Relation between production and biomass
Abstract
A series of mathematical models of cohorts in animal populations representing various combinations of several different simple growth and mortality functions is examined to investigate the ratio between mean biomass and production over unit time, and to compare this ratio with the mean age and mean life span of the animals in the cohort.For any cohort, the ratio of production per unit time to mean biomass is equal to the ratio of total production by the cohort to its total biomass integral by time. For populations consisting of a number of simultaneous, successive, or overlapping cohorts, the ratio of production per unit time to mean biomass is equal to the mean of the ratios for the individual cohorts weighted by the mean biomasses of the cohorts.If the cohorts are identical, the population ratio is the same as the cohort ratio and problems arising from the presence of more than one cohort may be ignored. Formulations for the total production per cohort, biomass integral, and, where they can be simplified, their ratios, are given.Comparison with mean age and mean life span shows that for constant exponential mortality, mean age and mean life span are both equal to the reciprocal of the production–biomass ratio. For other mortality functions, if growth in weight is linear, the production–biomass ratio equals the reciprocal of the mean age. For other models there is no simple relation. In general, mean age appears a better approximation than mean life span to the reciprocal of the production–biomass ratio.These methods are applied, as an example, to Antarctic krill, using a model having linear growth in length and four periods with different exponential mortality rates. For this model, annual production is 1.8 times the mean biomass so that assumption of equality leads to an underestimate of production. Mean age and mean life span are 0.21 and 0.037 years respectively. Thus, use of either of these as an approximation, and particularly mean life span, leads to severe overestimation of annual production.
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“Further, for groups in a steady-state system, the instantaneous rate of mortality is equal to the production/biomass ratio (P /B) (Allen, 1971).”
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“…. . , n functional groups; (P=B) i is the production/biomass ratio of (i) equal to total mortality rate ( Z i ) under the assumption of equilibrium (Allen, 1971); EE i is the ecotrophic efficiency, i.e. the fraction of the production (P i B i : (P=B) i ) that is consumed within the system or…”
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“On the other hand, production, in EwE models (Table 1), is usually estimated from biomass and turnover rates (P/B), which is equivalent, under equilibrium, to total mortality (Allen, 1971).”
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“P/B was usually estimated by assuming that it equals total mortality ( Z) under the assumption of population equilibrium ( Allen, 1971).”
BackgroundEnergy budget and ecological role of mangrove epibenthos in the Caeté estuary, North Brazil
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“Where individuals in a population die at a constant instantaneous rate Z, P/B is equivalent to the instantaneous total mortality rate Z (Allen 1971).”
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“Pi = Σ Bj * M2ij + Yi + Ei + BAi + Pi * (1-EEi) This equation can be re-expressed as: B * (P/B)i = Σ Bj * (Q/B)j * DCij + Yi + Ei + BAi + Bi * (P/B)i * (1-EEi) where (P/B)i indicates the production of (i) per unit of biomass and is equivalent to total mortality, or Z, under steady-state conditions (Allen, 1971); (Q/B)…”
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“Growth increments or net production (P) and elimination production (E) were estimated with a method derived from Allen (1971), as described in Dauvin (1986).”
MethodPredicting the carrying capacity of bivalve shellfish culture using a steady, linear food web model
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“modelling (Allen, 1971), which is the sum of fishing mortality (F) and natural mortality (M).”
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“Growth production (P) and elimination production (E) were calculated with a method derived from Allen (1971) with total values of P and E for the population of C.”
MethodLassalle, G., Lobry, J., Loc'h, F. L., Bustamante, P., Certain, G., Delmas, D., Dupuy, C., Hily, C., Labry, C., Pape, O. L., et al. (2011) · Progress In Oceanography · doi:10.1016/j.pocean.2011.09.002 · Cited by 117 · see citing works
“Under steady-state conditions, the P/B ratio is equal to instantaneous coefficient of total mortality (Z) (Allen, 1971): Z ¼ M þ F ð7Þ with M being natural and F fishing mortality.”
