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A stoichiometric exception to the competitive exclusion principle. Irakli Loladze∗, Yang Kuang Department of Mathematics Arizona State University Tempe AZ 85287-1804 [email protected], [email protected] James J. Elser and William F. Fagan Department of Biology Arizona State University Tempe AZ 85287-1501 [email protected], [email protected] September 26, 2001 Abstract The competitive exclusion principle (CEP) states that no equilibrium is possible if n species exploit fewer than n resources. This principle does not appear to hold in nature, where high biodiversity is commonly observed, even in seemingly homeogenous habitats. Although various mechanisms, such as spatial heterogeneity or chaotic fluctuations, have been proposed to explain this coexistence, none of them invalidates this principle. Here we evaluate whether principles of ecological stoichiometry can help in understanding the stable maintenance of biodiverse communities. Stoichiometric analysis recognizes that each organism is a mixture of multiple chemical elements (carbon, nitrogen, phosphorus) and that the relative relative proportions of such elements vary within and among species. We incorporate stoichiometric principles into a standard predator-prey model to analyze competition between two consumers on one autotrophic resource. The model tracks two essential elements, carbon and phosphorus, in each species. We show that a stable equilibrium is possible with two competitors on this single biotic resource. At this equilibrium both consumers are limited by the phosphorus content of the resource. This suggests that within a stoichiometric framework, the CEP may be violated because one biotic resource may support multiple competitors at a stable equilibrium. ∗ Current address: Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ 08544-1033 1 1 Introduction The competitive exclusion principle, one of the oldest and most intriguing paradigms in community ecology, states that at most n species can coexist on n resources (Volterra, 1926; Gause, 1934; Hardin, 1960; MacArthur & Levins, 1964; Levin, 1970). The principle relates to one of the central problems in ecology: finding mechanisms that maintain vast biodiversity. Since the time Hardin (1959, 1960) coined the term “the competitive exclusion principle,” subsequent advances in theoretical ecology (by no means a complete list) have identified various mechanisms that promote coexistence and demonstrated that the above formulation of the principle is too restrictive. First, if resources are not explicitly modelled, competitive exclusion is not mandatory. Strobeck (1973) showed that n species can coexist at a locally stable equilibrium in a homogeneous (spatially and temporally) classical Lotka-Volterra model with no resources. In such a model, however, competition is expressed in an ad hoc fashion via constant competition coefficients. In explicit resource models, exploitative competition among consumers is realized, as in real life, directly via the consumption of shared resources. Second, even for models that explicitly consider resources, the coexistence of n species on fewer than n resources is possible via internally produced limit cycles or chaotic fluctuations (Armstrong & McGehee, 1980; Huisman & Weissing, 1999). Third, other factors such as spatial (Tilman, 1994; Richards et al., 1999) or temporal (Butler et al., 1985; Lenas & Pavlou 1995) heterogeneity, light fluctuations (Litchman & Klausmeier, 2001), predation (Armstrong, 1994; Leibold, 1996), disturbance (Hastings, 1980), and interference between consumers (Vance, 1984, 1985) provide mechanisms for coexistence. In view of these findings, the competitive exclusion principle can be restated as following: a stable equilibrium is impossible in a homogeneous system with n species exploiting fewer than n resources. For the rest of this paper, by the competitive exclusion principle (CEP), we will mean this formulation. It assumes exploitative competition; that is, a situation in which consumers do not interfere directly but affect each other via the consumption of shared resources. It also assumes spatial and temporal homogeneity. Despite these restrictions, the CEP is extremely robust to various types of models: whether resources are abiotic such as chemical elements or biotic such as prey species (Armstrong and McGehee, 1980), essential or substitutable (Leon & Tumpson, 1975; Tilman, 1982; Grover, 1997) the CEP holds firmly. All these models share the same general structure: one set of equations describes resource dynamics and the second set describes the dynamics of consumers. The CEP follows as a rigorous mathematical result under an assumption that consumers and resources are in a predator-prey relationship. In other words, the partial derivative of consumer abundance with respect to resource abundance is positive or zero, while the partial derivative of resource abundance with respect to the consumer is negative or zero. (One notable exception to this rule is Levin’s (1970) extension of the CEP, where resources were replaced with more abstract “limiting factors.” Levin’s formulation requires only that species-specific growth rates be linear 2 functions of the limiting factors, an assumption that may not hold in reality. Under this assumption, neither a stable equilibrium nor a stable cycle is possible with n species on fewer than n limiting factors.) As it frequently happens in mathematical biology, the verbal version of a rigorous mathematical statement is often stated without some of its underlying assumptions. Such omission of assumptions can and often leads to widespread misconceptions by forcing the argument beyond the realm of its original formulation (e.g. the CEP or May’s complexity-stability argument). While this should not negate the versatility and effectiveness of mathematical applications in ecology, it highlights the importance of explicit statement of the precise conditions under which the statement holds. Whether resources are biotic or abiotic, similar assumptions have been made about consumer-resource interactions (Armstrong & McGehee, 1980). Hence, a conclusion follows that the CEP holds irrespective of the type of resources (biotic or abiotic) one models. However, any biotic resource (i.e. prey item), as all life, consists