a stoichiometric producer-consumer model is toallow for more species. For example, a variety of grazing species in a corn fieldwould require a model of one producer and multiple consumers [9]. Yet anecosystem of a garden of various flowers might be a model of multiple producers(flowers and green plants) and one consumer (a certain type of insect). In the pa-per “Herbivores, the Functional Diversity of Plants, and Cycling of Nutrients inEcosystems” by Pastor and Cohen [10], a model of one consumer, two producers,and a single currency (no stoichiometry) was addressed. The focus of the modelwas to determine the criteria of coexistence in terms of plants’ growth rates inopen and closed environments. An open environment means there are nutrientinputs from and exports to the surrounding environments for some components,and a closed environment corresponds to those being a fixed total amount ofnutrient cycling through the components in the model. Our model is open for


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carbon, but closed for its nutrient. It can be thought of as a generalization of thePastor-Cohen model, now with stoichiometry. Alternatively, it can be thoughtof as a generalization of the LKE model, with a second producer. Our stoichio-metric assumptions, however, differ in certain aspects from the LKE assumptions.

There are many interesting questions associated with a model of two producersand one consumer with stoichiometry: (i) Under what conditions can all threespecies stably coexist? In equilibrium? As a periodic limit cycle? (ii) How doesthe producer competition affect coexistence? What about other differences inproducer growth? (iii) What new transitions are possible when a system movesfrom one producer to two producers? This paper includes a bifurcation study– largely numerical – of the model of two producers and one consumer withstoichiometry, focusing on the questions above.

Much of the work presented in this paper was performed for the master’s thesisof the first author, under the supervision of the other three authors [7].

2. Model Construction

This model contains two populations of producers and a single population ofconsumers. Since this is a stoichiometric model, both quality and quantity ofproducers and consumers are measured. We choose to keep track of carbon as ameasure of quantity, and nutrient (later changed to the nutrient to carbon ratio)as a measure of quality. This leads to six varialbles: P1, P2, C,N1, N2, NC . We alsokeep track of the nutrient in the sediment, M . As we describe below, a sedimentclass for the carbon is not necessary. The model we will actually study is reducedfrom these seven variables to five by assuming a fixed amount of total nutrient,and a fixed stoichiometric ratio q = NC/C for the consumer. Details are providedbelow. Figure 1 illustrates the flow of carbon and cycling of a nutrient in thismodel. Variables of the model are summarized in table 1.

General Assumptions

The equations for our model are given below in equation (3). We now describethe assumptions which lead to the equations.

The biomass (carbon) of the producer is assumed to follow logistic growth.Note that we have chosen to make the producer growth independent of thenutrient. This independence assumption simplifies the analysis and focuses therole of stoichiometry on the food quality of the consumer. As is necessary instoichiometric models, the death terms are separated from the (logistic) growthterms in this model. This is the reason for the use of the “positive part” function,()+, in the first two equations of (3). Note also that there is no carbon sedimentcompartment in the model. It is possible to ignore this compartment becausewe assume that the rates of litter decomposition and nutrient mineralization areindependent of sediment nutrient concentration. Thus, mortality is consideredas an exit of carbon from the system. (See figure 1(a).)

This is a closed nutrient model. The amount of nutrient in the system is fixed.There are no external nutrient input and output in any components. In other


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words, all nutrients are cycling within the model. Let NT be the total sum ofall nutrient in system. Then, by this assumption, the amount of mineralizednutrient, M = NT −Nc −N1 −N2.

Lotka-Volterra competition is assumed between the two producers. The productof competition coefficients λ12λ21 is always less than the product of self-limitationcoefficients λ11λ22 [2]. Otherwise, the two producers could not even coexist inthe absence of the consumer.

The predation of both producers (consumer response) is assumed to followa Michaelis-Menten function. Both producers are food for the consumer; theprobabilities of producer P1 and P2 being eaten are assumed to be P1

P1+P2and

P2P1+P2

, respectively. That is, the producers are consumed in proportion to theirrelative abundance.

