the idea that animals are detrimental to their food ... · flow, and hence primary production, in...

Vol. 145, No. 1 The American Naturalist January 1995

CONSUMERS AS MAXIMIZERS OF MATTER AND ENERGY FLOW IN ECOSYSTEMS

MICHEL LOREAU*

Department of Animal Biology, C.P. 160/13, Free University of Brussels, 50 av. Roosevelt, B-1050 Brussels, Belgium

Submitted August 12, 1993; Revised February 18, 1994; Accepted March 14, 1994

Abstract.-This article investigates the conditions under which consumers maximize matter and energy flow in ecosystems, using a simple, general, abstract nutrient-limited ecosystem model. The model is analyzed for several functional forms of trophic interactions. The conditions for enhancement of energy flow by consumers are found to be qualitatively the same in all cases. The first and most important condition is that consumers act to accelerate the circulation of matter within the ecosystem; more precisely, the mean transit time of nutrients in the path consumers-decomposers-nutrient pool must be sufficiently smaller than their mean transit time in the path producers-decomposers-nutrient pool. The second condition is that the total quantity of nutrients in the ecosystem must be higher than some threshold value. The third condition is that consumption rate must be moderate. These conditions being usually met, it is likely that consumers as a whole do generally play a role of maximizers of matter and energy flow in stable natural ecosystems.

Coupled transformers are presented to us in profuse abundance, wherever one species feeds on another, so that the energy sink of the one is the energy source of the other.

A compound transformer of this kind which is of very special interest is that composed of a plant species and an animal species feeding upon the former. The special virtue of this combination is as follows. The animal (catabiotic) species alone could not exist at all, since animals cannot anabolise inorganic food. The plant species alone, on the other hand, would have a very slow working cycle, because the decomposition of dead plant matter, and its reconstitution into C02, completing the cycle of its transformations, is very slow in the absence of animals, or at any rate very much slower than when the plant is consumed by animals and oxidized in their bodies. Thus the compound transformer (plant and animal) is very much more effective than the plant alone....

For it must be remembered that the output of each transformer is determined both by its mass and by its rate of revolution. Hence if the working substance, or any ingredient of the working substance of any of the subsidiary transformers, reaches its limits, a limit may at the same time be set for the performance of the great transformer as a whole. Conversely, if any one of the subsidiary transformers develops new activity, either by acquiring new resources of working substance, or by accelerating its rate of revolution, the output of the entire system may be reflexly stimulated. (LOTKA [1925] 1956, pp. 330, 334-335)

The idea that animals are detrimental to their food resources is deeply en- * Present address: Institute of Ecology, Pierre et Marie Curie University, Bat. A, Case 237, 7 quai

Saint Bernard, F-75252 Paris Cedex 05, France.

Am. Nat. 1995. Vol. 145, pp. 22-42. ? 1995 by The University of Chicago. 0003-0147/95/4501-0002$02.00. All rights reserved.

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CONSUMERS AS MAXIMIZERS OF ENERGY FLOW 23

grained in our civilization both culturally and economically. The struggle for life is presented to us from our earliest days under the fascinating form of terrible predators devouring their innocent prey, and these images often continue to shape our perception of reality later. The need for a smooth functioning of the economy imposes a constant fight against other animal species feeding on our plant food resources, which are therefore viewed as undesirable pests from which we must protect ourselves. Ecology as a science has had to establish a more balanced view of nature, yet even in ecology consumer-resource interactions (predator-prey, herbivore-plant, etc.) have been studied most of the time under the aspect of the benefits it provides to the consumer and the losses it makes the resource incur (the traditional +, - interaction). In an immediate sense, and most of the time for the individuals engaged in the interaction, consumption is indeed a +, - interaction by its very nature. However, within the context of the whole ecosystem, consumers may fulfill a positive function, which may feed back to the resource as a positive indirect effect. In such a case, the net effect of consumers on their resources may even be positive.

Several authors have argued that consumers are not simply "parasitic" on autotrophs but play active, positive functions in ecosystems, either as ecosystem regulators or as accelerators of nutrient cycling (Chew 1974; Lee and Inman 1975; Mattson and Addy 1975; O'Neill 1976; Hayward and Phillipson 1979; Kitchell et al. 1979; Seastedt and Crossley 1984). As the introductory quotation shows, Lotka (1956) lucidly formulated the view that consumers may benefit the entire ecosystem by increasing the rate of material circulation. This view has been supported by numerous studies on saprophage communities in both terrestrial and aquatic ecosystems, which showed that not only decomposers but also higher-level consumers in the detrital food chain had a positive effect on nutrient cycling (see reviews in Verhoef and Brussaard 1990; DeAngelis 1992). Recently there has been a growing interest in the way plants may benefit from the herbivores that eat them. Owen and Wiegert (1976, 1981), McNaughton (1979, 1983), Owen (1980), Hilbert et al. (1981), and Petelle (1982) have suggested several ways by which moderate grazing can optimize plant productivity and increase plant fitness, one of them being increased nutrient cycling. Although the empirical evidence for this "grazing optimization hypothesis" has been debated (Belsky 1986, 1987; McNaughton 1986; Brown and Allen 1989), many studies continue to provide results supporting this hypothesis in both terrestrial and aquatic environments (DeAngelis 1992).