BackgroundRUMINATIONS ON THE DEVELOPMENT AND FUTURE OF POPULATION DYNAMICS MODELS IN FISHERIES
Quinn, T. J. (2003) · Natural Resource Modeling · doi:10.1111/j.1939-7445.2003.tb00119.x · Cited by 118 · see citing works
“The IWC Committees of Three and Four (Chapman, Allen, Holt, Gulland) successfully applied mathematical models to whale populations and showed that whale harvesting needed to be reduced. Pella and Tomlinson [1969] generalized the Graham-Schaefer production model by using a Richards function instead of the logistic. All…”
BackgroundMethodInfluentialCan Overfishing be Responsible for the Successful Establishment of in the Black Sea?
Gücü, A. C. (2002) · Estuarine Coastal and Shelf Science · doi:10.1006/ecss.2000.0657 · Cited by 116 · see citing works
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Distribution, Biomass and Demography of Antarctic Krill, Euphausia superba
Siegel, V., & Watkins, J. L. (2016) · Advances in polar ecology · doi:10.1007/978-3-319-29279-3_2 · Cited by 124 · see citing works
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“where (P/B)i is the production of (i) per unit of biomass and equals total mortality Z in steady-state ecosystems (Allen, 1971), (Q/B)i the consumption of (i) per unit of biomass and DCij the proportion of (i) in the diet composition of predator (j) in terms of biomass.”
BackgroundFood-web changes in the Adriatic Sea over the last three decades
Marta Coll, M., Santojanni, A., Palomera, I., & Arneri, E. (2009) · Marine Ecology Progress Series · doi:10.3354/meps07944 · Cited by 110 · see citing works
“…381: 17–37, 2009 where (P/B)i indicates the production of i per unit of biomass, and is equivalent to total mortality (Z ) under steady-state conditions (Allen 1971); (Q/B)i is the consumption of i per unit of biomass; and DCij indicates the proportion of i that is in the diet of predator j in…”
BackgroundAutomated parameter optimization for Ecopath ecosystem models
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“…(the latter pertaining to j, the consumers of i); P/Bi is the production/biomass ratio, equivalent to total mortality under most circumstances (Allen, 1971; Mertz and Myers, 1998); EEi is ecotrophic efficiency which is the fraction of production (P = B(P/B)) that is consumed within, or…”
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“This information can be used to estimate the energy 265 channeled through populations into biomass production [72], and is used, for example, to 266 quantify secondary production in aquatic systems and fisheries management [72, 73].”
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Coupled energy pathways and the resilience of size-structured food webs
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“P/B ratio was assumed to be equivalent to total mortality rate (per year) in keeping with Allen (1971).”
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“Owing to the intrinsic relationships between mortality rate and production-to-biomass ratio (Allen 1971; Brey 1999), exploitation per se does not put a population outside the empirical context considered here.”
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Torres, M. Á., Marta Coll, M., Heymans, J. J., Villy Christensen, A., & Sobrino, I. (2013) · Ecological Modelling · doi:10.1016/j.ecolmodel.2013.05.019 · Cited by 98 · see citing works
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“We estimated both growth (P) and elimination (E) production as derived by Allen [1].”
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“Ecopath uses the steady-state assumption that P/B = Z (Allen 1971).”
BackgroundPopulation consequences of reproductive decisions
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“Female size (Lt) was estimated each April using a Von Bertalan¡y growth equation (Allen 1971) (Lt ˆ L1(17exp [7 K(t ¡ t 0)])), with parameters derived by agecalibrated, length^frequency analysis using FiSAT (Gayanilo & Pauly 1997): L1 ˆ 62 mm, t0 ˆ 7 0.”
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“Generic list of model groups and input values for trophic flow budgets of upwelling systems (P, production; B, biomass; C, consumption; ss, system-specific; P/B, equals total mortality Z under steady-state conditions e Allen (1971); P/C, gross growth”
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“In steady state populations, mortality rate Z approximates the production-to-biomass ratio (Allen 1971, Brey 1999) and hence is an additional proxy for somatic productivity.”
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“, Z = P/B (Allen 1971).”