The consumer gains biomass through predation. The maximum conversionefficiency is assumed to be a constant, γ, without stoichiometry. However, itbecomes a minimum function of γ and Q1P1+Q2P2

q(P1+P2)when stoichiometry is taken

into account. The term Q1P1+Q2P2

q(P1+P2)is the ratio of the food source’s stoichiometry

to the consumer’s stoichiometry, N1+N2P1+P2

: q. When the food source containssufficient nutrient to support the maximum conversion efficiency γ, then therate γ is achieved. Otherwise, the conversion efficiency is reduced to Q1P1+Q2P2

q(P1+P2),

reflecting the maximum rate at which nutrient can be supplied to build struc-tural consumer biomass. Consumer death is assumed to be proportional to itspopulation size.

The producer gains nutrient via uptake from the mineralized nutrient, and losesnutrient due to predation and mortality. The nutrient uptake rate in this modelis proportional to the mineralized nutrient M = NT −Nc −N1 −N2. In contrastto other stoichiometric models we have seen, nutrient uptake is also assumedto be proportional to producer growth. This assumption appears to be realisticfor terrestrial plants [3]. Nutrient is transferred from producer to consumer viapredation, and returned to the nutrient pool when either the producers or theconsumer dies. The transpiration rates β1 and β2 are assumed to be constants.

Here are resulting differential equations for the model.

dP1dt = (b1 − λ11P1 − λ12P2)+P1 − dp1P1 − P1

P1+P2

α(P1+P2)h+(P1+P2)

C

dP2dt = (b2 − λ21P1 − λ22P2)+P2 − dp2P2 − P2

P1+P2

α(P1+P2)h+(P1+P2)

C

dCdt = min(γ, 1

qQ1P1+Q2P2

P1+P2) α(P1+P2)h+(P1+P2)

C − dcC

dN1dt = (NT −Nc −N1 −N2)β1(b1 − λ11P1 − λ12P2)+P1 − P1

P1+P2

α(P1+P2)h+(P1+P2)

CN1P1− dp1N1

dN2dt = (NT −Nc −N1 −N2)β2(b2 − λ21P1 − λ22P2)+P2 − P2

P1+P2

α(P1+P2)h+(P1+P2)

CN2P2− dp2N2

(3)


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To reduce the complexity of formulas (3), we use Qi = NiPi, i = 1, 2 to replace N1

and N2. Then calculating the derivatives of Q1 and Q2, we get the “stoichiometricfrom” of the system:

dP1dt = ((b1 − λ11P1 − λ12P2)+ − dp1 − αC

h+(P1+P2))P1

dP2dt = ((b2 − λ21P1 − λ22P2)+ − dp2 − αC

h+(P1+P2))P2

dCdt = (min(γ, 1

qP1Q1+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)+

(4)

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.

For our numerical investigrations, we fixed the following parameters:Self-limiting coefficients λ11 = λ22 = 0.5,Interference coefficients λ12 = λ21 = 0.2,Producer death rates dP1 = dP2 = 0.05,Consumer death rate dc = 0.17,Consumer stoichiometric ratio q = 0.05,The efficient conversation rate γ = 0.1,The maximal and half-saturation coefficients in the predation function α = 2.75and h = 0.75,Transpiration constants β1 = β2 = 0.3 (even if P1 = 0 or P2 = 0).

Parameters b1, b2, and NT are varied in our numerical experiments.

3. One-Producer-One-Consumer Model

In order to more easily understand the model (4), we first consider the special casewhere there is a single producer in the system. This one-producer-one-consumermodel is similar to the LKE model [8], but with different nutrient uptake assump-tions. We set P2 = 0 and ignore Q2 to obtain:

dP1dt = (b1 − λ11P1 − dP1 − αC

h+P1)P1

dCdt = (min(γ, Q1

q ) αP1h+P1

− dc)C

dQ1

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

(5)

The positive part functions from equations (4) have been dropped becausethe growth terms are non-negative in the invariant region of the following


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proposition.

Proposition 3.1: The following inequalities describe an invariant region of model (5):0 ≤ P1 ≤ b1

λ11, 0 ≤ C ≤ N

q , 0 ≤ Q1 ≤ NTβ, and (NT − qC −Q1P1) ≥ 0.