of multiple abiotic resources that are essential to its consumer. For example, one may consider a biotic resource as a mixture of macro-compounds such as carbohydrates, proteins, and lipids, which in turn are made of chemical elements such as carbon, nitrogen, phosphorus and so on. If a mathematical model incorporates such properties of biotic resources, will the CEP remain true as a rigorous mathematical result? It is the purpose of this paper to construct a simple plausible model to address this issue. A question arises regarding the appropriate “chemical scale” for modeling. The overwhelming complexity of organic compounds and their intricate metabolic pathways leave us no hope to follow them with a simple model. Fortunately, there is a powerful approach advocated by Alfred Lotka (1925), an originator of the famous Lotka-Volterra equations. Lotka, as a physical chemist, stressed that for biological systems, mass conservation with respect to all elements must hold: “for any self-contained system, we shall have an equation of the form X X1 + X2 + ... + Xn = X = A = const. (1) expressing the constancy of the total mass of the system; and a similar equation holds separately for every chemical element (except in those rare cases in which radioactive or other atomic disintegration occurs)”. Equation (1) is a part of the foundation of “stoichiometry”, a term that Lotka borrowed from chemistry and redefined for mathematical biology as a “branch of the science which concerns itself with material transformations, with the relations between the masses of the components.” Unfortunately, this approach has been largely neglected in the subsequent development of theoretical and mathematical population dynamics. This oversight of Lotka’s ideas is especially regrettable because the underlying assumptions of stoichiometry are among those few laws that can be firmly stated about biological systems. Incorporating such assumptions in mathematical models can be crucial for generating sound biological insights. In recent years, Lotka’s stoichiometric vision has been revived in such works as Reiners (1986), Sterner (1990), Andersen (1997), Hes3 sen (1997), Elser et al. 2000, Sterner & Elser (2001). Here we state five facts that, on our opinion, underline the foundation of stoichiometry: - biological systems, despite their overwhelming complexity, are not able to create or destroy chemical elements, nor they are able to convert one element into any other; - there are several chemical elements required by all life; - the proportions of these elements vary within and among species; - all such proportions have upper and lower bounds; - the amount of any element in any ecosystem is finite. Lotka’s equation (1) follows from these facts. Although Lotka favored a stoichiometric approach, his population dynamics equations did not incorporate stoichiometry. We formulate a simple model that captures the critical elements of stoichiometry. Our motivation is to find out if and when the CEP remains true in a model where consumers compete for a biotic resource in a stoichiometric setting. In the next section we construct the model. In section 3, we analyze it qualitatively. Numerical and graphical analysis of the model with Monod (or Michaelis-Menten) functional responses are in section 4. We will demonstrate that, in a stoichiometrically explicit model, two consumers exploiting one prey can coexist at a stable equilibrium. In the discussion, we will address the implications of these findings for our understanding of real ecosystems. 2 Model We model two consumers exploiting one biotic resource in a system with no spatial heterogeneity or external variability. The difference between our approach and the models mentioned above is that we do not assume, a priori, the existence of conventional predator-prey interactions (that is, those in which increased prey abundance never hurts predator growth). Instead, our model relies mainly on the stoichiometric principles listed in the previous section and the basic law of mass conservation. To have some concrete system in mind, we can think of the biotic resource as a single species of alga, while the consumers are two distinct zooplankton species, all placed in a well-mixed system open only to light and air. (One can imagine a continuously stirred culture in an open-top clear chamber.) In the model construction, we follow a course outlined in Loladze et al. (2000), which deals with one consumer-one biotic resource interactions. Let us start with a conventional model, which describes a system of two consumers feeding on one biotic resource, here assumed to be a photoautotroph at the base of the food web (e.g. phytoplanktonic algae): 4 ³ x´ dx = rx 1 − − f1 (x)y1 − f2 (x)y2 , dt K dy1 = e1 f1 (x)y1 − d1 y1 , dt dy2 = e2 f2 (x)y2 − d2 y2 . dt (2a) (2b) (2c) Here, x, y1 and y2 are the densities of the biotic resource and two consumers respectively (in milligrams of carbon per liter, mg C/l), r is the intrinsic growth rate of the resource (day−1 ), d1 and d2 are the specific loss rates of the consumers that include rates of respiration and death (day−1 ). f1 (x) and f2 (x) are the consumers’ ingestion rates, which we assume to follow Holling type II functional response. In other words, fi (x) is a bounded smooth function that satisfies the following assumptions: fi (0) = 0, fi0 (x) > 0 and fi00 (x) < 0 for x ≥ 0, i = 1, 2. (3) e1 and e2 are constant growth efficiencies (conversion rates or yield constants) of converting ingested resource into consumer biomasses. The second law of thermodynamics requires that e1 and e2 be less than 1. K represents a constant carrying capacity that we relate to light in the following way: suppose that we fix light intensity at a certain value, then let the resource (which is a photoautotroph) grow with no consumers but with ample nutrients. The resource density will increase until self-shading ultimately stabilizes it at some value, K. Thus, we might model the influence of higher light intensity as having the effect of raising K, all else being equal (Loladze et al., 2000). More mechanistic descriptions of light limited growth are possible (e.g. Huisman & Weissing, 1994), but these would significantly complicate our model and are not attempted here. Model (2) views the biotic resource, x, as a chemically homogeneous substance. Hence, this