A Bifurcation Analysis of a Differential Equations Model

Bulletin of Mathematical Biology (2006)DOI 10.1007/s11538-006-9070-3

ORIGINAL ARTICLE

A Bifurcation Analysis of a Differential EquationsModel for Mutualism

Wendy Gruner Gravesa,∗, Bruce Peckhamb, John Pastorc

aLake Superior College, Duluth, MN, USAbDepartment of Mathematics and Statistics, University of Minnesota Duluth, MN, USAcDepartment of Biology, University of Minnesota Duluth and NRRI, MN, USA

Received: date / Accepted: date / Published online: dateC© Society for Mathematical Biology 2006

Abstract We develop from basic principles a two-species differential equationsmodel which exhibits mutualistic population interactions. The model is similar inspirit to a commonly cited model [Dean, A.M., Am. Nat. 121(3), 409–417 (1983)],but corrects problems due to singularities in that model. In addition, we inves-tigate our model in more depth by varying the intrinsic growth rate for each ofthe species and analyzing the resulting bifurcations in system behavior. We are es-pecially interested in transitions between facultative and obligate mutualism. Themodel reduces to the familiar Lotka–Volterra model locally, but is more realisticfor large populations in the case where mutualist interaction is strong. In partic-ular, our model supports population thresholds necessary for survival in certaincases, but does this without allowing unbounded population growth. Experimen-tal implications are discussed for a lichen population.

Keywords

1. Introduction

Mutualism is defined as an interaction between species that is beneficial for bothspecies. A facultative mutualist is a species that benefits from interaction with an-other species, but does not absolutely require the interaction, whereas an obli-gate mutualist is a species that cannot survive without the mutualist species. Thereare many interesting examples in ecology of mutualist interactions, as well as anumber of mathematical models for two-species mutualism in the literature, al-though the volume of theoretical work on mutualism is dwarfed by the volumeof work dealing with predator-prey and competition interaction. For a review and

∗Corresponding author:E-mail addresses: [email protected].

Bulletin of Mathematical Biology (2006)

discussion of mutualism models through the mid 1980s—still frequently referredto—see Wolin (1985).

In this paper, we develop a model that can be used to describe both obligate andfacultative mutualism, as well as transitions between the two. These transitionsmay be of interest in understanding populations whose birth rates are influencedby controllable factors such as the environment (see, for example Hernandez,1998). Transitions between different types of mutualism are also important froman evolutionary perspective.

One commonly cited reference, developed to account for both facultative andobligate mutualism, was presented by Dean (1983). To limit population growth,Dean introduced a model for two mutualistic populations where each popula-tion’s carrying capacity saturated as the other population increased. This assump-tion prevented the positive feedback between the two mutualists from causing thesolutions to grow without bound.

We found Dean’s model appealing, but upon examination determined that therewas a problem with the derivation of the equations, and therefore in applying themodel to the case of obligate mutualism, and therefore also to the transition be-tween facultative and obligate cases. Briefly, as carrying capacity passed throughzero, a singularity in the model moved into the first quadrant of the phase spaceand made the model behavior biologically unreasonable when either populationwas obligate. See Appendix I for further explanation.

The model we present here is similar in spirit to Dean’s model. We call it the“limited per capita growth rate (LGR) mutualism model.” Like Dean’s model,it features saturating benefits to both populations but unlike Dean’s model it as-sumes that each mutualist asymptotically enhances the other’s growth rate ratherthan directly affecting carrying capacity. It produces results qualitatively similar toDean’s model when both populations are facultative, but eliminates the problemswith Dean’s model when either mutualist is obligate. Thus, our model can be usedto describe facultative–facultative, facultative–obligate, obligate–obligate mutual-ism as well as smooth dynamical transitions that may occur between and amongthese types of mutualism. It may therefore be useful in guiding further experimen-tal studies and in the theory of the evolution of different types of mutualism.

We analyze the models in this paper using a bifurcation point of view. Bifurca-tion analysis is relatively new to ecology (see Kot, 2001 for some examples), andto our knowledge has not been applied to understanding how different types ofmutualism relate to one another. In our approach we identify two of our parame-ters as “primary” and the remaining as “auxiliary.” We choose the two parameterswhich determine the birth rates of each of our populations in the absence of theother (their signs determining facultative vs. obligate), as our primary parameters.In general, we fix a set of auxiliary parameters, compute bifurcation curves whichdivide the primary parameter plane into equivalence classes, and provide repre-sentative phase portraits for each class. We then use the bifurcation diagrams andphase portraits to determine the implications for the ecology of the populations

The remainder of the paper is organized as follows. The LGR model is devel-oped in Section 2. In Section 3, we analyze the model. It turns out that, to lowestorder terms, our model reduces to the well-known Lotka–Volterra (LV) model.Thus, a byproduct of our analysis is a bifurcation analysis of the LV model. In


Section 4 we describe a lichen symbiosis to which our model can be applied andpossible experiments to parameterize and test our model. Results are summarizedin Section 5, we point out the problems in the original Dean model in Appendix I,and we include equations for nullclines of the LGR model as well as informationregarding equilibria and eigenvalues for both models in Appendix II.

