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How to calculate leaf litter consumption by saprophagous macrofauna?
Eur. J. Soil Bid.,
1998, 34 (3), 11 l-1 15 0 Elsevier,
How
to calculate
Paris
leaf litter
consumption
by saprophagous
macrofauna?
Jean-Francok Centre d ‘e’cologie fonctionnelle
David
et 6volutive, CNRS, 1919, route de Mende, 34293 Montpellier cedex 5, France. (e-mail: davidQce$e. cnrs-mop.fr) Received April 28, 1999; accepted June 14, 1999.
Abstract - Leaf litter, as food for saprophagous macrofauna, loses dry mass with time whether consumed or not, and this affects estimates of food consumption based on gravimetric methods. A model combining discontinuous consumption of food with continuous, exponential decomposition of litter makes it possible to compare the accuracy of various formulae used in calculating the amount of food consumed. Bocock’s formula yields an underestimate, unless consumption takes place at the very end of feeding tests. The Reiman formula yields an overestimate, unless consumption takes place at the very beginning of tests. A new formula giving an intermediate estimate is proposed and it is shown that this is the most accurate for most patterns of feeding. Calculations taking no account of decomposition (no control) may lead to substantial overestimates, especially if the amount of litter actually consumed is small in comparison with the initial amount. 0 Elsevier, Paris Sapropbagous
animals / food consumption
/ litter decomposition
R&urn6 - Comment calculer la consommation de litike de la macrofaune saprophage ? Les tests destines a mesurer la consommation de litiere de la macrofaune saprophage par des methodes gravimttriques sont toujours affect& par la perte de masse due a la decomposition. La precision des differentes formules utilisees pour calculer les quantites consommees a et6 comparee a l’aide d’un modele combinant d’une part une consommation discontinue par la faune et d’autre part une decomposition exponentielle de la lit&e non-consommde. La formule de Bocock sous-estime la consommation, a moins que celle-ci ne soit concentree a la fin des tests. La formule de Reiman surestime la consommation, a moins que celle-ci soit concentrte au tout debut des tests. Une nouvelle formule est proposee pour obtenir une estimation intermediaire, et l’on montre qu’elle est la plus precise pour la plupart des modalites de prise de nourriture. On montre enfin que les calculs qui ne tiennent pas compte de la decomposition (tests saris ttmoins) peuvent surestimer fortement la consommation, en particulier lorsque la quantite de litiere effectivement consommee est faible par rapport a la quantite initiale offerte aux animaux. 0 Elsevier, Paris Faune saprophage
I consommation
de litike
/ dkomposition
1. INTRODUCTION Although the literature abounds with gravimetric estimates of leaf or needle litter consumed by saprophagous macrofauna (macroarthropods, earthworms, molluscs), no consensus has been reached on the way to calculate consumption from the result of a feeding test. In a number of cases, the mass loss due to decomposition of litter during the test is simply ignored; the difference between initial dry mass (M, when t = 0) and final dry mass (A4, when t = n) of litter is wholly ascribed to consumption by macrofauna (e.g. [7, 9, Eur. J. Soil Bid.,
1164-5563/98/03/O
Elsevier,
Paris
141). Most often, however, the decrease in dry mass of unconsumed litter is taken into account. The proportion of litter that decomposes during the test is determined without macrofauna, for example in the form: D = (Mb -M;)IM’,
where Mb and Ml, are the initial and final dry masses of control litter, respectively, and the mean proportion is used to calculate consumption. The two most commonly used formulae are (i) the formula of Bocock [2]:
J.-F. David
112 where C* stands for the estimate of C, the. food ingested during a test (e.g. [4,6, 8, I I]); and (ii) the socalled Reiman formula [21 J: c;
= (M;, -M,)
MO/M;,
where M; actually corresponds to M, - h4@ (e.g. [ 13, 17, 181). It is easily seen that this formula is identical to that of Bocock divided by (1 - 0). Another formula used by Daniel [5], in which D is expressed as a proportion of the ‘final mass, yields the same estimates as that of Reiman. These various procedures lead to different results despite starting with the same experimental measurements. For example, assuming that the dry mass of litter decreased from M, = 200 to 84 mg in a 10-d test with D = 0.04, the amount consumed would be estimated to be 108 mg using Bocock’s formula, 112.5 mg using Reiman’s formula and 116 mg when working without a control. As the differences between estimates obviously increase with D, they may become substantial in certain feeding experiments using litter that decomposes rapidly (e.g. D > 0.24 in 10 d for ash leaf litter at 22 “C [lo]). The aim of this paper is to compare the accuracy of the various calculation procedures - assuming that D is a good estimate of the pro‘portion of unconsumed litter that decomposes in the presence of animals.