Proof : Show that all solutions which start on the boundary of the region will stayon the boundary or go to the interior of the region. See [7] for details.�

Equilibria

We calculate the following equilibria, with the variables listed in the order:(P1, C,Q1).

The no-life equilibrium (O): (0, 0, NTβ1), where neither producer nor consumerexist, but the (irrelevant) producer stoichiometric ratio is not necessarily zero.

The monoculture equilibrium (P1): ( b1−dp1λ11, 0, NTλ11β1

λ11+(b1−dp1 )β1). In the absence of

consumers, the population of producers follows the logistic growth and reachesits carrying capacity b1−dp1

λ11over time.

The coexistence equilibrium (P1C): There are two cases: H and L. In case H,the biomass conversion efficiency is a constant min(γ, Q1

q ) = γ. However, theconversion efficiency is min(γ, Q1

q ) = Q1

q in case L. We use subscripts to denotethe cases.

The coexistence equilibrium (P1C)H is ( dchαγ−dc ,

hγ(b1−dp1 )αγ−dc − γdcλ11h2

(αγ−dc)2 ,

β(NT (αγ−dc)−hqγ(b1−dp1 )+ kγqdch2

αγ−dc)

αγ−dc(hβ1−1) ). Note that, due to the consumption by con-sumers, the equilibrium population of the producer is independent of the growthrate b1.

Solving for the case L coexistence equilibrium, (P1C)L is more complicated. Bysetting the right hand sides of (5) to zero, solving in the third equation for Q1,solving in the second equation for C, and substituting these solutions into thefirst equation, it can be shown that the equilibria with nonzero P1 and C mustsatisfiy the following cubic equation in P1.

F (P1) = (P1)3c3 + (P1)2c2 + (P1)c1 + c0 = 0, (6)

wherec3 = −λ11

c2 = b1 − dp1 + dc − hλ11

c1 = (b1 − dp1 + dc)h− NTαq + dc

β1

c0 = hdcβ1

Thus, there can be up to three (P1C)L solutions. Along with one possible (P1C)Hsolution, there could exist up to four coexistence equilibria. Fewer are actuallyrealized, however, since the true coexistence solutions must also have positivepopulations, and satify the “consistency condition”: Q1

q ≥ γ for (P1C)H solutions,


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and Q1

q ≤ γ for (P1C)L solutions.

Bifurcations in b1

We provide representative bifurcation diagrams, varying b1, in figure 2. The totalnutrient NT is fixed at a relatively low value of 0.1. To illustrate the realizedequilibria, we plot Q1

q at the coexistence equilibria (P1C)H and (P1C)L againstb1 in figure 2(a). The portion of the cubic red curve representing the case Lequilibria is realized only when the curve is below γ = 0.1. Thus, the red dottedpart crossing regions I through IV and back across region IV is unrealized, whilethe dashed line in region III and the thick solid line in regions III, IV, and Vrepresent actual equilibria. Conversely, the portion of the slanted blue line rep-resenting the case H equilibria is realized only when the curve is above γ. Thus,the thin solid blue segment in region I, and the dashed part in regions II andIII represent realized equilibria, while the dotted blue segment part in regionsIV and V are not realized. The dashed blue segment in region II represents therepelling equilibrium, while the the two thin solid lines in region II represent therange of realized Q1

q along the attracting periodic orbit. The bold curves indicatethe biomass conversion efficiency on the attractor: γ at the case H equilibriumin region I, γ throughout most of region II — as long as the minimum value ofQ1

q along the periodic orbit stays above γ, a range of values between γ and theminimum realized Q1

q on the periodic orbit at the right hand part of region II,and at Q1

q for the case L equilibrium in regions III through V., solid to attracting,dashed to unstable, and dotted to unrealized. Red corresponds to case L, blue tocase H

Figures 2(b) and 3(c) are more traditional bifurcation diagrams with a fixedNT = 0.1. They correspond to figure 2(a). Bifurcations between regions areindicated on the figure, and identified as well in the chart for the two-parameterplane bifurcations presented in the next paragraph.