2. Development of the model

We develop our two species model with the following assumptions:

A1: The logistic assumption: Each species behaves according to the logistic model.A2: The growth rate assumption: Each species affects the other species’ per capita

growth rate, but not its self limitation.A3: The mutualism assumption: The increase in each species cannot harm the

other species.A4: The limited benefit assumption: There is a maximum per capita growth rate

attainable for each species.A5: The proportional benefit assumption: The marginal rate of change of the per

capita growth rate of each species due to the increase of the other speciesis proportional to the difference between the maximum growth rate and thecurrent growth rate.

Assumptions A1 and A2 lead to the following general form:

dxdt

= R1(y)x − a1x2

dydt

= R2(x)y − a2 y2 (1)

Assumption A3 can be stated mathematically as R′1(y) ≥ 0 and R′

2(x) ≥ 0. As-sumption A4 can be stated mathematically as the existence of maximum growthrates r11 for species x, and r21 for species y, satisfying R1(y) ≤ r11 and R2(x) ≤ r21.Assumption A5 can be restated as R′

1(y) = k1(r11 − R1(y)) and R′2(x) = k2(r21 −

R2(x)). These two linear differential equations can be easily solved to obtain

R1(y) = r10 + (r11 − r10)(1 − e−k1 y)

R2(x) = r20 + (r21 − r20)(1 − e−k2x) (2)

where the parameters r10 and r20 are the respective unaided growth rates of eachspecies: r10 = R1(0), and r20 = R2(0). The combination of Eqs. (1) and (2) aboveleads to the form of the limited per capita growth rate mutualism model studied inthis paper:

dxdt

= (r10 + (r11 − r10)(1 − e−k1 y))x − a1x2

dydt

= (r20 + (r21 − r20)(1 − e−k2x))y − a2 y2 (3)


Assumption A1 requires a1 > 0 and a2 > 0; assumption A3 requires r10 ≤ r11 andr20 ≤ r21; assumption A5 requires k1 > 0 and k2 > 0. Note that for species x(y) tohave any chance of survival, it must be true that r11 > 0(r21 > 0).

2.1. Discussion

The development of our model parallels the development in Dean (1983) withthe significant difference that we saturate the per capita growth rate instead ofsaturating the carrying capacity directly. An alternative model could have beendeveloped assuming that the mutualism acted through the quadratic term insteadof or in addition to the per capita growth rate term, but we chose to stay withthe growth rate term because it seemed to fit the population interaction we had inmind (see Sec. 4). In addition, the resulting model exhibited both key behaviorswe expected from realistic mutualist populations: bounded population growth andthe existence of threshold population values below which populations die out andabove which populations persist.

2.2. Parameter (non)reduction

A common mathematical technique at this point is to rescale both the x and y vari-ables to eliminate (that is, “make equal to one”) parameters a1 and a2. We choosenot to make this parameter reduction in order to retain the original interpretationof the parameters.

3. Analysis of the model

In this section we perform a bifurcation analysis of the limited per capita growthrate mutualism model in Eq. (3). For fixed values of the auxiliary parameters ourgeneral goal is to divide the r10–r20 parameter plane into “equivalence classes,”where two differential equations are defined to be equivalent if their “phaseportraits” are qualitatively the same. Formally, we are using “topological or-bital equivalence.” See, for example, Guckenheimer and Holmes (1983), Strogatz(1994), or Robinson (2004).

We display our main results for the LGR model in Figs. 4 and 5 via bifurcationdiagrams in the r10–r20 parameter plane that illustrate the equivalence classes, andaccompanying phase portraits in the x–y plane. The parameter plane curves areall standard codimension-one bifurcations: transcritical, saddle-node, Hopf, ho-moclinic, and heteroclinic bifurcations. See introductory dynamical systems texts(Guckenheimer and Holmes, 1983; Strogatz, 1994; Hirsch et al., 2004; Robinson,2004) for further explanation. Codimension-two points are discussed briefly in Sec-tion 3.3.5. The transcritical bifurcation is classified as codimension-one instead ofcodimension-two because of the coordinate axes being invariant. This is a com-mon feature of population models. Some bifurcation curves are determined ana-lytically, while others are numerically followed using the dynamical systems soft-ware To Be Continued . . . (Peckham, 1986–2004). Finally we use the bifurcationdiagrams to identify transitions which affect the first quadrant of the phase space,


and are therefore “ecologically significant.” The phase portraits include the fol-lowing: nullclines, equilibria, the stable and unstable manifolds of any saddles, andperiodic orbits.

We note that periodic orbits are impossible in the first quadrant of phase spacefor mutualism models in the general form of equation (1) and satisfying assump-tions A1, A2, and A3 in Section 2. This result can be proved geometrically bycontradiction: if there were a periodic orbit, the conditions on the dx/dt equa-tion would allow only a clockwise flow, while the conditions on the dy/dt equationwould allow only a counterclockwise flow. (It can also be proved algebraically thatequilibria cannot have complex eigenvalues by showing that the discriminant ofthe Jacobian derivative is always positive in the first quadrant of phase space. Thisprecludes the possibility of the birth of a first quadrant periodic orbit through aHopf bifurcation.)