resource can not support two consumers at fixed densities. As we pointed out in the Introduction, however, any biotic resource is not chemically homogeneous, but is made of multiple chemical elements. In particular, no life on Earth exists without carbon or phosphorus (we can add nitrogen, sulfur, potassium and other elements as well). The requirement for these elements was established early in evolution. For example, every nucleotide requires one atom of phosphorus. Millions of nucleotides linked together make DNA and RNA in every cell. Hence, at least some amount of phosphorus must be present in every known organism. The bulk of the dry weight of most organisms is carbon comprises, and phosphorus to carbon ratio, P:C, varies within and among species (e.g. Elser et al., 2000). The cellular physiology of autotrophs allows them to exhibit highly variable phosphorus content. For example, the green alga Scenedesmus acutus can have cellular P:C ratios (by mass) that range 5 from 1.6 · 10−3 to 13 · 10−3 , almost an order of magnitude. On the other hand, in animals P:C varies much less within a species. For example, the zooplankton Daphnia’s P:C stays within relatively narrow bounds around 31 · 10−3 by mass, decreasing only by around 30% when food P:C is more than 20-times lower than Daphnia’s P:C (Sterner & Hessen, 1994; DeMott, 1998). For simplicity, we will model only two elements, phosphorus and carbon. Though this is a gross simplification of reality, it is a qualitative improvement of the conventional models like model (2). We express the above considerations in the following assumptions: (A1): phosphorus to carbon ratio (P:C) in the resource varies, but never falls below a minimum q (mg P/mg C); the two consumers maintain a constant P:C ratio, s1 and s2 (mg P/mg C), respectively. (A2): the system is closed for phosphorus, with a total of P milligrams of phosphorus per liter (mg P/l), which is divided into two pools: phosphorus in the consumers and the rest as phosphorus potentially available for the biotic resource. From these two assumptions it follows that phosphorus available for the resource at any given time is (P − s1 y1 − s2 y2 ) (mg P)/l. This amount determines the “phosphorus carrying capacity” of the resource: (P − s1 y1 − s2 y2 ) /q (mg C)/l. In the spirit of Liebig’s Law of the Minimum, the combination of light and phosphorus limits the carrying capacity of the resource to µ ¶ P − s1 y1 − s2 y2 min K, . (4) q To determine the P:C ratio of the resource at any given time, we follow Andersen (1997) by assuming that the biotic resource can absorb all potentially available phosphorus. Then its P:C ratio in (mg P : mg C) is (P − s1 y1 − s2 y2 ) /x. (5) This assumption is not far from reality in freshwater systems, where algae can absorb almost all available phosphorus, bringing free phosphate levels in water below detection. However, it may not be suitable for terrestrial systems, where soil particles are a major stock of phosphorus, or for modeling competition among multiple biotic resources, where free phosphorus uptake affects competition outcome. While resource stoichiometry varies according to (5), the i-th consumer maintains its constant P:C, si . If resource P:C is more than si , the i-th consumer is able to maximally use the energy (carbon) content of consumed food with efficiency ei , and excretes any excess phosphorus it ingests. If P:C of the resource is below si , the i-th consumer wastes the excess of ingested carbon. This waste is assumed to be proportional to the ratio of resource P:C to i-th consumer’s P:C (Andersen, 1997). Hence, the growth efficiency in carbon terms, ei , is reduced. The following minimum function reflects growth efficiency under both good and 6 bad resource quality conditions: ¶ µ (P − s1 y1 − s2 y2 ) /x ei min 1, . si (6) The considerations of (4) and (6) are captured in the following model: µ ¶ dx x = rx 1 − − f1 (x)y1 − f2 (x)y2 dt min(K, (P − s1 y1 − s2 y2 ) /q) µ ¶ (P − s1 y1 − s2 y2 ) /x dy1 f1 (x)y1 − d1 y1 = e1 min 1, dt s1 ¶ µ (P − s1 y1 − s2 y2 ) /x dy2 f2 (x)y2 − d2 y2 = e2 min 1, dt s2 (7a) (7b) (7c) Note, that the growth rate of the biotic resource in the absence of consumers µ ¶ dx x = rx 1 − dt min(K, (P − s1 y1 − s2 y2 ) /q) can be equivalently represented as the minimum of a logistic equation and a biomass version of Droop’s (1974) model: µ ³ µ ¶¶ x´ x dx = min rx 1 − , rx 1 − . (8) dt K (P − s1 y1 − s2 y2 ) /q 3 Mathematical Analysis In this section, through rigorous mathematical derivations we obtain some analytical results on the solutions and aspects of positive equilibria of model (7). The interpretations of these results are presented in subsequent sections. Model (2) almost never has a positive equilibrium (that is, an equilibrium with all three species present). It is easy to see why this is the case. If such an equilibrium exists, then the resource density, x∗ , at such an equilibrium should simultaneously satisfy the following two equations to keep consumers’ net growth rates equal to zero: f1 (x∗ ) = d1 d2 and f2 (x∗ ) = , e1 e2 (9) which is almost impossible (the set of parameter values satisfying both equations is of measure zero). Moreover, model (2) belongs to the class of models studied by Armstrong & McGehee (1980), where resources are regarded as biotic. These authors have rigorously shown that an attracting equilibrium is impossible (not just almost impossible) with n species on fewer than n resources. Nevertheless, we are going to show that such an equilibrium exists in our stoichiometrically explicit model (7). 