1
0
a
I
1
b
description
I
n
Time Figure 1. Dry mass remaining as a function of time in an example of exponential decomposition of litter combined with the consumption of c,, at time u and c,, at time b. The values indicated with arrows are M, = Moemkii - c,, M,, = M,Ck” - c,em”h-“’ - ch and M,, = Moe-k” c e-“‘!, 0, constant). The dashed line - c,e A”’ ~” (k = dkomposition c&responds to the dry mass of litter decomposing with the same k value in the absence of consumption.
food at any given time. Such a model, with only two food intakes for simplicity, is illustrated in figure I. The final mass when t = 12is given by:
M,, = Moeek”- c yek(” - ‘)
2. METHODS 2.1. Mathematical in litter mass
I
(1)
of changes
The first stage consists of defining a reference model that describes the changes in litter mass resulting from decomposition and consumption by macrofauna as accurately as possible. For the decomposition process, various functions have been proposed. The most appropriate in short-term studies under constant laboratory conditions, as is the case for most feeding tests, is the exponential function:
M, = Moeekt where k is the decomposition constant (in timeunit-‘) and e the base of natural logarithms 13, 191. In an exponential decomposition model, a constant proportion of litter (d) is lost per unit of time, and the basic relationships between D, d and k are: Food consumption of macroarthropods, earthworms and molluscs cannot be considered as a continuous process. Resting periods alternate with feeding periods, and the latter are more or less discontinuous sequences of small food intakes [ 12, 16, 201. An exact description of changes in litter mass during a test should therefore combine continuous decomposition and instantaneous consumption of small amounts of
where cI is the mass of food ingested at time t. Equation (1) makes it impossible to calculate consumption during a test
c = Ccr simply by knowing MO, M,, and k. A solution requires specification of the times at which each amount of food (c,) was ingested - which is obviously unknowable. It is therefore impossible, in the context of an exponential decomposition model, to measure the exact consumption of macrofauna by means of gravimetric methods. The problem is to choose the formula that minimizes the bias.
2.2. Accuracy of the various formulae Substituting equation (1) into each formula, the amount actually measured by 6” can be calculated. The error made in the estimation of C is given by C* - C, and the relative error by (C - C)/C. As the error attached to each estimation depends on the times at which litter is consumed during the test, it has proved more convenient to define a mean time of feeding (“t) as follows. For any pattern of feeding Eur. J. Soil Biol.
Estimates
of leaf litter consumption
113
(e.g. the amounts ca and cb ingested at times a and b in figure 1) there is one 7 value within the range [a, b] such that the consumption of C = c, + cb at time 7 yields the same M,, value as the actual pattern. Therefore, substituting the consumption of C at time I for the actual pattern does not change either the estimate of the food consumed or the error of estimate. From equation (1) i must satisfy the relation: Ce-k(n
-7)
=
c e-k(n t
c
- t)
(2)
The mean time of feeding, as defined, varies from 0 to n according to the animals’ feeding pattern. Using equation (2), the expression for the relative error is simplified and its lower and upper limits can be determined when “t varies between 0 and 12.