Mees, J., Abdulkerim, Z., & Hamerlynck, O. (1994) · Marine Ecology Progress Series · doi:10.3354/meps109043 · Cited by 93 · see citing works
“Under certain conditions (in case of a steady state model and if individual growth is described by a von Bertalanffy function) the total mortality of a population is equal to the P/B ratio of the population (Allen 1971).”
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“Using the decomposition of total mortality in both fishing mortality (F) and natural mortality (M) we then calculated the P/B, which is equivalent to total mortality, or Z, under steady-state conditions (Allen, 1971; Christensen et al., 2008), as the sum of F and M.”
BackgroundEmpirical equations for estimating maximum length from length at first maturity
Binohlan, C., & Froese, R. (2009) · Journal of Applied Ichthyology · doi:10.1111/j.1439-0426.2009.01317.x · Cited by 91 · see citing works
“It is well known that most basic parameters of fish population dynamics are strongly related to maximum length (Allen, 1971; Pauly, 1980; Welcomme, 1999; Froese and Binohlan, 2000).”
BackgroundMarine Mammal Impacts in Exploited Ecosystems: Would Large Scale Culling Benefit Fisheries?
Morissette, L. G., Villy Christensen, A., & Pauly, D. F. (2012) · PLoS ONE · doi:10.1371/journal.pone.0043966 · Cited by 90 · see citing works
“Bi indicates the biomass of I; Pi/Bi indicates the production/biomass ratio, which is equivalent to total mortality; (Z) under the most circumstances [39];”
BackgroundThe trophic-level-based ecosystem modelling approach: theoretical overview and practical uses
Gascuel, D., Guénette, S., & Pauly, D. F. (2011) · ICES Journal of Marine Science · doi:10.1093/icesjms/fsr062 · Cited by 90 · see citing works
“(A10) Under equilibrium assumption, Allen (1971) demonstrated that P/B ¼ Z.”
BackgroundLake of flies, or lake of fish? A trophic model of Lake Malawi
Darwall, W. R. T., Allison, E. H., Turner, G. F., et al. (2009) · Ecological Modelling · doi:10.1016/j.ecolmodel.2009.11.001 · Cited by 91 · see citing works
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“Total mortalityZi was estimated for every group i wherePi/Bi was unknown since, according toAllen (1971), for marine populations under equilibrium,”
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“The P/B ratio is equivalent to total mortality (Z) under steady-state conditions (Allen, 1971), which is the sum of fishing mortality (F and natural mortality (M). F was calculated as the ratio of catch and biomass (F = Y/B) and M was generated using the equation in Pauly (1980) (Supplement 1B).”
BackgroundMethodPatrício, J., & Marques, J. C. (2006) · Ecological Modelling · doi:10.1016/j.ecolmodel.2006.03.008 · Cited by 90 · see citing works
“…Bi and Bj are biomasses of prey (i) and predators (j), espectively; P/Bi the production/biomass ratio, equivalent to otal mortality (Z) in most circumstances (Allen, 1971); EEi the cotrophic efficiency; the fraction of the total production of group utilised in the system; Yi the fisheries…”
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Severe coral loss shifts energetic dynamics on a coral reef
Morais, R. A., Depczynski, M., Fulton, C. J., et al. (2020) · Functional Ecology · doi:10.1111/1365-2435.13568 · Cited by 88 · see citing works
Effects of local fisheries and ocean productivity on the northeastern Ionian Sea ecosystem
Piroddi, C., Bearzi, G., & Villy Christensen, A. (2010) · Ecological Modelling · doi:10.1016/j.ecolmodel.2010.03.002 · Cited by 88 · see citing works
“The P/B and Q/B ratio were estimated using empirical equations (Allen, 1971; Pauly, 1980; Christensen et al., 2005) or were taken from literature and expressed as annual rates.”
MethodEcoTroph: Modelling marine ecosystem functioning and impact of fishing
Gascuel, D., & Pauly, D. F. (2009) · Ecological Modelling · doi:10.1016/j.ecolmodel.2009.07.031 · Cited by 88 · see citing works
Khan, M. F., & Panikkar, P. (2009) · Ecological Modelling · doi:10.1016/j.ecolmodel.2009.05.020 · Cited by 87 · see citing works
“The P/B ratio is equivalent to the instantaneous rate of total mortality (Z) used by fisheries biologists (Allen, 1971).”