Bifurcations in b1 and NT

To better place the one-parameter bifurcation diagrams in b1 of figure 2 incontext, we show a two-parameter bifurcation diagram in b1 and NT in figure3. Corresponding time series are shown in figure 4. The solid curves indicatethe bifurcations resulting in a change of attractor; the dashed curves indicatebifurcations which do not change the attractor. In each time series, solutionsof P1, C, and magnified Q1 are plotted against the time. The table 2 describesthe bifurcations between the adjacent regions. Note that regions A and F, Band E are connected. Region O (0 ≤ b1 ≤ dP1), which is too small to shownin figure 3(b) and 3(c), is on the left to region A. As a comparison, figure 3(c)shows a bifurcation diagram of a model without stoichiometry. It is, of course,independent of NT since the model reverts to a single currency model.


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4. The Two-Producer-One-Consumer Model

We now return to analyze the full model of equation (4), with both producers,one consumer, and stoichiometry. The variables and parameters were already de-scribed when the model was developed in section 2. Much of the analysis we didparallels the analysis performed in section 3 with a single producer.

Proposition 4.1: The following inequalilties describe an invariant region of model (4):0 ≤ P1 ≤ b1

λ11, 0 ≤ P2 ≤ b2

λ22, 0 ≤ C ≤ NT

q , 0 ≤ Q1 ≤ NTβ1, 0 ≤ Q2 ≤ NTβ2, and(NT − qC −Q1P1 −Q2P2) ≥ 0.

Proof : Show that all solutions, which start on the boundary of the region willstay on the boundary or go to the region. See [7] for details.�

Equilibria

Similar to the one producer model, we first calculate the possible equilibria, withcoordinates listed in the order: (P1, P2, C,Q1, Q2). There are still 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).

Note that equilibria with P2 = 0 or P1 = 0 are same as the equilibria in thesingle-producer-single-consumer model in the previous section. We list theseequilibria first. Since Qi is not ecologically relevant when Pi = 0, we denote thecorresponding mathematical equilibrium value by the unspecified value Q∗i .

• The no-life equilibrium (O): (0, 0, 0, Q1∗, Q2

∗)

• The monoculture equilibrium (P1): (b1−dP1λ11

, 0, 0, NT β1λ11λ11+(b1−dP1 )β1

, Q2∗)

• The monoculture equilibrium (P2): (0,b2−dP2λ22

, 0, Q1∗, NT β2λ22

λ22+(b2−dP2 )β2)

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

, 0,γh(b1−dP1 )

αγ−dc− γdch

2λ11(αγ−dc)2

,

β1(NT (αγ−dc)−hqγ(b1−dP1 )− dcqγh2λ11

αγ−dc)

αγ−dc(hβ1−1), Q2

∗).

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

,γh(b2−dP2 )

αγ−dc− γdch

2λ22(αγ−dc)2

,

Q1∗,

β2(NT (αγ−dc)−hqγ(b2−dP21)− dcqγh2λ22

αγ−dc)

αγ−dc(hβ2−1)).

The following are the new coexistence equilibria, involving both producers,in the two-producer-one-consumer model. Note that the coexistence equilib-rium solution exists only when the growth terms (b1 − λ12P2 − λ11P1)+ and(b2 − λ21P1 − λ22P2)+ both are positive.

• The coexistence equilibrium without consumer (P1P2):

Set C = 0 and assume all other variables are positive. The equilibriumsolution is

(λ22(b1−dP1 )−λ12(b2−dP2 )

λ11λ22−λ12λ21,λ11(b2−dP2 )−λ21(b1−dP1 )

λ11λ22−λ12λ21, 0,

NT β1(λ11λ22−λ12λ21)(b1−dP1 )(β1λ22−β2λ21)+(b2−dP2 )(β2λ11−β1λ12)+λ11λ22−λ12λ21

,

NT β2(λ11λ22−λ12λ21)(b1−dP1 )(β1λ22−β2λ21)+(b2−dP2 )(β2λ11−β1λ12)+λ11λ22−λ12λ21

) .