3. RESULTS
AND DISCUSSION
3.1. Bocock’s formula:
C*, = M, - M,D - M,
As M, - M,$ is equal to Moemk”, the amount actually measured by this formula is, according to equation (1): C*, = Moe-kn - M, =
ctepk(” - t,
c
which, using equation (2), can be written as:
This underestimates consumption when k # 0. This formula only gives the amount actually consumed when i = IZ, i.e. if the entire consumption took place at the end of the test. For all other feeding patterns (0 I i < n), there is an underestimate and the relative error resulting from Bocock’s formula varies within the range: CL-C e-kn - 1 < C
10
or, more simply: -D<-
c;-c C
which, using equation (2), can be written as:
Ci = Ceki This overestimates actual consumption when k f 0. This formula only gives the amount consumed when i = 0, i.e. if the entire consumption took place at the beginning of the test. For all other feeding patterns (0 < t I n), there is an overestimate and the relative error varies within the range:
c;-c OS-
3.3. Intermediate
D
-1-D
formulae
For most feeding patterns, it is preferable to have an estimate that is intermediate between that of Bocock (systematic underestimate) and that of Reiman (systematic overestimate). A good solution is to use the new formula : C; = (M,,-MOD-M,)/(m) which, using equations (1) and (2), can be rewritten c; = Ce-W2 -2)
(3) as:
The error of estimate is zero for any feeding pattern in which i = n/2. This is advantageous when there is no reason to suspect that food consumption is concentrated at the beginning or the end of tests. In particular, i varies around n/2 when there is an even uptake of food during a test, or when many bites of similar size are randomly distributed between 0 and 12.For feeding patterns in which I > n/2, equation (3) overestimates C, and for i < n/2, it underestimates C but the limits of the relative error are lower (in absolute value) than with the formulae of Bocock and Reiman (figure 2). D/(1 -0)
.
Dl2
50
It should be noted that the maximum error (D in absolute value) increases with the decomposition constant (k) and the duration of tests (n).
3.2. Reiman’s formula: c; = (MO -M,D - M,)l(l
-D)
As 1 -D is equal to ePkn, the amount measured by this formula is, according to equation (1):
Ci = Ccfekf f Vol.
34, no 3 - 1998
Mean feeding time $J Figure 2. Relationships between the relative error in the estimation of food consumption and i (the mean time of feeding defined in the text) for various estimation procedures (Reiman, equation (3). Bocock). Equation (3) is the most accurate when i is around n/2 (n = duration of the test, = mean proportion of litter that decomposes during the test).
D
J.-F. David
114 The error varies within the range: c; - c
e -kn/2 - ] 5 -
c
which, using equation (2), can be rewritten as:
Se kn/2 _ 1
which is not very different from: -- D
Ck = Mo( 1 - ewkn)+ Ce-k(n - t, This always overestimates C when ,k # 0, for it is possible to show that the difference C, - C is positive for all i (noting that M, > C for a feeding test to be valid). When i varies between 0 and IZ, the relative error in the estimation of C due to lack of a control varies within the range: M+DeD<-
2d[Mo( 1 - dy - MJ
c; =
C;-C
(2-d)[l-(1 -d)y where c\ estimatesthe massof food consumed per unit of time and d is the proportion of litter that decomposes per unit of time. This formula is based on a model in which (i) a constant amount of litter (c*,) is supposedto be ingested each day and (ii) the amount of litter that decomposesdaily is the product of d by the mean value of the massat the beginning of the day and this minus ci. The dry mass remaining after one day is therefore:
The limits of the error cannot be determined numerically becauseC is unknown and D - not measured- is known only roughly from experimental conditions. It is clear, however, that the relative error not only increaseswith D but is also directly proportional to the ratio MdC. Thus, estimates of consumption are all the more biased due to lack of a control when the amounts of food offered exceed the animals’ requirements. As excess food with a sufficient selection of dead leaves is often necessaryfor conducting reliable feeding experiments, the lack of a control may lead to substantial overestimates.
a recurrence relation which leads, after sol&ng for M,, to Soma and Saito’s formula. Alternatively, a constant consumption rate (c”,) can be combined with an exponential decomposition model. This is a $mplified version of the model used by Axelsson and Agren [I] for a growing population of phytophagous invertebrates. The final massof litter is given by the expression: *
REFERENCES
M, = MOepkn-!$(I
-epkn)
and the amount consumed per unit of time is:
CT;=
;(Mo-MOD-M,,)
Numerically, this formula and that of Soma and Saito give practically the same estimates as equation (3) divided by II. It can also be shown that the errors made with these formulae are zero when i = n/2, and that the relative errors vary approximately within the samerange as with equation (3). Equation (3) is however the simplest to use and makes it easy to correct estimates based on the formulae of Bocock and Reiman in the literature, provided D is known.