It turns out that locally (in variables x, y, r10, r20) our model reduces to the famil-iar Lotka–Volterra (LV) model of mutualism, so we provide bifurcation diagramsfor this better-known case for comparison with our LGR diagrams. A similar re-duction is mentioned by Goh (1979) for the phase variables x and y only. Thereare two subcases to consider: weak mutualism and strong mutualism. We will dis-tinguish these two subcases after presenting the reduction.

3.1. Local reduction to the Lotka–Volterra model.

By expanding the exponential terms in equation (3) in a Taylor series, one canrewrite the limited growth rate model—up through quadratic terms in the phasevariables and the primary parameters together—as

dxdt

= (r10 + k1r11 y)x − a1x2 + O(xy2, r10xy)

dydt

= (r20 + k2r21x)y − a2 y2 + O(x2 y, r20xy) (4)

Thus, at least for values of x, y, r10, and r20 close to zero, the dynamics of ourlimited growth rate model should match that of the familiar Lotka–Volterra (LV)interaction model:

dxdt

= (r10 + b1 y)x − a1x2

dydt

= (r20 + b2x)y − a2 y2 (5)

with parameters r10, r20, a1, a2, b1, b2 in the LV model corresponding to parametercombinations r10, r20, a1, a2, k1 r11, k2 r21 in our LGR model (Eq. 3), respectively.

Note that the LV model could have been derived on its own using assump-tions A1–A3 in Section 2, but replacing assumptions A4 and A5 with the as-sumption that the change in the per capita birth rate of each species dependslinearly on the other. This results in the same general form of equation (4), butwith R1(y) = r10 + b1 y and R2(x) = r20 + b2x. Assumption A4 no longer holdssince neither population’s growth rate is bounded. Because of our assumptions of


mutualism (A3) and self-limitation (A1), b1 and b2 are assumed to be nonnegative,and a1 and a2 are assumed to be positive.

3.2. Features of the Lotka–Volterra model

Since the LV model is a standard beginning model for interacting populations,and because it is clearly related to our LGR model, we begin with some commentsabout the LV model. Certain aspects of this analysis have been well-known for along time (see, for example, Vandermeer and Boucher, 1978; Goh, 1979; Wolin,1985; Kot, 2001), but the bifurcation diagrams we present here appear to be lesswidely known.

3.2.1. Weak vs. strongBefore presenting the bifurcation diagrams in our primary parameter space—(r10,r20)—we place the mutualism in context with better-known predator-prey andcompetition models by providing a bifurcation diagram of the “interaction param-eter space.” This space is determined by the interaction coefficients b1 and b2 fromEq. (5) and is illustrated in Fig. 1. The self-limitation coefficients a1 and a2 are as-sumed to be constant. In this paper, we are concerned with the first quadrant of theinteraction space, where each population benefits from the presence of the other.It turns out that there is a natural division between “weak” and “strong” mutu-alist interactions when a1a2 = b1b2. (See, for example, Vandermeer and Boucher,1978, or Goh, 1979.) The division can be seen by considering the nullclines. Lookahead to compare, for example, Fig. 2Bi (weak facultative–facultative LV mutu-alism) and 3Bi (strong facultative–facultative LV mutualism). In the weak case(a1a2 > b1b2), self-limitation dominates the mutualist interaction, and the slope ofthe dx/dt = 0 nullcline is steeper than the dy/dt nullcline; the reverse is true forthe strong (a1a2 < b1b2) case. Populations under weak LV mutualism always staybounded, and open first quadrant equilibria, when they exist, are always stable.Because strong positive feedbacks between the two species dominate each other’sgrowth when a1a2 < b1b2, strong LV mutualism always allows some populationsto increase without bound, and precludes the possibility of a stable equilibrium inthe first quadrant. This is a significant shortcoming of the LV model.

Interestingly, the curve a1a2 = b1b2 also appears in the third quadrant where itdivides the cases of weak competition (that need not exhibit competitive exclusion)from strong competition (that must exhibit competitive exclusion).

3.2.2. Simple LV bifurcation diagramsThe r10–r20 LV bifurcation diagrams are required to be relatively simple asguaranteed by the following lemma.

Lemma 1 The r10–r20 bifurcation diagrams for the Lotka–Volterra populationmodels are all straight line rays from the origin. (Aside: this lemma is true for com-petition and predator-prey models as well as for mutualism.)

The lemma can be proved by showing that any differential equation on a ray inthe parameter plane can be converted into any other differential equation on the


Fig. 1 Lotka–Volterra interaction space: b2 vs. b1. The curve for b1b2 = a1a2 appears in the firstand third quadrants and is calculated with a1 = 1 and a2 = 1. In the first quadrant, mutualismswith coefficient Cartesian pairs that lie below/left of the curve give rise to weak (stable) mutu-alisms whereas mutualisms with coefficient Cartesian pairs that lie above/right of the curve giverise to strong (unstable, with unbounded growth) mutualisms. In the third quadrant, competitivespecies with coefficient Cartesian pairs that lie above/right of the curve give rise to competitionthat need not exhibit competitive exclusion whereas competitive species with coefficient Carte-sian pairs that lie below/left of the curve give rise to competition that does exhibit competitiveexclusion.

ray by rescaling the two phase variables and time. Two systems being related bysuch a change of variables implies that the two systems are equivalent. (See textssuch as Strogatz (1994) or Hirsch et al. (2004)). The consequence is that we onlyneed to consider a small circle around the origin in the parameter plane in orderto understand the bifurcations in the full parameter plane. An interesting furtherconsequence of this rescaling is that the phase portrait on the “negative” of a raycan be obtained by reflecting the phase space about the x and y axes, and reversingthe direction of the time flow arrows.