7 First, we point out that model (7) is well defined as x → 0. The conditions (3) assure that for fi (x) /x, the following holds: µ ¶0 fi (x) fi (x) 0 < 0 for x > 0, i = 1, 2. (10) lim = f (0) < ∞ and x→0 x x This means that yi0 (t) is well defined as x → 0, since for i = 1, 2, µ µ ¶ ¶ P − s1 y1 − s2 y2 (P − s1 y1 − s2 y2 )fi (x) min 1, fi (x) = min fi (x), . x x The condition (10) ensures that system (7) is well defined and the functions at the right hand sides are locally Lipschitzian, which ensures that solutions to initial values are unique. Stoichiometric considerations provide natural bounds on densities of all species. We have the clear upper bound defined by phosphorus: the sum of phosphorus in the resource and the consumers cannot exceed P - the total phosphorus in the system. The following theorem provides an analytical result on the boundedness and invariance of solutions of model (7); its proof can be found in the appendix . Theorem 1 Let k = min(K, P/q). Then, solutions with initial conditions in ∆ ≡ {(x, y1 , y2 ) : 0 < x < k, 0 < y1 , 0 < y2 , qx + s1 y + s2 y < P } (11) remain there for all forward times. ˘ we denote the region ∆ with its entire boundary except of biologically By ∆, unrealistic edge γ = {(x, y1 , y2 ) : x = 0, P = s1 y1 + s2 y2 }. (At this edge consumers contain all phosphorus, thus there is no any resource in the system.) That is, ˘ = (∂∆\γ) ∪ ∆. ∆ To simplify our analysis, we rewrite the system (7) in the following form: x0 = xF (x, y1 , y2 ) y10 = y1 G1 (x, y1 , y2 ) y20 = y2 G2 (x, y1 , y2 ), (12a) (12b) (12c) where µ F (x, y1 , y2 ) = r 1 − x min(K, (P − s1 y1 − s2 y2 )/q) ¶ − 2 X fi (x) i=1 x ¶ µ (P − s1 y1 − s2 y2 ) /x fi (x) − di Gi (x, y) = ei min 1, si µ ¶ P − s1 y1 − s2 y2 fi (x) = ei min fi (x), − di , i = 1, 2. si x 8 yi , (13a) (13b) Condition (10) ensures that all partial derivatives of F and Gi exist almost ˘ : everywhere on ∆ ¶0 2 µ X ∂F fi (x) r yi (14) =− − ∂x min(K, (P − s1 y1 − s2 y2 )/q) i=1 x For i, j = 1, 2, we have fi (x) − <0 if K < P −s1 yq1 −s2 y2 ∂F x = , rqsi x fi (x) P −s1 y1 −s2 y2 ∂yi − − < 0 if K > q x (P − s1 y1 − s2 y2 )2 if P −s1 yx1 −s2 y2 > si e f 0 (x) > 0 µ ¶0 ∂Gi i i , = P − s1 y1 − s2 y2 fi (x) ei ∂x < 0 if P −s1 yx1 −s2 y2 < si si x if P −s1 yx1 −s2 y2 > si ∂Gi 0 = , s f (x) −ei j i ∂yj < 0 if P −s1 yx1 −s2 y2 < si si x (15) (16) (17) To determine the type of species interactions at equilibria and the stability properties of equilibria, it is convenient to examine the Jacobian of system (12): ∂F ∂F ∂F F+ x x x ∂x ∂y1 ∂y2 ∂G ∂G1 ∂G1 1 G1 + y1 y y1 J(x, y1 , y2 ) ≡ (aij )3×3 ≡ (18) ∂x ∂y1 ∂y2 ∂G2 ∂G2 ∂G2 y2 G2 + y2 y2 ∂x ∂y1 ∂y2 If the following system has a solution F (x, y1 , y2 ) = G1 (x, y1 , y2 ) = G2 (x, y1 , y2 ) = 0, (19) then system (12) has an internal (positive) equilibrium. We will show that for biologically realistic parameter sets this condition can be satisfied. Let us denote such an equilibrium as (x∗ , y1∗ , y2∗ ). Then the Jacobian at (x∗ , y1∗ , y2∗ ) is ∗ x 0 0 (20) J(x∗ , y1∗ , y2∗ ) = 0 y1∗ 0 E(x∗ , y1∗ , y2∗ ), 0 0 y2∗ where E(x∗ , y1∗ , y2∗ ) = ∂F ∂x ∂G1 ∂x ∂G2 ∂x 9 ∂F ∂y1 ∂G1 ∂y1 ∂G2 ∂y1 ∂F ∂y2 ∂G1 ∂y2 ∂G2 ∂y2 (21) is the ecosystem matrix of our system. Here, ij-th term in the matrix measures the effect of j-th species on i-th species growth rate. If (P − s1 y1∗ − s2 y2∗ ) /x∗ > si , i = 1, 2 (22) then the quality of the resource is good for both consumers, and, as (15-17) suggest, the signs in the ecosystem matrix are +/− − − + 0 0 , (23) + 0 0 meaning that a conventional predator-prey type interaction exists between the competitors and the resource. In this case, fi (x∗ ) = di /ei , i = 1, 2, an almost impossible situation. If (P − s1 y1∗ − s2 y2∗ ) /x∗ < si , i = 1, 2 (24) then the quality of the resource is bad for both consumers, and (15-17) indicate that the signs of the ecosystem matrix are +/− − − − − − . (25) − − − This means that all species compete with each other and the consumers endure self-limitation. In this case, A ≡ P − s1 y1∗ − s2 y2∗ = d1 s1 x∗ d2 s2 x∗ = e1 f1 (x∗ ) e2 f2 (x∗ ) and µ B ≡ rx∗ 1 − x∗ min(K, (d2 s2 x∗ )/(qe2 f2 (x∗ )) ¶ = f1 (x∗ )y1∗ + f2 (x∗ )y2∗ . This case is highly likely to happen with the value of x∗ determined by d1 s1 x∗ d2 s2 x∗ = , e1 f1 (x∗ ) e2 f2 (x∗ ) (26) and the values of y1∗ and y2∗ given by y1∗ = f2 (x∗ )P − f2 (x∗ )A − s2 B , s1 f2 (x∗ ) − s2 f1 (x∗ ) y2∗ = f1 (x∗ )A + s1 B − f1 (x∗ )P . (27) s1 f2 (x∗ ) − s2 f1 (x∗ ) It is straightforward to find that condition (24) is equivalent to di < ei fi (x∗ ), 10 i = 1, 2, (28) which is to say that if the resource at density, x∗ , would have been of good quality, then the growth rate of each consumer would have exceeded its death rate. At this equilibrium both consumers are limited by the quality (i.e. phosphorus content) of the resource. An intermediate scenario can also occur, where the quality is bad for one consumer, but good for the other. The coexistence of all species at a stable positive equilibrium is possible as well in this case (parameter values for this case are listed in Table 1 and K=0.65). Observe that (7) has E0 ≡ (0, 0, 0) and Ek ≡ (min(K, P/q), 0, 0) as its only axial equilibria. Clearly, E0 is always a saddle. It is easy to show that when any other steady state exists, then Ek is also a saddle. For general functions fi (x) satisfying (3), there can be many other nonnegative equilibria on the boundary surfaces (x − yi surface, i = 1, 2) of ∆ (Loladze et al., 2000). The stability of these equilibria and positive equilibria (if any) can be routinely studied via Routh-Hurwitz criteria (Edelstein-Keshet (1988), p. 234) when specific functions fi (x), i = 1, 2 are given. Persistent properties of the system can be obtained by applying existing theory (e.g. Thieme (1993) and the references cited there). 4 Numerical and Graphical Analysis To illustrate some of the above analytical findings, we run simulations on model (7) with both fi (x), i = 1, 2 as Monod (or Michaelis-Menten) functions of the form ci x fi (x) = , i = 1, 2. (29) ai + x With these functions, if a positive equilibrium exists with bad resource quality (conditions (24)), then equation (26) yields the following resource density at the equilibrium: d2 s2 a2 e1 c1 − d1 s1 a1 e2 c2 . (30) x∗ = d1 s1 e2 c2 − d2 s2 e1 c1 We provide practical graphical interpretations for some of the analytical and numerical results below. 