3.4. No control:
C*, = MO - M,,
According to equation (I), the amount thus measured is: Ch = Mo( 1 - edkn) + Cc,e-k(“-f)
[I] AxelssonB., Agren G.I., A correctionfor food respiration in balancingenergybudgets,Entomol. Exp. Appl. 25 (1979)260-266. [2] Bocock K.L., The digestionand assimilationof food by Glomeris, in: DoeksenJ., Van der Drift J. (Eds.), Soil Organisms, North-Holland, Amsterdam, 1963, pp. 85-91. [3] Cortez J., DemardJ.-M., Bottner I?, Jocteur-Monrozier L., Decomposition of Mediterranean leaf litters: a microcosm experiment investigating relationships between decompositionrates and litter quality, Soil Biol. Biochem.28 (1996) 443-452. DangerfieldJ.M., Effects of starvationand mechanical manipulationof leaf litter on faecal pellet production and assimilationin some millipedes from southern Africa: implicationsfor feeding strategies,Eur. J. Soil Biol. 31 (1995) 199-206. Daniel O., Leaf-litter consumptionand assimilationby juveniles of Lumbricus terrestris L. (Oligochaeta, Lumbricidae)underdifferent environmentalconditions, Biol. Fertil. Soils 12(1991)202-208. [6] David J.-F., Consommationannuelled’une lit&e de chCnepar une population adulte du Diplopode Cylin-, droiulus nitidus, Pedobiologia30 (1987)299-310. [7] DeleporteS., EtudeexpkrimentaledeI’ajustemententre le cycle de Brudysia confinis (Diptera : Sciaridae)et I’Cvolution du substrattrophique (lit&e de feuillus) : importancedes microorganismes, Acta Oecol. Oecol. Gen. 9 (1988) 13-35. Eur. J. Soil Bid.
Estimates
of leaf litter consumption
[8] Dickschen F., Topp W., Feeding activities and assimilation efficiencies of Lumbricus rubellus (Lumbricidae) on a plant-only diet, Pedobiologia 30 (1987) 3 l-37. ]9 ] Geoffroy J.-J., CClCrier M.-L., Garay I., Rherissi S., Blandin I?, Approche quantitative des fonctions de transformation de la mat&e organique par des macroarthropodes saprophages (Isopodes et Diplopodes) dans un sol forestier a moder. Protocoles experimentaux et premiers resultats, Rev. Ecol. Biol. Sol 24 (1987) 573590. [lo 11Gillon D., Joffre R., Ibrahima A., Can litter decomposability be predicted by near infrared reflectance spectroscopy? Ecology 80 (1999) 175-186. [ 1 l] Haimi J., Huhta V., Effects of earthworms on decomposition processes in raw humus forest soil: a microcosm study, Biol. Fertil. Soils 10 (1990) 178-183. [ 121 Kretzschmar A., Soil transport as a homeostatic mechanism for stabilizing the earthworm environment, in: Satchel1 J.E.‘(Ed.), Earthworm Ecology, Chapman and Hall, London, 1983, pp. 59-66. [ 131 Pobozsny M., uber Streuzersetzungsprozesse in Hainbuchen-Eichenwaldern unter Berticksichtigung der Diplopoden, Opusc. Zool. (Budap.) 22 (1986) 77-84. [ 141 Rushton S.P., Hassall M., Effects of food quality on isopod population dynamics, Funct. Ecol. 1 (1987) 359367.
Vol.