BackgroundMethodFisheries Impacts on North Atlantic Ecosystems : Models and Analyses
Guénette, S., Christensen, V., & Pauly, D. (2001) · Open Collections · doi:10.14288/1.0348145 · Cited by 97 · see citing works
“Bi and Bj are biomasses of prey (i) and predators (j) respectively; P/Bi is the production/biomass ratio, equivalent to total mortality (Z) in most circumstances (Allen, 1971); EEi is the ecotrophic efficiency—the fraction of the total production of a group that is utilized in the system; Yi is the fisheries catch per…”
Extreme environmental conditions reduce coral reef fish biodiversity and productivity
Brandl, S. J., Johansen, J. L., Casey, J. M., Tornabene, L. M., Morais, R. A., & Burt, J. A. (2020) · Nature Communications · doi:10.1038/s41467-020-17731-2 · Cited by 86 · see citing works
Simulating community effects of sea floor shading by plankton blooms over the West Florida Shelf
Okey, T. A. (2003) · Ecological Modelling · doi:10.1016/j.ecolmodel.2003.09.015 · Cited by 83 · see citing works
“…of prey (i) and predators (j), respectively; (P/B)i is the production/biomass ratio, equivalent to total mortality (Z) in most circumstances (Allen, 1971); EEi is the ecotrophic efficiency; the fraction of the total production of a group that is utilized in the system;Yi is the fisheries…”
BackgroundTrophic flow kinetics in marine ecosystems: Toward a theoretical approach to ecosystem functioning
Gascuel, D., Morissette, L. G., Palomares, M. L. D., & Villy Christensen, A. (2008) · Ecological Modelling · doi:10.1016/j.ecolmodel.2008.05.012 · Cited by 81 · see citing works
“In this case, the P/B ratios were generally estimated as the total mortality rate, Z (Allen, 1971), using catch curve analyses or empirical functions such as Pauly’s (1980) equation”
BackgroundMethodInfluentialShannon, L. J., Marta Coll, M., & Neira, S. (2009) · Ecological Indicators · doi:10.1016/j.ecolind.2008.12.007 · Cited by 80 · see citing works
Flows of biomass and structure in an exploited benthic ecosystem in the gulf of California, Mexico
Arreguı́n-Sánchez, F., Arcos, E., & Chávez, E. A. (2002) · Ecological Modelling · doi:10.1016/s0304-3800(02)00159-x · Cited by 77 · see citing works
A simplified formulation for fish production
Mertz, G., & Myers, R. A. (1998) · Canadian Journal of Fisheries and Aquatic Sciences · doi:10.1139/f97-216 · Cited by 79 · see citing works
“Clarke (1946) defined fish production as the amount of tissue elaborated per unit time per unit area, regardless of its fate.”
BackgroundGriffiths, S. P., Young, J. W., Lansdell, M. J., Campbell, R. A., Hampton, J., Hoyle, S., Langley, A. D., Bromhead, D., & Hinton, M. G. (2010) · Reviews in Fish Biology and Fisheries · doi:10.1007/s11160-009-9157-7 · Cited by 77 · see citing works
“These data were derived from stock assessments, scientific surveys, the literature, or estimated by Ecopath; P/Bi is the production/biomass ratio (year ), equivalent to natural mortality (M) for an unfished population, total mortality (Z) for a fished population (Allen 1971), or the inverse of maximum age (x).”