In the absence of the consumer, our model reduces to a Lotka-Volterra com-petition model. Our simplifying assumption that the producer growth beindependent of the stoichiometry renders Q1 and Q2 irrelevant. Note thatin order to keep the equilibrium value of P1 and P2 positive, the product


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of self limitation terms, λ11λ22, must be “large” compared to the product ofcompetition coefficients λ12λ21. Note also that if b1 is sufficiently large, withother parameters being fixed, the equilibrium value of P2 becomes negative;that is, ecologically, P2 cannot persist. An analogous statement is true for largeb2. Consequently, coexistence of both producers requires the two birth rates berelatively close to each other.

• The coexistence equilibrium (P1P2C)H :

Note that in order to keep the components of the all-species-coexistenceequilibrium positive, the sum of the self-limitations λ11 + λ22 is assumed to belarger than the sum of the competition-limitations λ12 + λ21.

Assuming all variables are positive, the equilibrium solution is

((b1−dP1−b2+dP2−

dch(λ22−λ12)αγ−dc

)

λ11+λ22−λ12−λ21,

(b2−dP2−b1+dP1−dch(λ11−λ21)

αγ−dc)

λ11+λ22−λ12−λ21,

γh((b−1−dP1)(λ22−λ21)−(b2−dP2

)(λ11−λ12))

αγ−dc+γdch

2(λ11λ22−λ12λ21)(αγ−d−c)2

λ11+λ22−λ12−λ21,

β1(NT−q

γh((b−1−dP1)(λ22−λ21)−(b2−dP2

)(λ11−λ12))αγ−dc

+γdch

2(λ11λ22−λ12λ21)(αγ−d−c)2

λ11+λ22−λ12−λ21)

1+β1(b1−dP1

−b2+dP2− dch(λ22−λ12)

αγ−dc)

λ11+λ22−λ12−λ21+β2

(b2−dP2−b1+dP1

− dch(λ11−λ21)αγ−dc

)

λ11+λ22−λ12−λ21

,

β2(NT−q

γh((b−1−dP1)(λ22−λ21)−(b2−dP2

)(λ11−λ12))αγ−dc

+γdch

2(λ11λ22−λ12λ21)(αγ−d−c)2

λ11+λ22−λ12−λ21)

1+β1(b1−dP1

−b2+dP2− dch(λ22−λ12)

αγ−dc)

λ11+λ22−λ12−λ21+β2

(b2−dP2−b1+dP1

− dch(λ11−λ21)αγ−dc

)

λ11+λ22−λ12−λ21

).

• The coexistence equilibrium (P1P2C)L :

The procedure of finding (P1C)L was described in the last section. Similar to theone-producer-one-consumer model, there are up to three (P1P2C)L equilibria inthe two-producer-one-consumer model. The P1 equilibrium value of (P1P2C)Lalso turn out to be the roots of a cubic equation F (P1) (7).

F (P1) = (NT − qα(1

2(b1 + b2 − dP1 − dP2 − λ12b1−b2−dP1+dP2−P1λ11+P1λ21

λ12−λ22

−λ11P1 − λ21P1 − λ22b1−b2−dP1+dP2−P1λ11+P1λ21

λ12−λ22))

(h+ P1 + b1−b2−dP1+dP2−P1λ11+P1λ21

λ12−λ22))

(β1P1 + β2b1−b2−dP1+dP2−P1λ11+P1λ21

λ12−λ22)

− qdcα (1 + P1β1 + β2

b1−b2−dP1+dP2−P1λ11+P1λ21

λ12−λ22)

(h+ P1 + b1−b2−dP1+dP2−P1λ11+P1λ21

λ12−λ22)

(7)


11

Fixing the total nutrient NT and the second producer growth rate b2, a plot ofQ1P1+Q2P2

q(P1+P2)at the coexistence equilibria (P1P2C)H and (P1P2C)L against b1 is

shown in figure 5(a). Notation and line types are analogous to those describedfor figure 2(a). In particular, the bold curves indicate the biomass conversionefficiency on the attractor.