34, no 3 - 1998
115
[ 151 Soma K., Saito T., Ecological studies of soil organisms with references to the decomposition of pine needles. II. Litter feeding and breakdown by the woodlouse, Porcellio scaber, Plant Soil 75 (1983) 139-l 5 1. [ 161 Szlavecz K., Food selection and nocturnal .behavior of the land snail Monadelia hillebrandi mariposa (Pulmonata: Helminthoglyptidae), Veliger 29 (1986) 183-190. [17] Szlavecz K., Needle litter consumption by two terrestrial isopods, Protrucheoniscus amoenus (C.L. Koch), and Cylisticus convexus (De Geer) (Isopoda, Oniscidea), Pedobiologia 37 (1993) 57-64. [ 181 Szlavecz K., Pobozsny M., Coprophagy in isopods and diplopods: a case for indirect interaction, Acta Zool. Fenn. 196 (1995) 124-128. [19] Taylor B.R., Parkinson D., Aspen and pine leaf litter decomposition in laboratory microcosms. I. Linear versus exponential models of decay, Can. J. Bot. 66 (1988) 1960-1965. [20] Wooten R.C., Crawford C.S., Riddle W.A., Behavioural thermoregulation of Orthoporus ornatus (Diplopoda: Spirostreptidae) in three desert habitats, Zool. J. Linn. sot. 57 (1975) 59-74. [21] Zicsi A., Pobozsny M., Einfluss des Zersetzungsverlaufes der Laubstreu auf die Konsumintensitat einiger Lumbriciden Arten, Ecol. Bull. 25 (1977) 229-239.
1998, 34 (3), 11 l-1 15 0 Elsevier,
How
to calculate
Paris
leaf litter
consumption
by saprophagous
macrofauna?
Jean-Francok Centre d ‘e’cologie fonctionnelle
David
et 6volutive, CNRS, 1919, route de Mende, 34293 Montpellier cedex 5, France. (e-mail: davidQce$e. cnrs-mop.fr) Received April 28, 1999; accepted June 14, 1999.
Abstract - Leaf litter, as food for saprophagous macrofauna, loses dry mass with time whether consumed or not, and this affects estimates of food consumption based on gravimetric methods. A model combining discontinuous consumption of food with continuous, exponential decomposition of litter makes it possible to compare the accuracy of various formulae used in calculating the amount of food consumed. Bocock’s formula yields an underestimate, unless consumption takes place at the very end of feeding tests. The Reiman formula yields an overestimate, unless consumption takes place at the very beginning of tests. A new formula giving an intermediate estimate is proposed and it is shown that this is the most accurate for most patterns of feeding. Calculations taking no account of decomposition (no control) may lead to substantial overestimates, especially if the amount of litter actually consumed is small in comparison with the initial amount. 0 Elsevier, Paris Sapropbagous
animals / food consumption
/ litter decomposition
R&urn6 - Comment calculer la consommation de litike de la macrofaune saprophage ? Les tests destines a mesurer la consommation de litiere de la macrofaune saprophage par des methodes gravimttriques sont toujours affect& par la perte de masse due a la decomposition. La precision des differentes formules utilisees pour calculer les quantites consommees a et6 comparee a l’aide d’un modele combinant d’une part une consommation discontinue par la faune et d’autre part une decomposition exponentielle de la lit&e non-consommde. La formule de Bocock sous-estime la consommation, a moins que celle-ci ne soit concentree a la fin des tests. La formule de Reiman surestime la consommation, a moins que celle-ci soit concentrte au tout debut des tests. Une nouvelle formule est proposee pour obtenir une estimation intermediaire, et l’on montre qu’elle est la plus precise pour la plupart des modalites de prise de nourriture. On montre enfin que les calculs qui ne tiennent pas compte de la decomposition (tests saris ttmoins) peuvent surestimer fortement la consommation, en particulier lorsque la quantite de litiere effectivement consommee est faible par rapport a la quantite initiale offerte aux animaux. 0 Elsevier, Paris Faune saprophage
I consommation
de litike
/ dkomposition
1. INTRODUCTION Although the literature abounds with gravimetric estimates of leaf or needle litter consumed by saprophagous macrofauna (macroarthropods, earthworms, molluscs), no consensus has been reached on the way to calculate consumption from the result of a feeding test. In a number of cases, the mass loss due to decomposition of litter during the test is simply ignored; the difference between initial dry mass (M, when t = 0) and final dry mass (A4, when t = n) of litter is wholly ascribed to consumption by macrofauna (e.g. [7, 9, Eur. J. Soil Bid.,
1164-5563/98/03/O
Elsevier,
Paris
141). Most often, however, the decrease in dry mass of unconsumed litter is taken into account. The proportion of litter that decomposes during the test is determined without macrofauna, for example in the form: D = (Mb -M;)IM’,
where Mb and Ml, are the initial and final dry masses of control litter, respectively, and the mean proportion is used to calculate consumption. The two most commonly used formulae are (i) the formula of Bocock [2]:
J.-F. David
112 where C* stands for the estimate of C, the. food ingested during a test (e.g. [4,6, 8, I I]); and (ii) the socalled Reiman formula [21 J: c;
= (M;, -M,)
MO/M;,
where M; actually corresponds to M, - h4@ (e.g. [ 13, 17, 181). It is easily seen that this formula is identical to that of Bocock divided by (1 - 0). Another formula used by Daniel [5], in which D is expressed as a proportion of the ‘final mass, yields the same estimates as that of Reiman. These various procedures lead to different results despite starting with the same experimental measurements. For example, assuming that the dry mass of litter decreased from M, = 200 to 84 mg in a 10-d test with D = 0.04, the amount consumed would be estimated to be 108 mg using Bocock’s formula, 112.5 mg using Reiman’s formula and 116 mg when working without a control. As the differences between estimates obviously increase with D, they may become substantial in certain feeding experiments using litter that decomposes rapidly (e.g. D > 0.24 in 10 d for ash leaf litter at 22 “C [lo]). The aim of this paper is to compare the accuracy of the various calculation procedures - assuming that D is a good estimate of the pro‘portion of unconsumed litter that decomposes in the presence of animals.