MethodComparative analysis of trophic structure and interactions of two tropical lagoons
Villanueva, M. C., Lalèyé, P., Albaret, J. ‐., Laë, R., de Morais, L. T., & Moreau, J. (2006) · Ecological Modelling · doi:10.1016/j.ecolmodel.2006.03.016 · Cited by 73 · see citing works
“…as: 113 114 (1) 115 116 where Bi is the biomass of group i; P/Bi is the production rate of i equal to the total 117 mortality coefficient (Z) (Allen, 1971); Q/Bi is the relative consumption rate; Bj is the 118 biomass of the predating group j; DCji, the proportion of the predated group i in…”
BackgroundZetina-Rejón, M. J., Arreguı́n-Sánchez, F., & Chávez, E. A. (2003) · Estuarine Coastal and Shelf Science · doi:10.1016/s0272-7714(02)00410-9 · Cited by 75 · see citing works
Brêthes, J. C., Ferreyra, G. A., & de la Vega, S. (1994) · Polar Biology · doi:10.1007/bf00240521 · Cited by 74 · see citing works
Variations in the abundance of fisheries resources and ecosystem structure in the Japan/East Sea
Zhang, C. I., Lee, J.-B., Seo, Y. I., et al. (2004) · Progress In Oceanography · doi:10.1016/j.pocean.2004.06.009 · Cited by 74 · see citing works
Balanced harvest: concept, policies, evidence, and management implications
Zhou, S., Kolding, J., Garcia, S. M., Plank, M. J., Bundy, A., Charles, A. T., Hansen, C., Heino, M., Howell, D., Jacobsen, N. S., et al. (2019) · Reviews in Fish Biology and Fisheries · doi:10.1007/s11160-019-09568-w · Cited by 73 · see citing works
“In Form (ii), current productivity (P/B or p) may be estimated from total mortality (Z) under steady state assumptions (Paloheimo and Dickie 1970; Allen, 1971).”
BackgroundUse of structured expert judgment to forecast invasions by bighead and silver carp in Lake Erie
Wittmann, M. E., Cooke, R. M., Rothlisberger, J. D., Rutherford, E. S., Zhang, H., Mason, D. M., & Lodge, D. M. (2014) · Conservation Biology · doi:10.1111/cobi.12369 · Cited by 71 · see citing works
“…questions (n = 14) about the biomass, production (ratio of production to biomass [P/B]), and consumption (ratio of consumption to biomass [Q/B]) (Allen 1971; Palomares & Pauly 1998) at peak and at equilibrium conditions of bighead and silver carp in Lake Erie relative to the Lake Erie food web.”
BackgroundCorrales, X., Marta Coll, M., Tecchio, S., Bellido-Millán, J. M., Fernández, Á. M., & Palomera, I. (2015) · Journal of Marine Systems · doi:10.1016/j.jmarsys.2015.03.006 · Cited by 71 · see citing works
Trophic structure of Lake Tanganyika: carbon flows in the pelagic food web
Sarvala, J., Salonen, K., Järvinen, M., et al. (1999) · Springer eBooks · doi:10.1007/978-94-017-1622-2_15 · Cited by 72 · see citing works
Koehn, L. E., Essington, T. E., Marshall, K. N., Kaplan, I. C., Sydeman, W. J., Szoboszlai, A. I., & Thayer, J. A. (2016) · Ecological Modelling · doi:10.1016/j.ecolmodel.2016.05.010 · Cited by 72 · see citing works
Trophic interactions and community structure in the upwelling system off Central Chile (33–39°S)
Neira, S., & Arancibia, H. (2004) · Journal of Experimental Marine Biology and Ecology · doi:10.1016/j.jembe.2004.07.011 · Cited by 72 · see citing works
Jh, H. (1985) · Australian Journal of Marine and Freshwater Research · doi:10.1071/mf9850219 · Cited by 71 · see citing works
Caddy, J. F., & Stamatopoulos, C. (1990) · Estuarine Coastal and Shelf Science · doi:10.1016/0272-7714(90)90030-u · Cited by 70 · see citing works
Lercari, D., Bergamino, L., & Defeo, O. (2010) · Ecological Modelling · doi:10.1016/j.ecolmodel.2010.08.027 · Cited by 69 · see citing works
Trophic analysis of Lake Awassa (Ethiopia) using mass-balance Ecopath model
Fetahi, T., & Mengistou, S. (2006) · Ecological Modelling · doi:10.1016/j.ecolmodel.2006.10.010 · Cited by 69 · see citing works
Brey, T., Arntz, W., Pauly, D. F., et al. (1990) · Journal of Experimental Marine Biology and Ecology · doi:10.1016/0022-0981(90)90162-6 · Cited by 68 · see citing works
Rosas‐Luis, R., Salinas‐Zavala, C. A., Koch, V., del Monte‐Luna, P., & Morales‐Zárate, M. V. (2008) · Ecological Modelling · doi:10.1016/j.ecolmodel.2008.06.036 · Cited by 67 · see citing works
Trophic structure of Lake Tanganyika: carbon flows in the pelagic food web
Sarvala, J., Salonen, K., Järvinen, M., et al. (1999) · Hydrobiologia · doi:10.1023/a:1003753918055 · Cited by 68 · see citing works
“In fish populations exhibiting exponential biomass growth, the instantaneous mortality rate equals P/B ratio (Allen, 1971).”