Yet there have been relatively few theoretical explorations of the potential role of consumers through increased material cycling in ecosystems using dynamic mathematical models. Most dynamic models have been used to investigate the effect of nutrient cycling on ecosystem stability (see reviews in DeAngelis et al. 1989; DeAngelis 1992; see also Loreau 1994). Several detailed models have been able to reproduce the beneficial effects of consumers in the detrital food chain on nutrient cycling (Barsdate et al. 1974; Douce and Webb 1978) or the grazing optimization hypothesis (Carpenter and Kitchell 1984; Dyer et al. 1986). These models, however, were system specific and leave unanswered the question to what extent and under what conditions their results can be generalized. Recently



24 THE AMERICAN NATURALIST

ConsumersQ u2r p DecomposersN X 2 F

o- D2 l

f2 (XsI X2)

I eX, eDS_

XI D

~~~N lErco 0

FIG. 1.-Flow diagram of the model (eqq. [1]-[5])

DeAngelis (1992) analyzed a simple abstract ecosystem model and again found maximum phytoplankton production at intermediate values of herbivory for certain parameter values, but he did not explore under what general conditions this maximization occurs.

In this article I explore Lotka's (1956) ideas explicitly. My objective is to investigate the conditions under which consumers maximize matter and energy flow, and hence primary production, in ecosystems, using a simple, general abstract ecosystem model.

THE MODEL

The model pictures a nutrient-limited ecosystem as simply as possible (fig. 1). In the absence of consumers, the ecosystem is made up of three functional compartments only: the available limiting nutrient pool (N), the producers or autotrophs (XI) (or, more exactly, the nutrients contained in their biomass), and their decomposers (D1). Consumers (X2) and their decomposers (D2) are then added to this basic structure. Nutrients are recycled to the nutrient pool by the decomposers. The latter are divided into the two functional compartments DI and D2 to allow different decomposition rates of the detritus from the producers and the consumers; this, however, does not preclude that part of the two decomposition processes may be caused by the same decomposer species. The size of all compartments is measured in the same unit of quantity of nutrients, which is assumed to be proportional to biomass.

Nutrient recycling by decomposers is assumed to be a linear donor-controlled process, but the material flows from the nutrient pool to the producers (fi(N, XI)) and from the producers to the consumers (f2(XI, X2)) are left unspecified at this point. The openness of the system with respect to matter is modeled under




the form of a flow of nutrients at a constant rate per unit mass e through the system. The assumption of a unique rate parameter e for all compartments is made both for simplicity and because it allows one to recover the case of a closed ecosystem for e = 0. This case is of special interest because its analysis leads to transparent expressions that help the interpretation of the results. It may be an acceptable approximation for some mature ecosystems that are almost closed with respect to some nutrients (Odum 1969; Howard-Williams and Allanson 1981; Wood et al. 1984).

The model thus reads

d= eQ + d1DI + d2D2 -f1(N,XI) - eN, (1)

dt dt fi (N, XI) - A (XI IX2) - b IXI - eXI (2)

dX2 =f2(XI, X2) - b2X2 - eX2, (3) dt

dD1 = b1XI - d1DI - eDI, (4) dt

and

dD2 = b2X2 - d2D2 - eD2. (5) dt

When the ecosystem is closed (e = 0), the first equation is replaced by the following relation expressing the conservation of mass:

N + X1 + X2 + D1 + D2 = Q (6)

In these equations, Q is the inflowing quantity of nutrients in a volume equivalent to that of the ecosystem considered (at equilibrium or in a closed ecosystem, it is also the total quantity of nutrients within the ecosystem); b, and b2 are the rates at which a unit quantity of nutrients in the producers and the consumers, respectively, is transferred to the decomposers as detritus; and d1 and d2 are the rates at which nutrients are released to the nutrient pool at the close of the decomposition processes through D1 and D2, respectively.

To analyze variations in matter and energy flow in this model ecosystem, the material flows or trophic functions f1(N, X1) and f2(X1, X2) now need to be formulated explicitly. The per capita uptake rate of nutrients by autrotrophs is usually modeled by a Michaelis-Menten function (DeAngelis 1992). In the cases analyzed here- after, f1(N, X1) will be expressed under the form of a Lotka-Volterra term, that is,

fi (N, X1) = aNX, . (7)

This form leads to the same qualitative conclusions as the equivalent form based on a Michaelis-Menten function but is simpler to analyze. Both forms imply that autotrophs actively control the size of the nutrient pool in the absence of consum-




ers. Note that a in equation (7) is a coefficient determined in part by the specific characteristics of the producers and in part by the amount of energy available to them. Energy is not considered explicitly in this model, but the flow of matter to the producers is directly dependent on a simultaneous flow of energy through photosynthesis. The amount of available energy is thus incorporated in the parameter a.

There has been considerable argument about the nature of the control of the material flows between successive trophic levels in ecosystems. In general, the control of trophic interactions is shared by the donor and recipient populations (Odum and Biever 1984; McQueen et al. 1986; Hunter and Price 1992; Power 1992), but it can vary from almost complete donor control to almost complete recipient control. At the ecosystem level, donor control may be a good approximation near equilibrium because of the superposition of a large number of individual interactions (Patten 1975; Strong 1992). For the sake of generality, two simple cases will be analyzed below for the trophic interaction between producers and consumers-first, a linear donor-controlled trophic function,

f2(XI, X2) = cX1I, (8)

and then a Lotka-Volterra recipient-controlled trophic function,

f2(XI, X2) = cX X2* (9)

Finally, the behavior of the model with the more general function proposed by DeAngelis et al. (1975), which can describe the whole range of situations from donor to recipient control, for both f1 and f2 will be investigated numerically.