1
0
a
I
1
b
description
I
n
Time Figure 1. Dry mass remaining as a function of time in an example of exponential decomposition of litter combined with the consumption of c,, at time u and c,, at time b. The values indicated with arrows are M, = Moemkii - c,, M,, = M,Ck” - c,em”h-“’ - ch and M,, = Moe-k” c e-“‘!, 0, constant). The dashed line - c,e A”’ ~” (k = dkomposition c&responds to the dry mass of litter decomposing with the same k value in the absence of consumption.
food at any given time. Such a model, with only two food intakes for simplicity, is illustrated in figure I. The final mass when t = 12is given by:
M,, = Moeek”- c yek(” - ‘)
2. METHODS 2.1. Mathematical in litter mass
I
(1)
of changes
The first stage consists of defining a reference model that describes the changes in litter mass resulting from decomposition and consumption by macrofauna as accurately as possible. For the decomposition process, various functions have been proposed. The most appropriate in short-term studies under constant laboratory conditions, as is the case for most feeding tests, is the exponential function:
M, = Moeekt where k is the decomposition constant (in timeunit-‘) and e the base of natural logarithms 13, 191. In an exponential decomposition model, a constant proportion of litter (d) is lost per unit of time, and the basic relationships between D, d and k are: Food consumption of macroarthropods, earthworms and molluscs cannot be considered as a continuous process. Resting periods alternate with feeding periods, and the latter are more or less discontinuous sequences of small food intakes [ 12, 16, 201. An exact description of changes in litter mass during a test should therefore combine continuous decomposition and instantaneous consumption of small amounts of
where cI is the mass of food ingested at time t. Equation (1) makes it impossible to calculate consumption during a test
c = Ccr simply by knowing MO, M,, and k. A solution requires specification of the times at which each amount of food (c,) was ingested - which is obviously unknowable. It is therefore impossible, in the context of an exponential decomposition model, to measure the exact consumption of macrofauna by means of gravimetric methods. The problem is to choose the formula that minimizes the bias.
2.2. Accuracy of the various formulae Substituting equation (1) into each formula, the amount actually measured by 6” can be calculated. The error made in the estimation of C is given by C* - C, and the relative error by (C - C)/C. As the error attached to each estimation depends on the times at which litter is consumed during the test, it has proved more convenient to define a mean time of feeding (“t) as follows. For any pattern of feeding Eur. J. Soil Biol.
Estimates
of leaf litter consumption
113
(e.g. the amounts ca and cb ingested at times a and b in figure 1) there is one 7 value within the range [a, b] such that the consumption of C = c, + cb at time 7 yields the same M,, value as the actual pattern. Therefore, substituting the consumption of C at time I for the actual pattern does not change either the estimate of the food consumed or the error of estimate. From equation (1) i must satisfy the relation: Ce-k(n
-7)
=
c e-k(n t
c
- t)
(2)
The mean time of feeding, as defined, varies from 0 to n according to the animals’ feeding pattern. Using equation (2), the expression for the relative error is simplified and its lower and upper limits can be determined when “t varies between 0 and 12.