3.3. The main results: r10–r20 bifurcation diagrams

The main results of this paper are the LGR bifurcation diagrams for Eq. (3) pre-sented in Figs. 4 (locally weak: a1a2 > k1r11k2r21) and 5 (locally strong: a1a2 <


Fig. 2 Weak Lotka–Volterra mutualism, (a1a2 > b2b1).: (A) The bifurcation diagram in r20 vs.r10. Key: thick lines = “ecologically significant,” tc = transcritical; the labels of the equilibriainvolved in the bifurcation (see Section 3.3) precede the bifurcation abbreviation. (B) Repre-sentative phase portraits for the corresponding regions in (A). Dashed lines are nullclines; solidlines are stable or unstable manifolds of saddle equilibria. By interchanging x and y, similar phaseportraits would result for regions labeled with primed numbers. Auxiliary parameters: a1 = 1,a2 = 1, b1 = 1, and b2 = 0.5. Primary parameters: r1 = 1, r2 = 1 for (i); r1 = 1, r2 = −0.25 for (ii);r1 = 1, r2 = −0.75 for (iii); r1 = 0.5, r2 = −1 for (iv); and r1 = −1, r2 = −1 for (v).

k1r11k2r21). The terminology and inequalities follow from Sections 3.1 and 3.2.Because they are intimately related to the “local” part of the LGR bifurcationdiagrams, and because the LV bifurcation diagrams are of interest in their ownright, we present the simpler LV diagrams for Eq. (5) first. We treat weak LV insubsection 3.3.1 (Fig. 2) and strong LV in Section 3.3.2 (Fig. 3). The description ofthe LGR diagrams follow in Sections 3.3.3 (locally weak) and 3.3.4 (locally strong).

Equilibria: The LV model admits four equilibrium points, although some orall of them could coincide at certain parameter values. We label them in Figs. 2and 3 as follows: 0—origin, 1—x-monoculture, 2—y-monoculture, 3—coexistence.Analogously, the LGR model can have between 3 and 5 equilibrium points: 0—origin, 1—x-monoculture, 2—y-monoculture, 3 and 4—coexistence. Unlike the LV


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3.3.1. Lotka–Volterra weak mutualism case self-limitation dominatesmutualism (a1a2 > b1b2)The dynamics of the weak LV mutualism case is summarized in the r10–r20 bifurca-tion diagram in Fig. 2. A convenient way to view the phase portraits in Fig. 2B is tostart with Fig. 2Bi. Think of the nullclines as titled from horizontal and vertical—where they would have been if there had been no mutualism (b1 = b2 = 0)—tooblique. The coexistence equilibrium occurs at the intersection of the two obliquenullclines. This results in coexistence population values which are greater for eachspecies than the respective carrying capacity for each species in the absence of theother. The remaining phase portraits can be understood by studying the transitionsfrom Fig. 2Bi.

The four transcritical bifurcation curve formulas, with the correspond-ing equilibrium labels in parentheses, are: r10 = 0(0–1), r20 = 0(0–2), r20 =−(b2/a1)r10(1–3), r20 = −(a2/b1)r10(2–3). The bifurcation curves in Fig. 2A whichare “ecologically significant” are the ones which effect changes in the first quad-rant of the x–y space. This results in four ecological classes, distinguished in thebifurcation diagram by shading, and separated by the thick curves. Stable coexis-tence occurs in regions i, ii, and ii′; a stable x-monoculture occurs in regions iii, iv;a stable y-monoculture occurs in regions iii′, iv′; and extinction in region v.

Note especially that in regions ii and ii′ the stable coexistence occurs even thoughone of the populations is an obligate mutualist. One could argue that cases ii and ii′

are ecologically different from case i, but we choose to combine them due to theirsimilar behavior in the open first quadrant. Compare with Fig. 1 in Vandermeerand Boucher (1978) where they list four sub cases for weak (called “stable” intheir paper) Lotka–Volterra mutualism. Their four cases S1–S4 correspond to ourcases i, v, ii, and iv, respectively. They do not, however, provide a parameter spacebifurcation diagram organizing the cases.

3.3.2. Lotka–Volterra strong mutualism case: mutualism dominatesself-limitation (a1a2 < b1b2)The strong LV mutualism case is summarized in Fig. 3. The four transcritical bi-furcations are similar to the weak mutualism case, but the transitions between iiiaand ivb in the fourth quadrant of the parameter space are quite different from theweak mutualism case. In some sense they are not so relevant to the ecology of thesystem since the dynamics in the first quadrant of phase space is unaffected in thissequence of transitions. On the other hand, the transitions involving other quad-rants of phase space are important in order to “set up” the first quadrant changes.And they are certainly of interest from a dynamical systems point of view.