BackgroundChagaris, D., Drew, K., Schueller, A. M., et al. (2020) · Frontiers in Marine Science · doi:10.3389/fmars.2020.606417 · Cited by 67 · see citing works
“In Ecopath, PB ratios and total mortality rates are used interchangeably because the two values are equal under the assumption of equilibrium (Allen, 1971).”
BackgroundEcosystem Modelling Using the Ecopath with Ecosim Approach
Marta Coll, M., Bundy, A., & Shannon, L. J. (2008) · Springer eBooks · doi:10.1007/978-1-4020-8636-6_8 · Cited by 72 · see citing works
“where (P/B)i indicates the production of (i) per unit of biomass and is equivalent to total mortality, or Z, under steady-state conditions (Allen 1971); (Q/B)i is the consumption of (i) per unit of biomass; and DCij indicates the proportion of (i) that is in the diet of predator (j) in terms of volume or weight units.”
BackgroundCabral, J. A., & Marques, J. C. (1999) · Acta Oecologica · doi:10.1016/s1146-609x(99)00102-2 · Cited by 65 · see citing works
“The average biomass was determined by the ratio of the biomass integral within a given time interval and the time interval [2].”
BackgroundMethodStrategic Management of Marine Ecosystems
Levner, E., Proth, J., & Linkov, I. (2005) · Springer eBooks · doi:10.1007/1-4020-3198-x · Cited by 64 · see citing works
Principles for estimating fish productivity on coral reefs
Morais, R. A., & Bellwood, D. R. (2020) · Coral Reefs · doi:10.1007/s00338-020-01969-9 · Cited by 64 · see citing works
“Validating productivity estimates and data quality Empirically measuring dynamic quantities, such as absolute productivity, requires closely controlled conditions (Allen 1971).”
BackgroundMethodInfluential
- Authors
- Allen, K. R.
- Corresponding author
- Allen, K. Radway
- Venue
- Journal of the Fisheries Research Board of Canada
- Publisher
- Canadian Science Publishing
- Published
- 1971-10-01
- Volume
- 28
- Issue
- 10
- Pages
- 1573-1581
- ISSN
- 0015-296X (ISSN-L)
- OpenAlex
- W1997649950
- Cited by
- 590
- References
- 0
- Citations by year
- 2026: 13 · 2025: 9 · 2024: 16 · 2023: 19 · 2022: 21 · 2021: 19 · 2020: 21 · 2019: 14 · 2018: 18 · 2017: 16 · 2016: 16 · 2015: 14 · 2014: 18 · 2013: 20 · 2012: 28
- FWCI
- 2.10
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Journal of the Fisheries Research Board of Canada (publishedVersion)
- Topics
- Health Sciences › Medicine › Public Health, Environmental and Occupational Health › Mathematical and Theoretical Epidemiology and Ecology Models
- Physical Sciences › Environmental Science › Global and Planetary Change › Marine and fisheries research
- Physical Sciences › Environmental Science › Environmental Engineering › Sustainability and Ecological Systems Analysis
- Keywords
- production to biomass ratio, cohort model, animal populations, growth functions, mortality function, mean age, Antarctic krill, production estimation, population dynamics
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