Figures 5(b) and 5(c) are two-parameter bifurcation diagrams for model (4)obtained by varying b1 and b2, but with a fixed NT = 0.1. The solid curves in-dicate the more significant bifurcations on the attractors, and the dashed curvesindicate some of the non-attractor bifurcations. The two-producer-one-consumercoexistence parameter combinations are highlighted in yellow. Figure 5(d) isa bifurcation diagram for a two-producer-one-consumer model without stoi-chiometry (min(γ, Q1P1+Q2P2

q(P1+P2)) = γ). The diagonal dashed line is a reference line

b1 = b2. The two dashed blue lines forming the wedge shape are the transcriticalbifurcations for two-producer coexistence and a single-producer coexistencewhen there is no consumer. The red solid curve is a transcritical bifurcation ofperiodic orbits: a periodic orbit passes through another periodic orbit in one ofthe PiC planes into the interior of the first “octant” of the coordinate plane. It isnumerically approximated by polynomial regression. The details of bifurcationsbetween regions are summarized in table 3.

Regions D, P, E, F, Z, N, and O are the “conjugates” of regions D, P , E, F, Z, N,and O, with the roles of P1 and P2 interchanged. Thus, the time series in theseconjugate regions are not shown. In each time series, solutions of P1, P2, and Care plotted against the time. Table 3 summarizes the bifurcations between theadjacent regions.We denote by W the region (0 ≤ b1 ≤ dP1)∪(0 ≤ b2 ≤ dP2). The region is too smallto resolve in figures 5(b)-(d). We do not provide time series for region W either;coexistence of the two producers is not possible here.

5. Experimental Implications

John - fill in here.

• One-producer - can vary b andNT independently. Two-producers - hard to varyb1 and b2 independently.

• Increase light - radial path through b1-b2 space.

6. Conclusions

The model developed and studied in this paper is a simplified model of twoproducers and one consumer with stoichiometry. It is a generalization of theRosenzweig and MacArthur model (see (1)), which is a single-producer-single-consumer model without stoichiometry. First, we introduced stoichiometry intoRosenzweig and MacArthur model. This left us with a model similar to the LKEmodel [8], but with somewhat different mechanisms for effecting stoichiometry.Afterward, we added a second producer.

For the single producer model (5), we chose to vary the growth rate, b1, and thetotal nutrient, NT , both which might be controlled in experiments. As illustratedin figure 3, except for extremely low values of NT (below approximately 0.2), the


12

model exhibits more than the three bifurcations typically observed in the Rosen-zweig and MacArthur model by varying either birth rate or carrying capacity.As the producer’s growth rate increases, with a fixed amount of nutrient in thesystem, there is a typical sequence of bifurcations: a first transcritical enablingthe producer to enter the system, a second transcritical enabling the consumerto enter the system, a Hopf bifurcation creating coexistence limit cycles, a firstsaddle-node destroying the cycle and replacing it with an attracting equilibrium,a second saddle-node taking the two “extra” saddle coexistence equilibria away,but leaving the attracting coexistence equilibrium, and a transcritical bifurcationafter which the consumer can no longer survive. See figures 2(a) and 3(a).

We note that this description is far from a complete analysis of model (5). Thereare many parameters which we have not varied and which could lead to differentbehavior. The bifurcation sequence described above differs from the sequence in[8] only by switching the last two bifucations. Similar models we have studiedhave the first saddle-node taking place off the limit cycle, resulting in a coexistingattracting limit cycle and attracting equilibrium. A later homoclinic destroys thelimit cycle and returns the equilibrium to the only attractor. Other scenarios arelikely present in the model as well.