CASE I: ECOSYSTEM WITHOUT CONSUMERS

To study the effect of consumers on matter and energy flow, let us first consider the situation in which consumers are absent. The nutrient pool, producer and decomposer compartments in model (1)-(7) then reach the following sizes at steady state or equilibrium (denoted by an asterisk):

N* = Ql, (10)

X*= (Q - QI)/(bIwI), (11)

and

D= [b1/(d1 + e)]X*, (12)

where

Qi (b1 + e)la (13)

and

I= l/b + 1/(d1 + e). (14)

Variable Q1 is the threshold quantity of nutrients necessary for the persistence of the producers, since equation (11) requires Q > Ql. This inequality may also




be written as

aQ > b, + e, (15)

which shows that persistence of the producers hinges on there being sufficient amounts of both energy (incorporated in a) and matter (Q) in the ecosystem.

Since l/b1 is the mean residence time in the producer compartment of a unit quantity of nutrients destined to be decomposed in the ecosystem, and 1/(d1 + e) is its mean residence time in the decomposer compartment w, represents the mean transit time of a unit quantity of nutrients through the path X1 - D1. When the ecosystem is closed (e = 0), w, is simply the mean residence time of a unit quantity of nutrients in the biomass.

Since all the energy flowing in the ecosystem enters through the producer compartment and drives the material flow from the nutrient pool to the producers, the energy flow (D1 in the ecosystem will be measured simply by the material flow from the nutrient pool to the producers, f1(N, X1), assuming an appropriate transformation of units (since the changes in energy flow will be investigated here only qualitatively, all this assumes is that energy flow is a monotonic increasing function of material flow). Thus, at equilibrium,

(DI = aN*X* (b1 + e)(Q - QI)/(bw11). (16)

When the ecosystem is closed (e = 0), this equation reduces to

(= (Q- QI)/1W< (17) Under this form, the flow of matter and energy appears clearly as the product of the quantity of matter in circulation in the biomass (which is Q - Q I) by its rate of revolution or circulation in the biomass (which is the inverse of W1), in accor- dance with the introductory quotation from Lotka (1956).

CASE 2: LINEAR DONOR-CONTROLLED INTERACTION BETWEEN PRODUCERS

AND CONSUMERS

When the consumers are now added to the above ecosystem and the material flow from producers to consumers has a linear donor-controlled form (eq. [8]), the system reaches a new equilibrium at the following compartment sizes:

N* = Q2, (18)

X* = (Q - Q2)/(blw) + cy2), (19)

X* = [cl(b2 + e)]X*, (20)

D* = [bll(d1 + e)]X*, (21)

and

D* = [b2l(d2 + e)]X*, (22)

where




Q2 = (c + b, + e)la (23)

and

Y2 = (b2 + d2 + e)l[(b2 + e)(d2 + e)]. (24)

In the same way as before, Q2 iS the threshold quantity of nutrients necessary for the persistence of the producers when the consumers are present. Variable Y2 has a clear interpretation only when the ecosystem is closed (e = 0). It then reduces to

W2= 1/b2 + lId2, (25)

which is the mean transit time of a unit quantity of nutrients through the path X2 - D2

The energy flow is now

(')2 = aN*X* = (c + b, + e)(Q - Q2)I(b1ll

+ cy2) (26)

When the ecosystem is closed (e = 0), this reduces to

D2= (Q - Q2)lw 9 (27)

where X- = (b1 1 + cW2)/(b1 + c) is the average transit time of a unit quantity of nutrients through the two paths X1 - D1 and X2 - D2, that is, in the biomass. Note that this expression is independent of the lengths of the two paths. The two decomposition processes have been assumed to be one-step processes each associated with a single compartment (D1 or D2) in the model for simplicity, but it can easily be shown that modeling these processes as chains with an arbitrary number of steps and compartments does not alter equation (27) and the interpretation of (' and (w2-

Since Q2 > Q1 for any c > 0, it can be seen immediately from comparison of equations (17) and (27) that, in a closed ecosystem, energy flow is increased in the presence of consumers (02 > (D) when XW is sufficiently smaller than WI, hence, when W2 iS sufficiently smaller than l. More precisely, the inequality (D2 > (DI provides the condition

b1 + c(w2/w1) < Q - Q2 (28) b + c Q -Qi'

This condition is qualitatively similar to condition (31), which will be discussed below.

One advantage of the linear donor-controlled form of the trophic interaction between producers and consumers is that there is a smooth transition from the case when consumers are absent (c = 0) to the case when consumers are present (c > 0). The parameter c is a straightforward measure of the extent of consumption in the ecosystem. Therefore, a detailed analysis of the effect of consumption on energy flow is easy by taking the partial derivative of 12 with respect to c:

aD2/aC = {(Q - Q2)[b1I1 - (b1 + e)y2] - Q2(b1Iw + cy2)}/(b1Iw + cy2)2. (29)




The value of a'D2/ac can be positive only if the first term in square brackets in the numerator is positive. The numerator then decreases monotonically with increasing c and is always negative for large c. Therefore, energy flow is maximized at intermediate values of c if and only if a'D2/c is positive at c = 0. This leads to the following condition:

(b, + e)y2 < Q - 2Q, (30)

Once again this condition is much more transparent for a closed ecosystem. It then reduces to

W2 Q - 2Q t)2< Q 2Q* (31)

This condition requires that, first, W2 be sufficiently smaller than w1-that is, the presence of consumers accelerate the circulation of matter in the ecosystem sufficiently-and, second, Q > 2Q1, which is a more stringent condition than condition (15); thus the quantity of matter in the ecosystem must be twice larger than the threshold quantity necessary for the persistence of the producers.

The curve of (12 as a function of c is illustrated in figure 2 for two values of the ratio W2/W1 in the case of an open system. As expected from conditions (30) and (31), for W2/W1 = 1 (fig. 2A) (2 decreases monotonically with increasing c, while for W2/W1 = 0.1 (fig. 2B) 12 iS maximized at an intermediate value of c. On the other hand, the equilibrium sizes of the various compartments always follow the same pattern as a function of c. The signs of their partial derivatives with respect to c remain unchanged by variations in the ratio W2/1l:

aN*Iac = 1la > 0; (32)

aX*/ac = -[(Q - Q2)ay2 + blwl + cy2]1[a(bIw1 + cy2)2] < 0; (33)

and

aX*/aC = [(Q - Q2)ablwl - c(bll + cy2)]1[a(b2 + e)(blwl + CY2)2], (34)

>Oatc = 0 and <0athighc.

Thus N* always increases with c because the control exerted by the producers becomes weaker, X1 (hence also D*) always decreases with increasing c because an increasing part of its biomass is consumed, and X* (hence also D*) always has a maximum at an intermediate value of c because it benefits more from the increasing consumption rate at low c while it suffers more from the declining abundance of its resource X1 at high c. Note that the ecosystem collapses when Q2 (which increases with c) becomes equal to Q, that is, when c = aQ - b, - e.

CASE 3: LOTKA-VOLTERRA INTERACTION BETWEEN PRODUCERS AND CONSUMERS

When the trophic interaction between producers and consumers has a Lotka- Volterra form (eq. [9]), the equilibrium sizes of the compartments are quite differ-




10

8

6

2

0 1 2 3 4 5 6

C

10

8

6

2

0 1 2 3 4 5 6

C

FIG. 2.-Energy flow ((D) and equilibrium compartment sizes as a function of the consumption rate, c, when the producer-consumer interaction has a linear donor-controlled form (eq. [8]). A, W2 = wl; consumption always decreases energy flow. B, W2 = 0.1 w; energy flow is maximized at intermediate consumption rates. Parameter values: Q = 8 and a = d= = e = 1 in both cases; b2 = d2 = 1 in A; b2 = 10 and d2 = 19 in B.

ent from those in the previous cases:

N* = (b, + e + cX*)/a, (35)

X* = (b2 + e)lc, (36)

X* = [(Q - Ql)ac - (b2 - e)abjw1]I[c(c + ab2w2)], (37)

D* = bl(b2 + e)l[c(d1 + e)], (38)

and




D* = [b21(d2 + e)]X*, (39)

where

W2 = 1/b2 + 1/(d2 + e) (40) has the same form and interpretation as w, (eq. [14]).

The energy flow is now found to be

= aN*X* = a(b2 + e)[cQ + (b1 + e)b2w2 (41)

- (b2 + e)bjw1]I[c(c + ab2W2)] 4

While in the previous cases the producers were the controlling compartment and there was accordingly a smooth transition from a system without consumers to a system with consumers, in this case the Lotka-Volterra interaction between producers and consumers gives the control to the consumer compartment. Ac- cordingly, there is an abrupt transition from a system without consumers to a system with consumers. This makes the direct comparison between the energy flows with and without consumers difficult. However, a comparison is possible by studying the curve of F2 as a function of c beyond the transition point. From equation (37) it can be seen that the consumers can persist only if c is greater than some minimum value, c mj, at which X* vanishes. This is found to be

Cmin = (b2 + e)blwl/(Q - Qj). (42)

It can readily be verified that at this value the system recovers its equilibrium with the consumers absent. Thus the transition occurs at this positive value of c instead of c = 0 in the previous case.

The analysis of the effect of consumption on energy flow can now be done again using the partial derivative of F2 with respect to c:

a'121aC = {(2c + ab2w2)[(b2 + e)blwl - (b1 + e)b22] - c2 Q}

x a(b2 + e)l[c(c + ab2W2)]2.

The value for a'121aC can be positive only if the first term in square brackets in the numerator is positive. The numerator then vanishes for a single positive value of c and is always negative for large c. Therefore energy flow is maximized at intermediate values of c if and only if a'I21IC is positive at c = cmin. This gives the inequality

[atl(b1 + e)](W2/1w)(1 - tW2/W1)(Q - Qi)2

+ 2(1 - (W2/W1)(Q - Q1) - Q > 0

where

= (b1 + e)b21[(b2 + e)bl]. (45)

Although equation (44) is fairly complicated, it is easy to see that it requires that two conditions be fulfilled. First, the ratio W2/W1 must be sufficiently smaller than t, or, when the ecosystem is closed (hence e = 1), W2 must be sufficiently




10

A 8

6 -G

4

2

x2

0 1 2 3 4 5 6 7 8 9 10

C

10

8

6

4

2

0 1 2 3 4 5 6 7 8 9 10

C

FIG. 3. -Energy flow ((D) and equilibrium compartment sizes as a function of the per capita consumption rate, c, when the producer-consumer interaction has a Lotka-Volterra form (eq. [9]). A, co2 = w1; consumption always decreases energy flow. B, W2 = 0.1 w1; energy flow is maximized at intermediate consumption rates. Parameter values are as in fig. 2.

smaller than wl. Second, when W2/W1 is as small as it can be, that is, when it is equal to zero, the inequality (44) reduces to Q > 2Q1, which is thus a second necessary condition to be fulfilled.

These conditions are qualitatively identical to those found in the previous case. It can also be shown, using the partial derivatives of the compartment equilibrium sizes with respect to c, that these compartment equilibrium sizes behave in the same way as in the previous case when c is increased above the threshold value Cmin* The only qualitative difference is that in this case there is no upper threshold value at which the ecosystem collapses. The curves of energy flow and compartment equilibrium sizes as a function of c are illustrated in figure 3 for the same parameter values as in figure 2.




CASE 4: INTERACTION BETWEEN PRODUCERS AND CONSUMERS: GENERAL CASE

To verify whether the above results derived analytically hold under more general conditions, the model was investigated numerically using the general trophic function of DeAngelis et al. (1975) to describe both the material flows from the nutrient pool to the producers and from the producers to the consumers:

fi(N, X1) = aNX1I(k1 + p1N + q1X1) (46) and

f2(X1, X2) = cX1X2I(k2 + p2X1 + q2X2) (47) This functional form allows one to recover most other usual trophic functions as special cases: the linear donor-controlled form is obtained by letting ki = pi = 0; the Lotka-Volterra form is obtained by letting pi = qi = 0; the Michaelis- Menten or Holling Type II form is obtained by letting qi = 0; and the ratio- dependent form (Arditi and Ginzburg 1989) is obtained by letting ki = 0.

At equilibrium, the compartment sizes and energy flow are then

N* = y- tX*, (48)

X = [B + (B2 + 4AC)05]/2A, (49)

X2* = {[c - P2(b2 + e)]I[q2(b2 + e)]}X* - k2lq2, (50)

D* = [bI1(d1 + e)]X*, (51)

D* = [b2l(d2 + e)]X2*, (52)

and

()2 = 3X* - k2(b2 + e)lq2, (53)

where

A = aq2ct - q2 p(p1co - ql) (54)