4.1 Numerical Analysis Table 1 lists biologically realistic parameter values that are based on literature sources (Urabe & Sterner, 1996; Andersen, 1997; Elser & Urabe, 1999). These values indicate that the second consumer requires more phosphorus (s2 > s1 ) but, as a trade-off, has a higher growth rate (e2 f2 (x) > e1 f1 (x) for x > 0) when resource quality is good. Such a trade-off is consistent with “the growth rate hypothesis” (Elser et al., 2000) and has been empirically supported in studies of zooplankton (Main et al., 1997) First, we analyze the system when light intensity is low, so that self-shading of the resource bounds its density at K = 0.35 (mg C)/l. Under such low 11 P r K c1 c2 a1 a2 e1 e2 d1 d2 s1 s2 q P Description intrinsic growth rate of the resource resource carrying capacity determined by light maximal ingestion rate of the 1st consumer maximal ingestion rate of the 2nd consumer half-saturation constant of the 1st consumer half-saturation constant of the 2nd consumer maximal conversion rate of the 1st consumer maximal conversion rate of the 2nd consumer loss rate of the 1st consumer loss rate of the 2nd consumer constant P:C of the 1st consumer constant P:C of the 2st consumer minimal possible P:C of the resource total phosphorus in the system V 0.93 0.35 − 1.0 0.7 0.8 0.3 0.2 0.72 0.76 0.23 0.2 0.032 0.05 0.004 0.03 Units day−1 (mg C)/l day−1 day−1 (mg C)/l (mg C)/l day−1 day−1 (mgP)/(mg C) (mgP)/(mg C) (mgP)/(mg C) (mg P)/l Table 1: The parameters (P) of the model and their values (V) used for numerical simulations. light conditions, resource quality is good (due to low algal biomass relative to phosphorus) and both consumers are limited by resource quantity (carbon). Model (7) behaves like conventional resource based models. In other words, the CEP holds and the second consumer (whose break-even resource concentration is lower) wins. The only attracting equilibrium is (x, y1 , y2 ) = (0.098, 0, 0.23), (31) where we rounded these values (in mg C/l) and all values hereafter to a thousandth. Next, we analyze the system under high light intensity, so that self-shading of the resource occurs at a higher level, with K = 1.0 (mg C)/l. Under such high light conditions, resource quality can be bad for both consumers. The CEP does not hold anymore, because the system has the following positive equilibrium with all species present: (x, y1 , y2 ) = (0.59, 0.26, 0.17) . (32) All eigenvalues at this equilibrium are negative, so the equilibrium is locally asymptotically stable. It can be shown that this is the only positive equilibrium, all boundary equilibria are unstable and the system appears to be persistent. (That is, all species, if present, are expected to coexist. This implies that the invasions of either consumer species will be successful.) Moreover, all our numerical simulations (Fig. 4) indicate that trajectories with all positive initial conditions in ∆ converge to this equilibrium. 12 4.2 Graphical analysis Since system (7) involves three differential equations, its phase space is three dimensional. To facilitate the visualization of the dynamics, let us recall a two dimensional case: a single consumer on one biotic resource. This case was rigorously analyzed in Loladze et al. (2000) using the method of nullclines. The nullcline of a species on the phase plane (the coordinate plane with axes representing species densities) is the set of points where the species’s growth rate is zero . If nullclines of all species intersect at some common point, then at such a point the growth rate of every species is zero, making this point an equilibrium. The stability type of such an equilibrium can sometimes be determined by the way nullclines intersect. As we see in Fig.1, the resource nullcline is hump shaped, a usual feature of many predator-prey models. The consumer nullcline, however, has an unusual shape of a right triangle (in conventional consumerresource models, it is a vertical line or an increasing curve). Here, multiple equilibria are possible because two nullclines with such shapes can intersect more than once. The vertical segment of the consumer nullcline lies in a region where the quantity of the resource limits the consumer and where the system behaves like a conventional predator-prey model. The slanted segment of the consumer nullcline lies in a region where the resource quality limits the consumer and qualitatively novel dynamics arise. Introducing the second consumer adds a third dimension to Fig. 1 and nullclines on the phase plane become nullsurfaces in the phase space. For example, system (7), with biologically realistic parameter values as in Table 1 and K = 0.35, has nullsurfaces as shown in Fig. 2a. (It can be shown that for Monod functional responses, consumer species nullsurfaces always take the shape of two i folding planes: a portion of the plane vertical to x-axis, x = ciaeii d−d , i = 1, 2, and i di si a portion of the plane slanted to x-axis, s1 y1 +s2 y2 + ci ei x = P − dci si ei ai i , i = 1, 2.) For K = 0.35, the resource’s nullsurface lies entirely in the region where the consumers are limited by the quantity of the resource. The reason for this is that a low value of K corresponds to low light intensity which keeps the resource density low (in mg C/l), which in turn assures high P:C in the resource. In this case, the system is no different from standard resource based models, the CEP holds and the only attracting equilibrium is (31). Fig. 2b shows the slice of the phase space in Fig. 2a cut out by y1 = 0 plane. The second consumer has a lower break-even concentration (the vertical segment is closer to the origin) and, as resource competition theory predicts, this consumer wins. The picture changes drastically when we increase K to 1.0 and keep all other parameter values unchanged. This K value corresponds to high light intensity, which lowers P:C ratio of the resource to such levels that consume growth becomes phosphorus-limited. Fig. 3 shows that all three nullsurfaces intersect at one point, equilibrium (32). Its negative eigenvalues prove that this equilibrium is locally attracting. For conventional model (2), which ignores stoichiometry, consumers’ nullsurfaces are parallel planes, so they cannot intersect and an equilibrium is impossible, which provides a graphical interpretation of condition (9). To simplify Fig. 3, we slice the phase space through equilibrium (32) with plane 13 y1 = 0.26 (mg C)/l (slicing with plane y2 = 0.17 yields a qualitatively similar picture). This slice is shown in Fig. 4. Drawing an analogy from Fig. 1, we see that in Fig. 4 both consumers are limited by resource quality. Indeed, at equilibrium (32) inequalities (24) hold, implying that both consumers are limited by the same element - phosphorus. Hence, both consumers coexist on one biotic resource and are limited by the same chemical element! In our study, we consider two essential chemical elements, carbon and phosphorus, in the composition of a biotic resource. For resource competition for two nutrients, Leon & Tumpson (1975) introduced nullcline analysis on the plane with axes representing nutrient availability. Tilman (1982) greatly expanded this approach in developing resource-competition theory. When essential nutrients are considered in this theory, each consumer nullcline is L shaped. Neither the vertical legs of these two L’s can intersect each other nor can the horizontal ones. Hence, the stable coexistence of two consumers on the same nutrient is impossible. For system (7), such nullclines on the carbon-phosphorus plane have the shape of tilted V, where the angle of the tilt is species specific as seen on Fig. 5. The resulting intersection of two V-nullclines can yield a stable equilibrium where two consumers are limited by the content of the same essential element in the prey (Fig. 3, Fig.4 and Fig.5). 