The transition from iiia (not shown) to iiib is a change from real eigenvalues tocomplex eigenvalues for equilibrium 3, denoting a visual change from nonspiral-ing orbits to spiraling ones, but no formal topological change. The transition fromiiib to iva is actually a degenerate bifurcation: a “Hopf” curve and a heterocliniccurve. As the parameters are varied from region iiib to region iva equilibrium 3changes from repelling to attracting, its eigenvalues passing through pure imag-inary. Simultaneously, there is a crossing of the right hand branch of the stablemanifold of equilibrium 2 with the left hand branch of the unstable manifold ofequilibrium 1. On the bifurcation curve, the two branches coincide, creating theheteroclinic connection and equilibrium 3 appears to be a center. The coincidenceof the two bifurcation curves is not typical of nonlinear models. It is present be-cause of the truncation of the differential equation at the quadratic terms, and itseparates into two distinct curves—a heteroclinic curve and a nondegenerate Hopfcurve—in the bifurcation diagram for the locally strong LGR model. We againhave an equal eigenvalue transition for equilibrium 3 between iva and ivb (notshown). This leaves the eigenvalues of equilibrium 3 real in preparation for the2–3 transcritical bifurcation between ivb and v. The formulas for the transcriticalcurves are the same as in the weak mutualism case. The other bifurcation curveswere numerically continued using the software TBC (Peckham, 1986–2004).

All phase portraits for this case have orbits in the first quadrant for which bothpopulations grow without bound. No region exhibits a stable, open first quadrant(feasible in the terminology of Goh (1979)) coexistence equilibrium. The shadingin Fig. 3A illustrates the four ecologically distinct regions of behavior: in regionsi, ii, ii′, iii, iii′, iv, and iv′, all orbits in the open first quadrant are unbounded; inregion v, there is a threshold curve (the stable manifold of equilibrium 3, which isa saddle) which separates first quadrant initial conditions which have unboundedorbits from those that approach an x-monoculture (and are therefore bounded);in region v′, the threshold curve separates unbounded orbits from those for whichpopulations approach a y-monoculture; and in region vi, the threshold curve sep-arates unbounded orbits from those for which both populations die out. We haveindicated regions exhibiting ecologically significant thresholds by adding the pat-terned dots over the shading in this and the remaining bifurcation diagrams.

Compare these results with Fig. 1 in Vandermeer and Boucher (1978), this timewith the four cases for strong (called “unstable” in their paper) Lotka–Volterramutualism. Their four cases U1–U4 correspond to our cases i, vi, ii, and v, respec-tively.

3.3.3. Locally weak LGR model: a1a2 > k1r11k2r21

The behavior of the LGR model in this case is illustrated in Fig. 4. Due to the localreduction of Section 3.2, the bifurcation scenario near the origin in the r10–r20 planeis qualitatively the same as the weak Lotka–Volterra bifurcation scenario in Fig.2. The similarity actually extends to large positive population values because thedominance of the self-limitation over the mutualism, assumed for small populationvalues, still holds for large population values—even more so since the mutualismbenefit in the LGR model is limited. The labels of the bifurcations are adjustedbecause they involve equilibrium 4, instead of the equilibrium labeled 3 in theLotka–Volterra analysis.


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There are, however, small differences between Figs. 2 and 4. Recall from thederivation of the model in Section 2 that r10 ≤ r11 and r20 ≤ r21, so only the“quadrant” of the r10–r20 parameter plane determined by these inequalities isrelevant in Fig. 4A. In the phase plane, the shape of the two nonlinear nullclinesfor small values of r10 and r20 forces them to intersect a second time in the thirdquadrant—at equilibrium 3 in the fourth quadrant as well as at equilibrium 4 nearthe origin. Compare, for example, Figs. 2Bi with 4Bi. These two equilibria canbe lost in a saddle-node bifurcation, but it cannot happen in the first quadrant ofphase space, so while this bifurcation is mathematically significant, it is not ecolog-ically significant. We still obtain the same four ecologically distinct phase portraits,indicated by the shading in Fig. 4A: stable coexistence in regions i, ii, ii′; a stablex-monoculture in regions iii, iv, vi, vii; a stable y-monoculture in regions iii′, iv′, vi′,vii′; extinction in regions v and viii.

We comment toward the end of Section 3.3.5 on the codimension-two point la-beled “sn-tc” in Fig. 4A.

3.3.4. Locally strong LGR model: a1a2 < k1r11k2r21

The behavior of the LGR model in this case is illustrated in Fig. 5. It is the most“interesting” of our bifurcation diagrams, both mathematically and ecologically.Due to the local reduction of Section 3.2, the bifurcation scenario near the ori-gin in the r10–r20 plane is (almost) qualitatively the same as the strong Lotka–Volterra bifurcation scenario in Fig. 3. The only difference is that the degenerate“Hopf/heteroclinic” bifurcation curve of Fig. 3A has been resolved in Fig. 5A intotwo distinct, nondegenerate bifurcation curves, tangent to each other at the origin.This allows for the existence of a limit cycle (in the fourth quadrant of phase space)for parameter values in between the two curves.