B = aq2Y - q2 P(p1Y + kl) - k2(b2 + e)(pcLa - ql), (55)

C = k2(b2 + e)(p1y + kl), (56)

a = bll + [c - p2(b2 + e)]y2lq2, (57)

@ = b, + e + Ic - P2(b2 + e)]lq2, (58)

and y, = Q + k2b2W2/q2, (59)

and w1, W2, and Y2 are defined as before by equations (14), (40), and (24), respectively.

Although formulated explicitly, these equilibrium compartment sizes and energy flow are too complex to provide an insight into the conditions under which consumers maximize energy flow. Consequently, they were investigated numeri-




10

8

S~~~~4 N

6

4

2

10

4~~~~~~~~~~~~~~~4

2

0 1 2 3 4 5 6 7 8 9 10

FIG. 4. -Energy flow ((D) and equilibrium compartment sizes as a function of the per capita consumption rate, c, when the nutrient-pool-producer and producer-consumer interactions are described by DeAngelis et al.'s (1975) function (eqq. [46], [47]). A, co2 = @1; consumption always decreases energy flow. B, W2 = 0.1 w1; energy flow is maximized at intermediate consumption rates. Parameter values: k1 = 0.75, Pi = 0.05, q1 = 0.01, k2 = 0.5, P2 = 0.05, and q2 = 1. Other parameter values are as in fig. 2.

cally. The results were in agreement with the above conclusions derived analytically; one example is illustrated in figure 4. In all cases, maximization of energy flow by consumers at intermediate values of c hinge on (w2 being sufficiently smaller than 01. The second condition, that the quantity of matter Q be greater than some threshold value, is valid generally, except in the limiting cases where ki 0, that is, in completely linear donor-controlled or ratio-dependent models. This can readily be proved analytically and is easily understood since linear donor-controlled and ratio-dependent models share the same basic property of having equilibrium compartment sizes that are all proportional to each other.




CASE 5: TWO CONSUMER TROPHIC LEVELS

A last issue that may be raised concerns the robustness of the above conclusions to the addition of a second consumer trophic level to the model. Two other equations for the secondary consumers (X3) and their decomposers (D3) then need to be added to model (1)-(6):

dX3 dt =f3(X2,X3) - b3X3 - eX3 (60)

and

dD3 b dt= b3X3 - d3D3 - eD3 (61)

The equations for X2 and N need, of course, to be modified accordingly; that is, f3(X2, X3) must be subtracted from the right-hand side of equation (3) and d3D3 must be added to the right-hand side of equation (1).

I shall only mention the end results of the analysis for two cases generalizing cases 2 and 3 investigated above. That is, first, both producer-consumer and consumer-consumer trophic interactions have a linear donor-controlled form:

f3(X2,X3) = c'X2, (62)

andf2(XI, X2) is defined as above by equation (8); and, second, both producer-consumer and consumer-consumer trophic interactions have a Lotka-Volterra form:

f3(X2,X3) = C'X2X3, (63)

and f2(XI, X2) is defined as above by equation (9). In the first case, the addition of another consumer trophic level does not modify

the results because the material flow to the primary consumers is controlled by the producers and thus only the distribution of this flow between the two consumer trophic levels can be affected. The energy flow is indeed found to be

(I3 = (c + b1 + e)(Q - Q2)I(b1w1 + cz2,3), (64)

where

Z2,3 - [(b3 + e)b2W2 + b3c'W3]1[(b3 + e)(b2 + c' + e)] (65)

and

(X3 = 1/b3 + 1I(d3 + e). (66)

Equation (64) is identical to equation (26) except that Y2' which contains parameters from the single path X2 - D2, is now replaced by z2,3, which contains parameters from the two paths X2 - D2 and X3 - D3.

As before, this expression becomes much more transparent for a closed ecosystem:

(I3 = (Q - Q2)lI, (67) where

X = (blwl + c 2,3)I(b1 + c) (68)




and 2,3 = (b2W2 + C'W3)1(b2 + c'). (69)

Thus, as before, -X is the average transit time of a unit quantity of nutrients in the biomass, while Wi2,3 is the average transit time of a unit quantity of nutrients in the consumers. The condition for energy flow to be increased in the presence of consumers (43 > (DI) is again that -X be sufficiently smaller than x,, hence, that 2 3 (instead of W2) be sufficiently smaller than wl.

In the second case, when the trophic interactions have the Lotka-Volterra form, the addition of a second consumer trophic level modify the results more substantially because it changes the pattern of control: the secondary consumers now control the primary consumers and thus partly release the producers from their control by the primary consumers. In this case, the energy flow is

()3 = [(b3 + e)clc' + b, + e](Q - Qi - W)/(b1wl + b3W3c/c'), (70)

where

W = (b3 + e)cl(ac') + b3d3b2W21[c'(d3 + e)] - b2d2b3W3/[c'(d2 + e)]. (71) For simplicity, let us consider the case of a closed ecosystem (the results are

similar when the system is open). The condition for energy flow to be increased in the presence of consumers (I3 > (D) is then, from equations (17) and (70),

b, + (b3clc')(W31w1) Q - Q - W b1 + b3clc' Q- Q (

This inequality is similar to the corresponding inequality (28) for the donor- controlled interaction between producers and consumers. By partly releasing the producers from their control by the primary consumers, the addition of a second consumer trophic level results in an interaction between producers and consumers that is akin to a donor-controlled interaction.

A major difference from equation (28) is that in equation (72) W3 instead of W2 must be sufficiently smaller than wl in the most usual case where W 2 0. The condition then concerns the secondary consumers, even though these have no direct interaction with the producers. There is, however, the possibility that W < 0. In a closed ecosystem, this occurs when W3 > W2 + cl(ab2). Under these conditions, energy flow can be increased by the consumers even when W3 > 1l But it is possible to prove that for this to be true and at the same time for X* to be feasible, it is also necessary that wl > W2 + cl(ab2). Thus, in any case either W3 or W2 (or both) must be sufficiently smaller than wl. These two cases are illustrated in figure 5. Note that when it is W2 that is smaller than wx (fig. 5B), energy flow is maximum for low c when only primary consumers persist; secondary consumers actually decrease energy flow but maintain it above its level without the consumers (at c = 0).

DISCUSSION