5 Discussion We showed that an attracting equilibrium is possible for two consumers on one biotic resource. This result disagrees with the competitive exclusion principle (CEP) that an attracting equilibrium is impossible in a homogeneous system with n species exploiting fewer than n resources (whether resources are biotic or abiotic). Since the CEP is a rigorous mathematical result (Armstrong & McGehee 1980) and one of the seemingly solid pillars in ecological theory, one may ask why such a result is possible at all. Our model, unlike conventional resource based models, does not assume a (+,-) relationship between consumers and resources nor does it have linear growth assumptions as in Levin’s (1970) “limiting factor” model. Instead, it relies on general stoichiometric principles. In particular, we use the fact that all species share common essential chemical elements, but in different proportions. Consumer species differ in their chemical constitution, both from each other and from the resources they consume. Model (7) captures this by considering different consumer P:C ratios, while allowing the P:C ratio of the autotroph to vary, as it does in nature. It is a fact that any biotic resource contains several chemical elements that are essential to all of its consumers (e.g. carbon, nitrogen, phosphorus, sulfur, and calcium.) This suggests that biotic resources can be viewed and can be modelled as channels through which multiple nutrients flow to their consumers. All three species in our system, and in fact, all species in any ecosystem, share a common need for carbon and phosphorus. Such shared requirements for essential elements connect population level properties to those on the level of the entire ecosystem, and create feedbacks between these two levels of organizational 14 complexity that lead to dynamics that may seem paradoxical or impossible from conventional points of view. Stoichiometric considerations provide biologically meaningful bounds on population densities: every organism requires at least some minimum amount of an essential chemical element. Therefore, the total amount of this element puts a ceiling on overall biomass in the system. We rigorously showed that in Theorem 1 by proving that all dynamics in the system is confined to a region defined by the interplay between ecosystem properties (total phosphorus and light intensity) and species-specific stoichiometries (P:C ratio of the consumers and the resource). That is why, the carrying capacity of the resource is not static, as it is usually assumed in the logistic equation, but instead is a dynamic quantity affected by consumer densities, energy inputs and phosphorus in the system (see eq. 4). In addition to imposing natural bounds on the dynamics, the consideration of multiple elements and their ratios reveals rich dynamics within these bounds. When the quality of the resource is good, the sheer quantity of the resource limits consumers. Their dynamics are completely predicted by resource competition theory: the consumer with the lowest break-even concentration wins (see eq. (9). The signs of ecosystem matrix (23) indicate the usual (+,-) interaction between consumers and the resource. However, when resource quality is bad, a novel attracting equilibrium can arise. For example, realistic parameter values listed in Table 1 (with K = 1) yield a locally attracting equilibrium (32). At this equilibrium, not only do two consumers coexist on one biotic resource, but both are limited by the same chemical element, phosphorus, which is in low concentration in the resource. The ecosystem matrix (25) takes an unusual form for consumer-resource models: negative signs indicate that the common need for limited essential element fosters competition between the consumers and the resource as well as between the consumers. Such an outcome is impossible in conventional resource-based models because the (+,-) relationship between consumers and resources is mandated by the model assumptions. In the Graphical Analysis section, we analyzed the model using nullsurfaces and nullclines on the coordinate system with each axis representing species densities. However, in resource competition theory (Leon & Tumpson 1975; Tilman, 1982; Grover, 1997), nullclines are drawn on the plane with each axis representing nutrient concentrations. For essential nutrients each consumer nullcline is L shaped. This shape preclude the possibility of an equilibrium with two competitors being limited by the same nutrient. For system (7), however, nullclines on the carbon-phosphorus plane have the shape of a tilted V, where angle of the tilt is species specific. The intersection of two Vs with different inclinations, as seen in Fig. 5, results in an equilibrium where two competitors are limited by the content of the same chemical element (phosphorus) in the prey. The key observation obtained here is that stoichiometric considerations provide and expand the possibility for coexistence in the form of a stable equilibrium - the most restrictive form of coexistence. Limit cycles, chaotic fluctuations, and persistence more generally, provide other forms for coexistence. How stoichiometry affects these other forms of coexistence remains to be examined. This question brings up Hutchinson’s (1961) famous “paradox of the plankton”: how 15 so many phytoplankton species can coexist on relatively few resources. Huisman & Weissing (1999, 2000, 2001), using numerical simulations, showed that coexistence of phytoplankton can be achieved through chaotic fluctuations: for example, by twelve species on five resources. They used