The similarity of the strong LV behavior and the locally strong LGR behavior,however, does not extend to large positive population values. This is because thelimited growth rate assumption requires that the self-limitation dominates overthe mutualism for sufficiently large populations. This is corroborated in the shapeof the nonlinear nullclines, which, for small values of r10 and r20, forces them tointersect a second time in the first quadrant (in addition to the first intersection atequilibrium number 3 near the origin), resulting in the stable coexistence equilib-rium number 4. The unbounded population growth from the LV model has beenreplaced in the LGR model by an approach to a stable coexistence equilibrium.Compare, for example, Figs. 3Bi with 5Bi.

There are also significant differences between Figs. 3A and 5A away from theorigin in the r10–r20 plane. One key feature (which we have already seen in the lo-cally weak mutualism case) is the saddle-node bifurcation curve. Equilibria 3 and4, which exist for parameter values above the saddle-node curve, come together onthe curve, and cease to exist below the curve. This transition – ecologically signif-icant for the portion of the saddle-node curve between the two “sn/tc” points—isillustrated in the corresponding adjacent phase portraits (from v to ix and from vito viii).

More significantly, we now have regions of parameter space (regions v, vi) whichallow behavior not seen in the Lotka–Volterra model: the existence of thresholdsbetween stable coexistence and extinction of one or both of the species. There are


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seven ecologically distinct regions in Fig. 5A. They are distinguished by the thicklines, and by the shading in each region. In regions i, ii, iii, iv, vii, ii′, iii′, iv′, and vii′

all open first quadrant initial populations eventually approach a stable coexistenceequilibrium; in region v, there is a threshold (the stable manifold of equilibrium 3)between a stable x-monoculture and the stable coexistence equilibrium; in regionv′ (not labeled)? there is a similar threshold between a stable y-monoculture andthe stable coexistence equilibrium; in all other regions of the fourth quadrant, allfirst quadrant phase orbits approach the x-monoculture equilibrium; similarly, inall other regions of the second quadrant phase, all first quadrant phase orbits ap-proach the y-monoculture equilibrium; in region vi, there is a threshold betweenorbits which approach the extinction of both species and the stable coexistenceof both species; in region viii, both populations become extinct. Once again, theregions with ecologically significant thresholds are indicated with the addition ofpatterned dots over the region shading.

There are several other bifurcations in Fig. 5 that are significant and interest-ing from a bifurcation point of view, but not directly significant from an ecologicalpoint of view. Some of these bifurcations curves are outside the range of parame-ters plotted in Fig. 5A. As mentioned in the Lotka–Volterra case, these “insignif-icant” bifurcations are necessary in order to set up the “significant” bifurcations(see Graves, 2003 or Graves et al., 2005 for a more complete bifurcation diagram).

3.3.5. Further bifurcation discussionOur strategy of dividing parameters into primary and auxiliary has a more formalinterpretation: we are studying bifurcations (in the auxiliary parameter space) ofbifurcation diagrams (in the primary r10–r20 parameter plane). Two points in theauxiliary parameter space are defined to be equivalent if their corresponding bi-furcation diagrams in the primary parameters “look the same.” “Look the same”is made formal by the existence of a homeomorphism of primary parameter planesthat map corresponding bifurcation curves in one primary parameter plane to bi-furcation curves of the same type in the other primary parameter plane.

Our local analysis guarantees that the primary parameter plane bifurcationdiagrams for our limited growth rate model must be divided into at least twoequivalence classes: auxiliary parameters that correspond to locally weak versuslocally strong mutualism. It turns out that there are further subdivisions of theauxiliary parameter space when we consider the global r10–r20 parameter space.For example, we found an example of a locally weak mutualism set of auxiliaryparameter values (r11 = 20, r21 = 2, k1 = 0.5, k2 = 0.25, a1 = 1, a2 = 1) for whichthe two transcritical curves cross two times (instead of not intersecting as in Fig.4) in the fourth quadrant. Thus it is not true that all locally weak bifurcationdiagrams are equivalent. On the other hand, it has been shown (Graves et al.,2005) that locally weak mutualism (the first inequality in the statement of theTheorem below) along with an additional condition (the second inequality) willguaranty that the two transcritical curves will not cross in the fourth quadrant,suggesting that a large class of locally weak mutualism models have r10–r20

bifurcation diagrams which might be equivalent:

Theorem 1 (Graves, 2003) If a1a2r21r11k1k2

> 1, and if a1r11k2

≥ 1, the transcritical bifur-cation curves displayed in Fig. 4 will not intersect in the fourth quadrant.


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In the locally strong mutualism case, although it is clear that the two transcrit-ical curves in the fourth quadrant of Fig. 5 must cross at least once, as they dofor our choice of auxiliary parameters, we have not analytically checked to seewhether other numbers of crossings are possible. In other words, we do not knowwhether the locally strong auxiliary parameters are all in the same equivalenceclass.

Higher codimension points: Except for the fact that we have been interestedin parts of both phase and parameter space far from the origin, this study couldbe thought of as a study of a codimension-three bifurcation point for differentialequations of the form

dxdt

= r10x + b1xy − x2 + h.o.t

dydt

= r20 y + xy − y2 + h.o.t.

(The coefficients a1, a2 and b2 from equation (5) can be assumed to be one byrescaling x, y and t.) The codimension-three point is at b1 = 1 and all other vari-ables (r10, r20, x, y) equal one. As b1 passes thru one, the system changes betweenweak and strong mutualism. Because of the degenerate Hopf/heteroclinic curvein the strong LV model, the normal form needs at least some cubic order termsin order to be determined. We plan to study this codimension-three point in thefuture.