The present theoretical study shows that consumers can play an important functional role as maximizers of matter and energy flow in nutrient-limited eco-




40

A 30 \

N*

20

10 \ s_''-@ 10

0 1 2 3 4 5 6 7 8 9 10 11 12 13

C

40

B 30

20 X. M X3

1 0 X2

10 , _0 ,F J t!H e IV

0 1 2 3 4 5 6 7 8 9 10 11 12 13

C

FIG. 5.-Energy flow ('D) and equilibrium compartment sizes as a function of the per capita consumption rate, c, when there are two consumer trophic levels with Lotka-Volterra interactions (eqq. [60]-[63]). A, 13 = 0.1 ( = 0.1 w2; B, w2 = 0.1 ( = 0.1 (1)3. In both cases energy flow is maximized at intermediate consumption rates. Parameter values: Q = 40, a = 5, and b, =d = e = c' = 1 in both cases; b2 = d2 = 1, b3 = 10, and d3 = 19 in A; b2 = 10, d2 = 19, and b3 = d3 = 1 in B. Values of energy flow have been divided by three.

systems and that the qualitative conditions under which they play such a role are quite general. The first and most important condition is that consumers act to increase the circulation of matter within the ecosystem through accelerated nutrient cycling. Although the quantitative details of this condition change depending on the specific functional forms in the model, in all cases (linear donor-dependent, Lotka-Volterra, or DeAngelis et al. 's [1975] trophic interactions, one or two consumer trophic levels), maximization of energy flow by consumers occurs provided that the mean transit time of nutrients in the path consumers-decomposers-




nutrient pool is sufficiently smaller than their mean transit time in the path producers-decomposers-nutrient pool. The second condition is that the total quantity of nutrients in the ecosystem must be higher than some threshold value. For a very low quantity of nutrients, the producers simply cannot persist at steady state; for a quantity of nutrients above the threshold for persistence of producers but below twice this threshold, the producers and consumers can persist, but consumers always reduce energy flow-only for a higher quantity of nutrients can consumers increase energy flow. This second condition, however, is more dependent on the functional form of the nutrient uptake by the producers and disappears when this has a linear or ratio-dependent form.

In all cases, maximization of matter and energy flow by consumers occurs at moderate consumption rates. The equilibrium producer biomass always decreases as a result of consumption. When consumption rates are too high, the negative effect of decreased producer biomass prevails over the positive effect of accelerated material circulation, so that matter and energy flow, which depends both on the quantity of matter and its rate of circulation, is decreased.

These results were obtained at equilibrium and thus are valid only when the ecosystem reaches a stable equilibrium. Ecosystem stability has been the object of a number of investigations already (see DeAngelis et al. 1989; DeAngelis 1992; Loreau 1994) and was not the focus in this work. It is known that model ecosystems with nutrient cycling have a stable equilibrium when trophic interactions have a Lotka-Volterra form (Nisbet and Gurney 1976) or a linear donor-controlled form (Hearon 1968; Mazanov 1976). Since the model considered here combines these two forms in the particular cases analyzed in detail, the equilibrium should be stable in these cases. A formal proof is given in the appendix. In the more general case in which trophic interactions are described by DeAngelis et al.'s (1975) function, local instability of the equilibrium is possible for some parameter values, which leads to a limit cycle. Whether consumers can also maximize matter and energy flow in periodic systems is an interesting though more difficult issue, which was not addressed here.

The above conclusions confirm Lotka's (1956) seminal views and provide a rigorous theoretical foundation for empirical studies on the effects of consumers on nutrient cycling and ecosystem functioning. In an extensive review of the contribution of soil animals to decomposition in terrestrial ecosystems, Verhoef and Brussaard (1990) concluded that the faunal contribution to nitrogen mineralization typically amounts to about 30%, while the proportion of soil animals in total biomass is an order of magnitude lower. This suggests that secondary consumers in the detrital food chain accelerate nutrient cycling by roughly an order of magnitude. Similar values have been found in aquatic ecosystems (Buechler and Dillon 1974; Gallepp 1979). Under such conditions, the model predicts almost invariably (except for very low amounts of nutrients or energy in the ecosystem) that these consumers contribute to increase energy flow and hence overall productivity of the ecosystem. There is no reason why this would not be true for herbivores too. Although Belsky (1986, 1987) criticized the empirical data in support of the grazing optimization hypothesis, increase of plant productivity by herbivores is theoretically as likely as it is by decomposers. From an ecosystem




viewpoint, as long as grazing is moderate and does not impair the physiological capacities of autotrophs, decomposition and grazing are but alternative pathways for the flow of matter and energy from the autotrophs. Grazing can also accelerate nutrient cycling (Ruess and McNaughton 1987), and under these conditions the present model predicts that it contributes to increase energy flow and plant production. The model with a linear donor-controlled producer-consumer interaction (fig. 2) exactly reproduces the grazing optimization curve of Hilbert et al. (1981). In fact, increased production by as much as a factor of two was sometimes found as a result of faster nutrient cycling due to herbivory (Doering et al. 1986). Note, however, that this conclusion applies to the plant trophic level as a whole; the way the possible benefit of grazing is shared among individuals and species and turns into individual selective pressure is a distinct and more complex issue, which depends on such factors as plant spatial distribution, grazing distribution and selectivity, and plant adaptive strategies (M. Loreau and D. L. DeAngelis, unpublished manuscript). Thus the present results do not imply that every single plant-herbivore interaction would represent a form of indirect mutualism.