a conventional resourcebased model with fixed chemical composition of all species. But in reality, phytoplankton can exhibit highly variable chemical composition (Elser & Urabe, 1999) suggesting that stoichiometric considerations may provide a fresh and alternative avenue to the resolution of “paradox of the plankton.” We should caution, however, that stoichiometric considerations provide limited possibilities for coexistence at an equilibrium. For example, equilibrium is impossible when we add a third competitor to system (7) but continue to track only two elements, carbon and phosphorus. One can use this limitation to argue that our result does not disagree with the CEP, because still at most two consumers can coexist on two chemical elements (abiotic resources) that are just packaged in one biotic resource. Abrams (1988), discussing the problem of counting resources in the context of the CEP, stated that “it would clearly be inappropriate to define resources in such a way that the theory [of the CEP] is simply false”. Given our result then, if one wishes to save the CEP, one should not count a biotic resource as a single resource. The idea that one biotic resource can represent more than one resource is not novel. For example, different parts of plants can provide distinct resources to herbivores (Tilman, 1982, Abrams, 1988). What is novel in our approach is its reliance not on a particular species-specific interpretation of the biotic resource but on the undisputable fact that all biotic resources are always a package of multiple chemical elements to all their consumers. This fact, which is independent of the species considered, invalidates the CEP with respect to biotic resources, unless one views a single biotic resource as multiple abiotic resources (chemical elements). It is intriguing, however, that a stable equilibrium can exist when two consumers are simultaneously limited by one essential element in the prey (in our specific case, phosphorus). In fact, at such an equilibrium, the two consumers and the resource are all competitors (as the signs in the ecosystem matrix (25) suggest), and one can argue that three competitors coexist on just two essential elements. Next, we discuss possible extensions of stoichiometric model (7). Though phosphorus is of special interest because it appears tightly linked to growth rates of individual organisms through ribosomal function and protein synthesis (Elser et al., 1996), other elements (such as N, S, or Ca) may also have important but unappreciated effects on consumer-resource interactions (Sterner & Elser, 2001; Williams & Frausto da Silva, 2001). To explore how such multi-element stoichiometry may affect coexistence and other ecological dynamics, one may analyze the following version of system (7) that models k consumers exploiting 16 one biotic resource consisting of n essential chemical elements: x x0 (t) = bx 1 − à à ! ! k k X X min(K, N − s y , ..., N − s y ) /q /q 1 1i i 1 n ni i n i=1 i=1 m X − fi (x)yi i=1 à à ! ! k k X X N1 − Nn − s1i yi sni yi i=1 i=1 0 y (t) = e min 1, , ..., fi (x)yi − di yi i i xs xs 1 n (33) where Ni is the total amount of i-th nutrient in the system, qi is the resource’s minimal i-th nutrient content, sij is the j-th consumer’s constant (homeostatic) i-th nutrient content and the other parameters are as in the model (7). Our experience with model (7) suggests that the ultimate dynamics of model (33) will depend on both model parameters and initial population densities. Straightforward algebraic study and numerical work indicate that stable positive steady states are possible for some set of parameters if k < n. This suggests that within a multi-element stoichiometric framework, the CEP may be thoroughly violated. It also suggests a new metric by which to judge the maintenance of biodiversity in competitive systems. In particular, if one biotic resource provides n essential chemical elements for competing consumers, it may support up to n different competitors at a stable equilibrium. The formulation of a plausible and tractable mathematical version of model (33) with more than one biotic resource is challenging. This difficulty stems from the competition among resource species, which forces one either to consider free pools of essential elements (i.e., not bound in any biomass) and model their uptake, or to find some other way to distribute each essential element among all resource species. A mathematically easier case can be imagined in a patchy habitat, where specialist consumers in each patch exploit a distinct biotic resource that provides n essential chemical elements. Then, m distinct biotic resources can support up to m × n consumer species at a stable equilibrium. In our model (7), we assumed no spatial heterogeneity and no external fluctuations in the system. Instead, chemical heterogeneity within and among species alone provides mechanisms for coexistence. Can puzzling aspects of biodiversity on Earth be, at least partially, explained by such chemical heterogeneity? We note that the worsening quality of the resource weakens consumer-resource interactions by limiting carbon flow into consumer biomass. As we have shown, such weakening promotes coexistence, a result that agrees with “the weak interaction strength” hypothesis (McCann et al., 1998). This suggests that stoichiometric considerations can play an important role in the biodiversity-stability debate 17 (McCann, 2000). There are other areas of population dynamics and ecology where stoichiometry may provide new insights. For example, chemostat theory is one of the most thoroughly studied parts of population dynamics and one where nutrients play a crucial role (e.g. Hsu et al., 1981; Smith & Waltman, 1995; Li et al., 2000). This theory is particularly suitable for extensions that encompass stoichiometric principles. Another interesting avenue is a coupling of stoichiometric effects with spatial ones. Spatial models are central to ecology (e.g. Durrett & Levin, 1994; Lewis, 1994); how the interplay between multiple patches and the ratios of multiple nutrients results in observed ecological patterns remains to be thoroughly evaluated, but some work in this area emerged recently. Work by Fagan and Bishop (2000) on Mount St. Helens suggests that nutritional quality of invading plants may depend on their spatial position, leading