3. RESULTS
AND DISCUSSION
3.1. Bocock’s formula:
C*, = M, - M,D - M,
As M, - M,$ is equal to Moemk”, the amount actually measured by this formula is, according to equation (1): C*, = Moe-kn - M, =
ctepk(” - t,
c
which, using equation (2), can be written as:
This underestimates consumption when k # 0. This formula only gives the amount actually consumed when i = IZ, i.e. if the entire consumption took place at the end of the test. For all other feeding patterns (0 I i < n), there is an underestimate and the relative error resulting from Bocock’s formula varies within the range: CL-C e-kn - 1 < C
10
or, more simply: -D<-
c;-c C
which, using equation (2), can be written as:
Ci = Ceki This overestimates actual consumption when k f 0. This formula only gives the amount consumed when i = 0, i.e. if the entire consumption took place at the beginning of the test. For all other feeding patterns (0 < t I n), there is an overestimate and the relative error varies within the range:
c;-c OS-
3.3. Intermediate
D
-1-D
formulae
For most feeding patterns, it is preferable to have an estimate that is intermediate between that of Bocock (systematic underestimate) and that of Reiman (systematic overestimate). A good solution is to use the new formula : C; = (M,,-MOD-M,)/(m) which, using equations (1) and (2), can be rewritten c; = Ce-W2 -2)
(3) as:
The error of estimate is zero for any feeding pattern in which i = n/2. This is advantageous when there is no reason to suspect that food consumption is concentrated at the beginning or the end of tests. In particular, i varies around n/2 when there is an even uptake of food during a test, or when many bites of similar size are randomly distributed between 0 and 12.For feeding patterns in which I > n/2, equation (3) overestimates C, and for i < n/2, it underestimates C but the limits of the relative error are lower (in absolute value) than with the formulae of Bocock and Reiman (figure 2). D/(1 -0)
.
Dl2
50
It should be noted that the maximum error (D in absolute value) increases with the decomposition constant (k) and the duration of tests (n).
3.2. Reiman’s formula: c; = (MO -M,D - M,)l(l
-D)
As 1 -D is equal to ePkn, the amount measured by this formula is, according to equation (1):
Ci = Ccfekf f Vol.
34, no 3 - 1998
Mean feeding time $J Figure 2. Relationships between the relative error in the estimation of food consumption and i (the mean time of feeding defined in the text) for various estimation procedures (Reiman, equation (3). Bocock). Equation (3) is the most accurate when i is around n/2 (n = duration of the test, = mean proportion of litter that decomposes during the test).
D
J.-F. David
114 The error varies within the range: c; - c
e -kn/2 - ] 5 -
c
which, using equation (2), can be rewritten as:
Se kn/2 _ 1
which is not very different from: -- D
Ck = Mo( 1 - ewkn)+ Ce-k(n - t, This always overestimates C when ,k # 0, for it is possible to show that the difference C, - C is positive for all i (noting that M, > C for a feeding test to be valid). When i varies between 0 and IZ, the relative error in the estimation of C due to lack of a control varies within the range: M+DeD<-
2d[Mo( 1 - dy - MJ
c; =
C;-C
(2-d)[l-(1 -d)y where c\ estimatesthe massof food consumed per unit of time and d is the proportion of litter that decomposes per unit of time. This formula is based on a model in which (i) a constant amount of litter (c*,) is supposedto be ingested each day and (ii) the amount of litter that decomposesdaily is the product of d by the mean value of the massat the beginning of the day and this minus ci. The dry mass remaining after one day is therefore:
The limits of the error cannot be determined numerically becauseC is unknown and D - not measured- is known only roughly from experimental conditions. It is clear, however, that the relative error not only increaseswith D but is also directly proportional to the ratio MdC. Thus, estimates of consumption are all the more biased due to lack of a control when the amounts of food offered exceed the animals’ requirements. As excess food with a sufficient selection of dead leaves is often necessaryfor conducting reliable feeding experiments, the lack of a control may lead to substantial overestimates.