The analysis of the two-producer-one-consumer model (4) provided the mostinteresting bifurcation scenario (see figures 5(b) and 5(c)). We varied the totalnutrient NT in our numerical experiments, but presented the two-parameterbifurcation diagram in b1 and b2 for only one fixed value of NT . As in the one-producer model, there are many other parameters which we kept fixed, so thebifurcation scenario we found is again far from a complete analysis of (4). Evenwith this limitation, however, we have observed new and interesting bifurcationsin this two-parameter space. Of special note are the parameter region (coloredyellow) in which all three species can coexist. This region is, of course, inside thewedge-shaped region which describes the coexistence region of two producersin the absence of any consumer. Also noteworthy is the existence of transcriticalbifurcations of peroidic orbits which occur at different parameter values fromtranscritical bifurcations of equilibria. Sufficiently high growth rates in anydirection in the b1-b2 parameter plane results in the extinction of the consumer.This is the paradox of energy enrichment.

To summarize, the introduction of a stoichiometry to our producer-consumermodel causes the loss of large amplitude limit cycles. Because the total nutrientNT is fixed, a sufficiently high growth in producer(s) results in the extinction ofthe consumer. Adding second competing producer creates additional interestingbifurcations, allowing us to see the somewhat complicated criteria for coexistenceof all three species.

Acknowledgements

The authors would like to thank the referees very much for their valuable com-ments and suggestions.


13

Table 1. Table of VarialesVariables DefinitionsP1 Population Density of

Producer P1 (Carbon)P2 Population Density of

Producer P2 (Carbon)C Population Density of

Consumer C (Carbon)N1 Nutrient of Producer P1

N2 Nutrient of Producer P2

NC Nutrient of Consumer CM Mineralized Nutrient

Q1 =NP1P1

Stoichiometric Ratio of P1

Q2 =NP2P2

Stoichiometric Ratio of P2

Table 2. Bifurcations Between The Adjacent Regions in Figure 3(a)Regions Type of Bifurcation Involved Attractors Non-Attractors TransitionsO→ A Transcritical (O)→ (P1)A→ B Transcritical (P1)→ (P1C)HB→ C Hopf (P1C)H

→ limit cycleC→ D Saddle-node limit cycle Unstable (P1C)L also born.

→ Stable (P1C)LD→ E Non-Smooth Stable (P1C)L Unstable (P1C)H and unstable (P1C)L

Saddle-node → Stable (P1C)L are eliminated.E→ F Transcritical (P1C)L→ (P1)

Table 3. Bifurcations Between The Adjacent Regions in Figures 5(b) and 5(c)Regions Type of Bifurcation Involved Attractors Non-Attractors TransitionsW→ A Transcritical (O)→ (P1P2)W→ D Transcritical (O)→ (P1)D→ A Transcritical (P1)→ (P1P2)A→ B Transcritical (P1P2)→ (P1P2C)HD→ E Transcritical (P1)→ (P1C)HE→ B Transcritical (P1C)H → (P1P2C)HB→ C Hopf (P1P2C)H

→ Interior limit cycleE→ F Hopf (P1C)H

→ P1C plane limit cycleF→ Z Transcritical Limit cycle in P1C plane

→ the interior limit cycleZ→ C Transcritical Interior limit cycle (P1C)H → (P1P2C)H

→ Interior limit cycleC→ J Saddle-node Interior limit cycle Unstable (P1P2C)L is born.

→ Stable (P1P2C)LZ→ J Saddle-node Interior limit cycle Unstable (P1P2C)L is born.

→ Stable (P1P2C)LF→ N Saddle-node P1C plane limit cycle Unstable (P1C)L is born.

→ Stable (P1C)LN→ J Transcritical (P1C)L→ (P1P2C)LJ→ K Non-Smooth Stable (P1P2C)L Unstable (P1P2C)H and

Saddle-node → Stable (P1P2C)L unstable (P1P2C)L areeliminated.