Also, the fact that consumers can maximize energy flow and production in ecosystems is, of course, no guarantee that they necessarily do so. Well-known cases of sudden devastation of plants by herbivores in agriculture and forestry attest to the contrary. In the long run, however, natural selection should tend to optimize the functioning of natural ecosystems and maximize energy flow or power output (Lotka 1922; Odum and Pinkerton 1955; Odum 1983). Increased production in an ecosystem is beneficial to all its functional components, and traits that result in increased productivity can be selected for in heterogeneous environments even when these traits entail costs for those individuals that bear them (Wilson 1976, 1980). Thus, it is likely that consumers as a whole do generally play a role of maximizers of matter and energy flow in stable natural ecosystems.

This view might seem to be contradicted by the tendency of ecosystems to evolve toward larger biomass, larger organism size, and lower production/biomass ratios during succession (Margalef 1963; Odum 1969). A lower production per unit biomass entails a longer mean transit time of nutrients in the biomass; if the circulation of nutrients tends to be slower in mature ecosystems, does this not indicate that maximization of energy flow by consumers through faster nutrient cycling is irrelevant? This is not so, for two main reasons. First, the trend toward lower production-biomass ratios and hence slower circulation of nutrients during succession may not be as general as thought originally; this trend is not apparent in marine ecosystems, which prompts the suggestion that, in terrestrial ecosystems, it represents an adaptation to the physical environment (Baird et al. 1991). As an example, terrestrial plants need to invest nutrients and biomass in slow-turnover woody structures to be able to further increase biomass and take up nutrients because of the physical constraints imposed by the largely two- dimensional terrestrial environment. Another factor that may come into play is the increased homeostasis allowed by a larger size. In any case, a lower production per unit biomass and a slower circulation of nutrients may be the price to pay for an increase in biomass and homeostasis rather than traits that are beneficial per se. Second, total production and energy flow is determined both by the




total biomass and by its rate of revolution, as shown particularly clearly in equations (17), (27), and (67). If, during succession, biomass increases at a faster rate than does its mean turnover time, total production increases even though production per unit biomass decreases, as is usually observed (Odum 1983). An increase in biomass should then be favored even at the cost of a slower circulation of nutrients. Despite this trend in ecosystem development through time, any process that would accelerate nutrient cycling without decreasing biomass sig- nificantly would still be favorable for overall ecosystem productivity. This should be especially true at steady state, when biomass no longer increases. In fact, moderate consumption by heterotrophs may be viewed as a process that contributes to enhance a vital ecosystem function (nutrient cycling) that tends to slow down for other reasons during the course of ecosystem development. This only reinforces the positive aspect of its contribution to ecosystem functioning.

ACKNOWLEDGMENTS

I thank D. L. DeAngelis for his interest in this work and his comments on the manuscript. This work was done in part at Oak Ridge National Laboratory, with support from the Belgian National Fund for Scientific Research, the Royal Acad- emy of Belgium, and the U.S. National Science Foundation's Ecosystem Studies Program through Interagency Agreement DEB-9013883 with the U.S. Department of Energy. Oak Ridge National Laboratory is managed by Martin Marietta Energy Systems, Inc., under contract DE-AC05-84OR21400 with the U.S. Department of Energy.

APPENDIX STABILITY OF THE MODEL ECOSYSTEM

The stability of the system described by equations (l)-(5) can be analyzed along the same lines as those described elsewhere (Loreau 1994).

Let us first replace N by the total quantity of nutrients within the system, T = N + X1 + X2 + D1 + D2, as a dynamic variable. Equation (1) is then changed into

-= eQ-eT. (Al) dt

The system then decomposes into equation (Al), which governs the dynamics of the total quantity of nutrients, and the set of equations (2)-(5), which govern the dynamics of the biological components of the system (where N has been replaced by T - X- X2 - D, - D2). From equation (Al), it is easily seen that T reaches the equilibrium T* = Q, which is stable for any e > 0. When the ecosystem is closed (e = 0), the total quantity of nutrients becomes neutrally stable, which merely expresses the constraint of matter conservation.

The local stability of the subsystem (2)-(5) at equilibrium is determined by the eigenval- ues of its Jacobian matrix, which is, in cases 2 and 3 (trophic interactions described by eqq. [7]-[9]),

[al1 a12 - aX* - aX*

a21 a22 0 0 A =. b1 0 -dl - e 0

L 0 b2 0 -d2 - e




In the first case, when f2(XI,

X2) = cX1, all = a12 = -aX*, a2l = c, and a22 = b- e. In the second case, when f2(XI, X2) = cX1X2, all = -aX*, a12 = -aX* - b2 - e, a2l = cX*, and a22 = 0. In both cases, it is not difficult to verify that the classical Routh-Hurwitz criteria or, equivalently, the qualitative loop analysis criteria (Puccia and Levins 1985) are met, which ensures local stability of the equilibrium.

LITERATURE CITED

Arditi, R., and L. R. Ginzburg. 1989. Coupling in predator-prey dynamics: ratio-dependence. Journal of Theoretical Biology 139:311-326.

Baird, D., J. M. McGlade, and R. E. Ulanowicz. 1991. The comparative ecology of six marine ecosystems. Philosophical Transactions of the Royal Society of London B, Biological Sci- ences 333:15-29.

Barsdate, R. J., R. T. Prentki, and T. Fenchel. 1974. Phosphorus cycle of model ecosystems: signifi- cance for decomposer food chains and effect of bacterial grazers. Oikos 25:239-251.

Belsky, A. J. 1986. Does herbivory benefit plants? a review of the evidence. American Naturalist 127:870-892. 1987. The effects of grazing: confounding of ecosystem, community, and organism scales. American Naturalist 129:777-783.