a recurrence relation which leads, after sol&ng for M,, to Soma and Saito’s formula. Alternatively, a constant consumption rate (c”,) can be combined with an exponential decomposition model. This is a $mplified version of the model used by Axelsson and Agren [I] for a growing population of phytophagous invertebrates. The final massof litter is given by the expression: *
REFERENCES
M, = MOepkn-!$(I
-epkn)
and the amount consumed per unit of time is:
CT;=
;(Mo-MOD-M,,)
Numerically, this formula and that of Soma and Saito give practically the same estimates as equation (3) divided by II. It can also be shown that the errors made with these formulae are zero when i = n/2, and that the relative errors vary approximately within the samerange as with equation (3). Equation (3) is however the simplest to use and makes it easy to correct estimates based on the formulae of Bocock and Reiman in the literature, provided D is known.
3.4. No control:
C*, = MO - M,,
According to equation (I), the amount thus measured is: Ch = Mo( 1 - edkn) + Cc,e-k(“-f)
[I] AxelssonB., Agren G.I., A correctionfor food respiration in balancingenergybudgets,Entomol. Exp. Appl. 25 (1979)260-266. [2] Bocock K.L., The digestionand assimilationof food by Glomeris, in: DoeksenJ., Van der Drift J. (Eds.), Soil Organisms, North-Holland, Amsterdam, 1963, pp. 85-91. [3] Cortez J., DemardJ.-M., Bottner I?, Jocteur-Monrozier L., Decomposition of Mediterranean leaf litters: a microcosm experiment investigating relationships between decompositionrates and litter quality, Soil Biol. Biochem.28 (1996) 443-452. DangerfieldJ.M., Effects of starvationand mechanical manipulationof leaf litter on faecal pellet production and assimilationin some millipedes from southern Africa: implicationsfor feeding strategies,Eur. J. Soil Biol. 31 (1995) 199-206. Daniel O., Leaf-litter consumptionand assimilationby juveniles of Lumbricus terrestris L. (Oligochaeta, Lumbricidae)underdifferent environmentalconditions, Biol. Fertil. Soils 12(1991)202-208. [6] David J.-F., Consommationannuelled’une lit&e de chCnepar une population adulte du Diplopode Cylin-, droiulus nitidus, Pedobiologia30 (1987)299-310. [7] DeleporteS., EtudeexpkrimentaledeI’ajustemententre le cycle de Brudysia confinis (Diptera : Sciaridae)et I’Cvolution du substrattrophique (lit&e de feuillus) : importancedes microorganismes, Acta Oecol. Oecol. Gen. 9 (1988) 13-35. Eur. J. Soil Bid.
Estimates
of leaf litter consumption
[8] Dickschen F., Topp W., Feeding activities and assimilation efficiencies of Lumbricus rubellus (Lumbricidae) on a plant-only diet, Pedobiologia 30 (1987) 3 l-37. ]9 ] Geoffroy J.-J., CClCrier M.-L., Garay I., Rherissi S., Blandin I?, Approche quantitative des fonctions de transformation de la mat&e organique par des macroarthropodes saprophages (Isopodes et Diplopodes) dans un sol forestier a moder. Protocoles experimentaux et premiers resultats, Rev. Ecol. Biol. Sol 24 (1987) 573590. [lo 11Gillon D., Joffre R., Ibrahima A., Can litter decomposability be predicted by near infrared reflectance spectroscopy? Ecology 80 (1999) 175-186. [ 1 l] Haimi J., Huhta V., Effects of earthworms on decomposition processes in raw humus forest soil: a microcosm study, Biol. Fertil. Soils 10 (1990) 178-183. [ 121 Kretzschmar A., Soil transport as a homeostatic mechanism for stabilizing the earthworm environment, in: Satchel1 J.E.‘(Ed.), Earthworm Ecology, Chapman and Hall, London, 1983, pp. 59-66. [ 131 Pobozsny M., uber Streuzersetzungsprozesse in Hainbuchen-Eichenwaldern unter Berticksichtigung der Diplopoden, Opusc. Zool. (Budap.) 22 (1986) 77-84. [ 141 Rushton S.P., Hassall M., Effects of food quality on isopod population dynamics, Funct. Ecol. 1 (1987) 359367.
Vol.
34, no 3 - 1998
115
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