The “straight line lemma” for the bifurcation curves in the LV model ensuresthat there are no other codimension-two points (besides the origin) in Figs. 2 or 3.Most of the crossings of codimension-one curves in Figs. 4 and 5 are trivial in thesense that they involve bifurcations which occur for the same parameter values, butare separated in phase space. There are two “nontrivial” types of codimension-twopoints which do appear. Each of Figs. 4 and 5 exhibit two codimension-two pointslabeled as “sn/tc.” A model for this bifurcation appears to be dx/dt = x + ax +bx2 ± x3, which exhibits a tangency of transcritical and saddle-node bifurcationcurves. The locally strong LGR model has one other codimension-two point, butit is in the 4th quadrant of parameter space and outside the range of what we haveshown in Fig. 5A. It is a crossing of a transcritical curve with a heteroclinic curve(see Graves et al., 2005).

4. Experimental implications

The basic assumption of our model which distinguishes it from other modelsof mutualism is that each of the mutualists asymptotically increases the other’sgrowth rate rather than their asymptotic biomass or carrying capacity. Our lim-ited growth rate assumption allows a smooth transition between facultative andobligate mutualisms without singularities, in contrast to the case in the Dean(1983) model in which each species asymptotically increased the other’s carryingcapacity .

There is considerable evidence that at least some important symbioses (mu-tualist interaction) operate through asymptotic increases in growth rates rather


than direct increases in carrying capacities. Some of the most-studied suchcases are nitrogen-fixation symbioses between higher plants and a nitrogen-fixingmicroorganism such as Rhizobium (with soybean) or Frankia (with alder). It haslong been known that rates of photosynthesis in the plant increase asymptoticallywith increased abundance of the nitrogen-fixing microorganism and that nitrogenfixation rates in the microorganism are in turn aided by supply of photosynthatefrom the plant. Nitrogen fixation rates in the symbiotic microorganism are aidedby supply of photosynthate because the energetic cost of nitrogen fixation enzymesystems is high (12 electrons per atom of atmospheric N reduced to amino acidform) and carbohydrate reductants are required in the N-fixing reactions (Nut-man, 1976). This suggests that mutualism in nitrogen-fixing symbioses operates byeach participant increasing the other’s growth rate (the assumption of our model)through enhancing reaction rates by providing substrate or enzymes rather than di-rectly increasing the other’s carrying capacity (the assumption of the Dean (1983)model and others).

Lichens, composed of nitrogen-fixing cyanobacteria in symbiotic associationwith a fungus, comprise a particularly interesting class of systems with nitrogen-fixing symbiosis, These symbioses are especially relevant to our model sincethey can be facultative–facultative, facultative–obligate, and obligate–obligateassociations. Lichens containing the fungal genus Peltigera may be an especiallygood model system for experimental investigation of our model because they growrelatively fast (Ahmadjian, 1967). The Peltigera fungus is obligate whereas theN-fixing cyanobacteria (usually Nostoc, but also Gloeocapsa, and Chroococcidiop-sis; Stewart, 1980; Budel, 1992; Pandey et al., 2000) are usually facultative and canbe separated from the fungus to form free living colonies. When environmentalnitrogen supply is low, less than 10% of free-living Nostoc fix nitrogen fromthe atmosphere. However, when in association with Peltigera fungus, the rate ofN-fixation in lichenized Nostoc increases (see review by Rai, 1988), suggesting thatthe Peltigera fungus increases growth rate and metabolism of the Nostoc symbiontcompared with free-living Nostoc. In some species of Peltigera fungus, greaterthan 90% of the fixed atmospheric N ends up not in the Nostoc but in the fungus,where it is used in protein synthesis (Rai, 1988). Thus, available data indicates thatthe symbiosis between Peltigera and Nostoc operates by each mutualist increasinggrowth rates and metabolism of the other, which supports the major assumptionof our model.

In addition, new molecular methods offer promise in further quantifyingthe effect of Peltigera and Nostoc on each other’s growth rates. Sterner andElser (2002) have presented strong evidence that growth rate in many algaeand other species is correlated with ribosomal RNA content, ribosomes be-ing the site of protein synthesis. Both rDNA and ribosomal RNA markershave recently been identified for both Peltigera fungus and the algal Nostocsymbiont (Miadlikowska and Lutzoni, 2004; Miadlikowska et al., 2003). Isola-tion and quantification of RNA markers of Nostoc and of Peltigera could pro-vide indices of their growth rates. Positive and asymptotic correlations of thesegrowth rates with each other and also with independent measurements of N-fixation by the Nostoc would be a strong test of the basic assumptions of ourmodel.