Brown, B. J., and T. F. H. Allen. 1989. The importance of scale in evaluating herbivory impacts. Oikos 54:189-194.

Buechler, D. G., and R. D. Dillon. 1974. Phosphorus regeneration in fresh-water paramecia. Journal of Protozoology 21:339-343.

Carpenter, S. R., and J. F. Kitchell. 1984. Plankton community structure. and limnetic primary production. American Naturalist 124:159-172.

Chew, R. M. 1974. Consumers as regulators of ecosystems: an alternative to energetics. Ohio Journal of Science 74:359-370.

DeAngelis, D. L. 1992. Dynamics of nutrient cycling and food webs. Chapman & Hall, London. DeAngelis, D. L., R. A. Goldstein, and R. V. O'Neill. 1975. A model for trophic interaction. Ecology

56:881-892. DeAngelis, D. L., P. J. Mulholland, A. V. Palumbo, A. D. Steinman, M. A. Huston, and J. W.

Elwood. 1989. Nutrient dynamics and food-web stability. Annual Review of Ecology and Systematics 20:71-95.

Doering, P. H., C. A. Oviatt, and J. R. Kelly. 1986. The effects of the filter-feeding clam Mercenaria mercenaria on carbon cycling in experimental marine mesocosms. Journal of Marine Re- search 44:839-861.

Douce, G. K., and D. P. Webb. 1978. Indirect effects of soil invertebrates on litter decomposition: elaboration via analysis of a tundra model. Ecological Modelling 4:339-359.

Dyer, M. I., D. L. DeAngelis, and W. M. Post. 1986. A model of herbivore feedback on plant productivity. Mathematical Biosciences 79:171-184.

Gallepp, G. W. 1979. Chironomid influence on phosphorus release in sediment-water microcosms. Ecology 60:547-556.

Hayward, G. F., and J. Phillipson. 1979. Community structure and functional role of small mammals in ecosystems. Pages 135-211 in D. M. Stoddart, ed. The ecology of small mammals. Chap- man & Hall, London.

Hearon, J. Z. 1968. Theorems on linear systems. Annals of the New York Academy of Sciences 108:38-68.

Hilbert, D. W., D. M. Swift, J. K. Detling, and M. I. Dyer. 1981. Relative growth rates and the grazing optimization hypothesis. Oecologia (Berlin) 51:14-18.

Howard-Williams, C., and B. R. Allanson. 1981. Phosphorus cycling in a dense Potamogeton pectina- tus L. bed. Oecologia (Berlin) 49:56-66.

Hunter, M. D., and P. W. Price. 1992. Playing chutes and ladders: heterogeneity and the relative roles of bottom-up and top-down forces in natural communities. Ecology 73:724-732.

Kitchell, J. F., R. V. O'Neill, D. Webb, G. W. Gallepp, S. M. Bartell, J. F. Koonce, and B. S. Ausmus. 1979. Consumer regulation of nutrient cycling. BioScience 29:28-34.




Lee, J. J., and D. L. Inman. 1975. The ecological role of consumers-an aggregated systems view. Ecology 56:1455-1458.

Loreau, M. 1994. Material cycling and the stability of ecosystems. American Naturalist 143:508-513. Lotka, A. J. 1922. Contribution to the energetics of evolution. Proceedings of the National Academy

of Sciences of the USA 8:147-151. (1925) 1956. Elements of physical biology. Reprinted as Elements of mathematical biology, Dover, New York.

Margalef, R. 1963. On certain unifying principles in ecology. American Naturalist 97:357-374. Mattson, W. J., and N. D. Addy. 1975. Phytophagous insects as regulators of forest primary produc-

tion. Science (Washington, D.C.) 190:515-522. Mazanov, A. 1976. Stability of multi-pool models with lags. Journal of Theoretical Biology 59:429-442. McNaughton, S. J. 1979. Grazing as an optimization process: grass-ungulate relationships in the

Serengeti. American Naturalist 113:691-703. 1983. Compensatory plant growth as a response to herbivory. Oikos 40:329-336. 1986. On plants and herbivores. American Naturalist 128:765-770.

McQueen, D. J., J. R. Post, and E. L. Mills. 1986. Trophic relationships in freshwater pelagic ecosystems. Canadian Journal of Fisheries and Aquatic Sciences 43:1571-1581.

Nisbet, R. M., and W. S. C. Gurney. 1976. Model of material cycling in a closed ecosystem. Nature (London) 264:633-634.

Odum, E. P. 1969. The strategy of ecosystem development. Science (Washington, D.C.) 164:262-270. Odum, E. P., and L. J. Biever. 1984. Resource quality, mutualism, and energy partitioning in food

chains. American Naturalist 124:360-376. Odum, H. T. 1983. Systems ecology. Wiley, New York. Odum, H. T., and R. C. Pinkerton. 1955. Time's speed regulator: the optimum efficiency for maximum

power output in physical and biological systems. American Scientist 43:331-343. O'Neill, R. V. 1976. Ecosystem persistence and heterotrophic regulation. Ecology 57:1244-1253. Owen, D. F. 1980. How plants may benefit from the animals that eat them. Oikos 35:230-235. Owen, D. F., and R. G. Wiegert. 1976. Do consumers maximize plant fitness? Oikos 27:488-492.

. 1981. Mutualism between grasses and grazers: an evolutionary hypothesis. Oikos 36:376-378. Patten, B. C. 1975. Ecosystem linearization: an evolutionary design problem. American Naturalist

109:529-539. Petelle, M. 1982. More mutualisms between consumers and plants. Oikos 38:125-127. Power, M. E. 1992. Top-down and bottom-up forces in food webs: do plants have primacy? Ecology

73:733-746. Puccia, C. J., and R. Levins. 1985. Qualitative modeling of complex systems. Harvard University

Press, Cambridge, Mass. Ruess, R. W., and S. J. McNaughton. 1987. Grazing and the dynamics of nutrient and energy regulated

microbial processes in the Serengeti grasslands. Oikos 49:101-110. Seastedt, T. R., and D. A. Crossley, Jr. 1984. The influence of arthropods on ecosystems. BioScience

34:157-161. Strong, D. R. 1992. Are trophic cascades all wet? differentiation and donor-control in speciose ecosys-

tems. Ecology 73:747-754. Verhoef, H. A., and L. Brussaard. 1990. Decomposition and nitrogen mineralization in natural and

agro-ecosystems: the contributions of soil animals. Biogeochemistry 11:175-211. Wilson, D. S. 1976. Evolution on the level of communities. Science (Washington, D.C.) 192:1358-1360.

1980. The natural selection of populations and communities. Benjamin/Cummings, Menlo Park, Calif.

Wood, T., F. H. Bormann, and G. K. Voigt. 1984. Phosphorus cycling in a northern hardwood forest: biological and chemical control. Science (Washington, D.C.) 223:391-393.

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the idea that animals are detrimental to their food ... · flow, and hence primary production, in...

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