N→ O Non-Smooth Stable (P1C)L Unstable (P1C)H andSaddle-node → Stable (P1C)L unstable (P1C)L are

eliminated.O→ J Transcritical (P1C)L→ (P1P2C)LO→ K Transcritical (P1C)L→ (P1P2C)LK→M Transcritical (P1P2C)L→ (P1P2)O→ P Transcritical (P1C)L→ (P1)P→M Transcritical (P1)→ (P1P2)


14

(a) Flow of Carbon (b) Nutrient Cycling

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

(a) Plot of Q1q

at Coexistence Equilibria (b) Producer Bifurcation Diagram with afixed NT = 0.1

(c) Consumer Bifurcation Diagram witha fixed NT = 0.1

Figure 2. One-parameter bifurcation diagrams for One-Producer-One-Consumer Model. (a) Plot of Q1q

at po-

tential equilibria (and range of Q1Q

along periodic orbit in region II). The realized conversion efficiencies ofrealized equilibria and periodic orbits are in bold. Red corresponds to case L; blue corresponds to case H. Solidindicates stable; dashed indicates unstable; dotted indicates unrealized. (b) Equilibrium values of P1 versus b forequilibrium solutions; range of P1 values for periodic orbits in region II. Solid lines represent stable equilibria.Dashed lines represent unstable equilibria. Dotted lines represent unrealized equilibria. (See text.) (c) Same as(b) but for C versus b.


15

(a) Bifurcation Diagram With Stoichiom-etry

(b) Bifurcation Diagram Without Stoi-chiometry

Figure 3. Two-parameter bifurcation diagrams in b and NT for the One-Producer-One-Consumer Model. Seefigure 4 for representative time series corresponding to each region.

High N:C

20 40 60 80 100t

0.2

0.4

0.6

0.8

1.0

population�ratio

Producers:Red, Consumers:Blue,1 * Producer Nutrient-Carbon Ratio: Green

(a) Region A

High N:C

50 100 150 200t

0.5

1.0

1.5

2.0

population�ratio


(b) Region B

High N:C

50 100 150 200t

1

2

3

4

population�ratio


(c) Region C

High N:CLow N:C

20 40 60 80 100t

5

10

15

20

25

population�ratio


(d) Region D

High N:CLow N:C

10 20 30 40t

5

10

15

20

25

population�ratio


(e) Region E

High N:CLow N:C

20 40 60 80 100t

5

10

15

20

25

population�ratio


(f) Region F

Figure 4. Representative time series corresponding to parameter regions in figure 3.


16

(a) Plot of Q1P1+Q2P2q(P1+P2)

at CoexistenceEquilibria

(b) Bifurcation Diagram

(c) Bifurcation Diagram (Zoom) (d) Bifurcation Diagram Without Stoi-chiometry

Figure 5. Two-parameter bifurcation diagrams for Two-Producer-One-Consumer Model. (a) Plot ofQ1P1+Q2P2q(P1+P2)

at potential equilibria (and range of Q1P1+Q2P2q(P1+P2)

along periodic orbit in region II). The realizedconversion efficiencies of realized equilibria and periodic orbits is in bold. Red corresponds to case L; blue cor-responds to case H. Solid indicates stable; dashed indicates unstable; dotted indicates unrealized. (b) and (c)Two-parameter bifurcation diagrams in b1 and b2 for the Two-Producer-One-Consumer Model. See figure 4 forrepresentative time series corresponding to each region.


17

50 100 150 200 250 300t

0.5

1.0

1.5

2.0

PopulationsP1:Blue P2:Green C:Red

(a) Region A

100 200 300 400t

0.5

1.0

1.5

2.0


(b) Region B

100 200 300 400t

0.5

1.0

1.5

2.0

2.5

3.0


(c) Region C

20 40 60 80 100t

0.5

1.0

1.5

2.0

2.5


(d) Region J

5 10 15 20t

2

4

6

8

10


(e) Region K

2 4 6 8 10t

5

10

15


(f) Region M

100 200 300 400t

0.5

1.0

1.5

2.0


(g) Region D

100 200 300 400t

0.5

1.0

1.5

2.0


(h) Region E

100 200 300 400t

1

2

3


(i) Region F

50 100 150 200 250 300t

1

2

3

4


(j) Region Z

5 10 15 20t

0.5

1.0

1.5

2.0

2.5

3.0


(k) Region N

5 10 15 20t

2

4

6

8

10

12

14Populations

P1:Blue P2:Green C:Red

(l) Region O

2 4 6 8 10t

5

10

15

20

25

30


(m) Region P

Figure 6. Representative time series corresponding to parameter regions in figure 5.


18 REFERENCES

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