5. Summary and discussion

In addition to the development of our limited per capita growth rate mutualismmodel, we are clearly advocating in this paper the bifurcation approach to analysisof ecological models because it gives us great power to infer physical/ecologicalbehavior, especially the conditions that separate qualitatively different behaviors,and for the insights it gives to design experiments. For example, Fig. 5A conciselysummarizes the pairings of initial per capita growth rates that allow for stable co-existence of mutualist pairs. As an example of ecological inference, we note thatthe bifurcation diagram of Fig. 5A suggests that facultative–obligate mutualismcoexistence might be fairly abundant in nature. This conclusion is supported bynoting the fairly large part of the fourth quadrant of primary parameter space cor-responding to coexistence. On the other hand, the region of the 3rd quadrant al-lowing coexistence is tiny compared to the rest of the 3rd quadrant, suggesting thatobligate–obligate mutualisms might be rare in nature. Of course, Fig. 5A is for onespecific choice of auxiliary parameters, but the suggestion is still evident from thebifurcation diagram.

As suggested in Section 4, the bifurcation perspective can be of use in designingexperiments, especially when the primary parameters can be controlled in the ex-periment. Models can be validated by setting up experiments in which parametervalues are manipulated to cross over bifurcation curves in the parameter space.For example, our model implies that successful formation of facultative–obligateor obligate–obligate pairs requires proper parameter values for growth rates, etc.We discussed in Section 4 how lichen symbioses could be used to parameterize themodel and test the limited growth rate assumption upon which it is based. Com-parative studies of strains of cyanobacteria and fungi with genetically determinedgrowth rates that lie to one side or another of the bifurcations displayed in Fig. 5could also be used as a test of the model.

In conclusion, the new LGR model for 2-dimensional mutualism proposed hereappears to be a better model than either Dean’s (1983) model or the logistic modelwith Lotka–Volterra interaction terms (the LV model). The LGR model has theability to describe a variety of interactions which seem ecologically logical, includ-ing the possibility of thresholds and the impossibility of unbounded growth. It isalso able to account for both facultative and obligate mutualisms as well as smoothtransitions between these two types of mutualism.

Appendix I: The Singularity in Dean’s Model

Dean (1983) introduced the following two species eight-parameter model of mu-tualism:

dxdt

= rx x(kx−x)kx

dydt

= ry y(ky−y)ky

(A.1)


where

kx = Kmax x{1 − e−((ax y+Cx)/(Kmax x))}ky = Kmax y{1 − e−((ayx+Cy)/(Kmax y))}. (A.2)

Briefly, the problem with this model occurs when the expressions for the carry-ing capacity in Eq. (A.2) are not positive. By inspection, kx = 0 when y = −Cx/ax

and ky = 0 when x = −Cy/ay. The differential Eq. (A.1) is therefore singular alongthese lines in the phase space. When Cx or Cy is negative, the respective singularline passes through the first quadrant of phase space and is therefore significant inthe ecological interpretation of the model. Consequently, all such figures in Dean(1983) (Figs. 2b, 2c, 3c, 4a, and 4b), while ecologically correct, are at odds withmodel (A.1).

In Fig. 6 we illustrate with two phase space figures from Dean (1983), onefor facultative–facultative mutualism (6A), and one for obligate–obligate mutual-ism (6B). In Fig. 6C and 6D, respectively, we redraw Fig. 6A and 6B, this timewith the singularities. Since the singular lines in Fig. 6C do not enter the firstquadrant, Fig. 6A, which includes only the first quadrant of phase space, doesmatch Dean’s model. The singular lines do, however, enter the first quadrant inFig. 6D, so the phase portrait of Fig. 6B does not match Dean’s model. Fig. 6Eshows the corrected phase portrait for Dean’s model for the obligate–obligatecase. This phase portrait clearly does not represent the ecology which Deanintended.

Appendix II: Formulas for Nullclines, Equilibria and Their Eigenvalues

LV oblique nullclines: x = r10 + b1 ya1

andy = r20 + b2x

a2.

Table 1 LV equilibria and eigenvalues.

Figures 2 and 3equilibrium label Equilibrium Eigenvalues

0 (0,0) r10, r20

1(

r10a1

, 0)

−r10, b2r10a1

+ r20

2(

0,r20a2

)−r20,

b1r20a2

+ r10

3(

a2r10+b1r20a1a2−b1b2

,b2r10+a1r20a1a2−b1b2

)Complicated expression.

LGR nonlinear nullclines:

x = r10 + (r11 − r10)(1 − e−k1 y)a1

and y = r20 + (r21 − r20)(1 − e−k2 y)a12

.


Table 2 LGR equilibria and eigenvalues.

Figures 4 and 5equilibrium label Equilibrium Eigenvalues

0 (0,0) r10, r20

1(

r10a1

, 0)

−r10, r20

2(

0,r20a2

)r10, −r20

3 Lowerleft intersection Both real and negativeof nonlinear nullclines, when in the first quadrantwhen an intersection exists. (stable coexistence!),Location determined other stabilities innumerically. other quadrants

4 Upper right intersection One positive and oneof nonlinear nullclines, negative when in the firstwhen an intersection exists. quadrant (threshold!),Location determined other stabilities in othernumerically. quadrants

Acknowledgments

The authors acknowledge helpful conversations with Harlan Stech in developingand analyzing the LGR model.

References

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Dean, A.M., 1983. A simple model of mutualism. Am. Nat. 121(3), 409–417.Goh, B.S., 1979. Stability in models of mutualism. Am. Nat. 113(2), 261–275.Graves, W.G. 2003. A comparison of some simple models of mutualism. Master’s Project, Univer-

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A Bifurcation Analysis of a Differential Equations Model