to differential effects on herbivores and on the dynamics of primary succession. Codeco & Grover (2001) provided another starting point by considering the effects of various P:C supply ratios on competition between algae and bacteria in a gradostat (interconnected multiple chemostats). Understanding how the balance of chemical elements affect population and ecosystem dynamics becomes ever more important as human activities profoundly change global cycles of carbon and phosphorus as well as other elements essential for all life, like nitrogen and sulfur. In model (7) with just two chemical elements, variation in stoichiometry of one prey species qualitatively alters food web dynamics. It is plausible that similar effects take place in diverse ecosystems where multiple elements and many species are at play. 6 Acknowledgements We thank Chris Klausmeier for insightful comments. This research has been partially supported by NSF grant DMS-0077790 and IBN-9977047. I.L. also has been partially supported by NSF grant DEB-0083566. 7 Appendix Here is the proof of the Theorem 1 on boundedness and invariance of solutions of model (7). Proof. Let us prove by contradiction, i.e. assume that there is time t∗ > 0, s.t. a trajectory with initial conditions in ∆ touches or crosses the boundary ∂∆ for the first time. Since the boundary consists of at most five surfaces (four if K > P/q), we divide the crossing into five possible cases: x(t∗ ) = 0, x(t∗ ) = k = min(K, P/q), y1 (t∗ ) = 0, y2 (t∗ ) = 0 and qx(t∗ ) + s1 y1 (t∗ ) + s2 y2 (t∗ ) = P. Case 1: Assume x(t∗ ) = 0 (here we exclude the edge P = s1 y1 − s2 y2 lying on x = 0 plane). Let Pmin = mint∈[0,t∗ ] (P − qx(t) − s1 y1 (t) − s2 y2 (t)) > 0, 18 ỹi = maxt∈[0,t∗ ] yi (t) and note that fi (x) ≤ fi0 (0)x. Then, µ x (t) = rx(t) 1 − 0 x(t) min(K, (P − qx(t) − s1 y1 (t) − s2 y2 (t))/q) ! à µ ¶ X 2 x(t) fi0 (0)ỹi x(t) ≡ αx(t) ≥ r 1− − min(K, Pmin ) i=1 ¶ − 2 X fi (x(t))yi (t) i=1 where α is some constant. Thus, x(t) ≥ x(0)eαt > 0, which implies that x(t∗ ) > 0. Case 2: Assume x(t∗ ) = k. Then µ ¶ µ ¶ x (t) x (t) 0 x (t) ≤ rx (t) 1 − ≤ rx (t) 1 − . min(K, P/q) k The standard comparison argument yields that x(t) < k for all t ≥ 0. Cases 3 and 4: Assume that y1 (t∗ ) = 0 or y2 (t∗ ) = 0. Since yi0 (t) ≥ −di yi (t) it follows that yi (t) ≥ yi (0)e−dt > 0, i = 1, 2 for t ≥ 0. Case 5 (includes the edge excluded in case 1) : Assume qx(t∗ ) + s1 y1 (t∗ ) + s2 y2 (t∗ ) = P. (34) Since for all t ∈ [0, t∗ ), qx(t) + s1 y1 (t) + s2 y2 (t) < P, it follows that qx0 (t∗ ) + s1 y10 (t∗ ) + s2 y20 (t∗ ) ≥ 0 (35) First, let us show that x(t∗ ) 6= 0 (to avoid division by 0 later). If x(t∗ ) = 0, then s1 y1 (t∗ ) + s2 y2 (t∗ ) = P. However, ¶ 2 µ X fi (x) (s1 y1 + s2 y2 ) ≤ ei (P − s1 y1 + s2 y2 ) yi − di yi ≤ x i=1 µ ¶ 2 2 X X s1 y1 + s2 y2 (ei (P − s1 y1 + s2 y2 ) fi0 (0)yi ) = (ei fi0 (0)yi ) P 1 − , P i=1 i=1 0 and the standard comparison argument yields that s1 y1 (y) + s2 y2 (t) < P for all t ≥ 0. Hence, x(t∗ ) 6= 0. Observe that at t∗ , the producer reaches its carrying capacity allowed by available phosphorus and has lowest quality, q, since (34) implies that (P − s1 y1 (t∗ ) − s2 y2 (t∗ ))/q = x(t∗ ). Substituting it to (7 a) yields: µ x0 (t∗ ) = rx (t∗ ) 1 − ¶ X 2 2 X x(t∗ ) ∗ ∗ f (x(t ))y (t ) ≤ − fi (x(t∗ ))yi (t∗ ). − i i min(K, x(t∗ )) i=1 i=1 (36) 19 From (34) it also follows that (P − s1 y1 (t∗ ) − s2 y2 (t∗ ))/x(t∗ ) = q. 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The Biological Chemistry of the Elements : The Inorganic Chemistry of Life. 2nd edition. Oxford University Press, New York. 8 Figure Legends Figure 1. Stoichiometric properties confine dynamics to a trapezoid-shaped area and divide the phase plane into two regions. In Region I, like in classical LotkaVolterra model, food quantity limits consumer growth. In shaded Region II food quality (that is, the phosphorus content of the resource) constrains consumer growth. Competition for the limiting chemical element between the consumer and the resource alters their interactions from (+, -) in Region I to (-, -) in Region II. This bends the consumer nullcline in Region II. This shape of the consumer nullcline, with two x-intercepts, creates the potential for multiple positive steady states. A solid circle denotes stable equilibrium, clear circles unstable equilibriums. (The figure is taken from Loladze et al. (2000).) Figure 2. a) The phase space of model (7) with parameter values listed in Table 1 and K = 0.3 mg C/l. The low value of K means low light intensity, which assures good quality (high P:C) of the resource. Consumer dynamics are governed by resource quantity and the competitive exclusion principle (CEP) holds; that is, the second consumer, whose nullsurface has the lowest x-intercept, wins. b) Slicing Fig.2a with plane y1 = 0 at the equilibrium shows its similarity with Fig. 1. Figure 3. Same as Fig. 2a, but with K=1.0 mg C/l. The high value of K means high light intensity, which leads to bad quality (low P:C) of the resource. The dynamics is governed by the resource quality and the CEP does not apply. Both consumers are limited by phosphorus content of the resource and coexist at an internal attracting equilibrium. 23 Figure 4. Slicing the phase space in Fig. 3 at the internal equilibrium (0.59, 0.26, 0.17) with plane y1=0.26. Unlike Fig. 2, where consumer-resource interactions are (+, -) at the boundary equilibrium, consumer-resource interactions in Fig. 3 and Fig. 4 change to (-,-) at the internal equilibrium. Figure 5. Analogous to resource competition theory, we draw consumer nullclines on the carbon-phosphorus phase plane. Instead of the usual L shape, the nullclines take the shape of tilted V, where the angle of the tilt is determined by the stoichiometric properties of each species. This shape can lead to an internal attracting equilibrium, where the two consumers coexist, even though their growth is limited by the same chemical element in the prey. 24 y II Nullclines: resource consumer I 0 Figure 1. K x Slice produced by plane y1=0.26 0.35 trajectory resource 1st consumer 2nd consumer 0.3 y2 (mg C/l) 0.25 0.2 . 0.15 0.1 0.05 0 Figure 4 0 0.2 0.4 0.6 x (mg C/l) 0.8 1 1.2 Nullclines on Carbon-Phosphorus plane 0.04 Total phosphorus, P (mg P/l) 0.035 0.03 0.025 0.02 1st consumer 2nd consumer 0.015 Figure 5 0 0.1 0.2 0.3 0.4 Carbon, x (mg C/l) 0.5 0.6 0.7 0.8