Aquarium Fish Stocking & Bioload Calculator

Introduction to aquarium bioload as a nitrogen budget

Bioload is not a mystical property of a fish. It is a mass flow of nitrogen. Protein in the food is digested, the amino acids are deaminated, and the resulting ammonia leaves the fish mostly across the gills as total ammonia nitrogen (TAN). Nitrifying bacteria in the filter oxidise that TAN through nitrite to nitrate. Nitrate is comparatively harmless but it is chemically a dead end in an aerobic aquarium: nothing in a normal tank destroys it. It leaves only in the water you pour down the drain, or in plant tissue you physically remove.

That gives an aquarium exactly three places where stocking can fail. Ammonia can outrun the biofilter, in which case fish are poisoned within days. The biofilter can be too small for the daily feed load, in which case ammonia and nitrite never fall to zero. Or nitrate can outrun the water changes, in which case nothing looks acutely wrong but the tank drifts towards chronically elevated nitrate, algae and poor long-term health. This calculator sizes all three, plus a fourth reference point: what the popular rule of thumb would have told you.

Nothing here replaces a test kit. Treat every number as a planning ceiling, stock to roughly half of it while a new tank matures, and let ammonia, nitrite and nitrate readings tell you whether the model matched your tank.

What the inch-per-gallon rule gets wrong

The "one inch of fish per gallon" rule has no basis in any published standard. No fisheries agency, no aquaculture extension service and no water-quality criterion expresses carrying capacity that way, and the reason is straightforward: waste output scales with how much a fish eats, feeding scales with body mass, and body mass scales with roughly the cube of length. A weight–length relationship of the form has been the accepted description of fish body mass since Keys published it in 1928, and Froese's 2006 meta-analysis of 3 929 relationships across 1 773 species found a median exponent of .

Doubling a fish's length therefore multiplies its bioload by about eight, not by two. Worse, the coefficient depends strongly on body shape. Froese's robust multiple regression of body form gives, at :

Formula: log ⁡ a = − 1.358 ⁢ b + 2.322 − 1.137 D_eel − 0.3377 D_elong − 0.1331 D_fusi

Evaluated at with lengths in centimetres and mass in grams, that yields for eel-like fishes, for elongated ones, for fusiform ones and for short, deep-bodied ones. A stocky fish therefore carries about fourteen times the mass of an eel-shaped fish of the same length, and roughly fourteen times the bioload. Counting inches treats those two animals as identical. That single error is why the rule cheerfully permits six adult common goldfish in a 55-gallon tank, a stocking level this calculator rejects outright.

How to use this stocking model with your own tank

Every field maps to a physical quantity you can measure or look up, and each one has a unit selector where more than one convention is common.

• Tank water volume. Enter the actual water volume, not the box on the label. Substrate, rock and the gap below the rim commonly remove 10–20 per cent of the nominal figure, and the model is linear in volume, so a 15 per cent error here is a 15 per cent error in your answer.
• Temperature and pH. These do not affect how much ammonia is produced, only how toxic it is. They set the un-ionised fraction, and the effect is dramatic: the un-ionised share roughly doubles for every 0.3 unit of pH and rises about 60 per cent from 20 °C to 30 °C.
• Body shape and adult length. Choose the shape class that matches the adult fish and enter the adult total length, not the size in the shop. Tetras, danios and rasboras are elongated; barbs, cichlids and most catfish are fusiform; goldfish, angelfish and discus are short or deep.
• Number of fish. The plan you want to test. The calculator reports the maximum count each constraint allows alongside it.
• Daily ration and feed protein. Ration is the mass of dry food fed per day as a percentage of live fish mass; 1–3 per cent is typical for adult ornamental fish. Crude protein is printed on the tin, usually 32–48 per cent for tropical flake and pellets.
• Water change size and interval. The pair sets nitrate export. Note that a 25 per cent change every 7 days and a 50 per cent change every 14 days are not equivalent; the second exports the same nitrogen but lets the peak concentration run substantially higher.
• Biological media volume. The settled volume of ceramic rings, sponge, bio-balls or matrix in the filter, in litres. The filter preset fills in a representative figure that you can overwrite.
• Nitrate ceiling and tap nitrate. The ceiling is the highest nitrate you are willing to see just before a water change. Tap nitrate is your own supply water, which in agricultural regions can already sit above 20 mg/L and eats directly into your budget.
• Plant nitrogen export. Leave this at zero unless you genuinely trim and discard plant growth. Nitrogen locked into a leaf that stays in the tank has not left the system.

Press Calculate stocking capacity. The result panel names the binding constraint, the chart projects nitrate through several change cycles, and the two secondary buttons copy a permalink that reproduces the exact scenario or download a CSV of every intermediate value.

The stocking formula: feed nitrogen, nitrate export and ammonia equilibrium

Step 1 — fish mass. Each fish's live mass comes from the weight–length relationship with the exponent fixed at 3 and the coefficient taken from its body form:

Formula: M = n ⁢ a ⁢ L^3

with the total stock mass in grams, the number of fish and adult total length in centimetres.

Step 2 — ammonia nitrogen from feed. Daily feed mass is the ration fraction times stock mass, and the standard recirculating-aquaculture expression converts feed and its crude protein fraction into total ammonia nitrogen:

Formula: R_TAN = φ ⁢ M ⁢ P ⁢ 0.092

The 0.092 factor is the fraction of dietary protein excreted as TAN. It is worth sanity-checking against an independent statement of the same physics: SRAC Publication 452 puts ammonia production at about 2.2 pounds of ammonia nitrogen per 100 pounds of feed, i.e. 0.022 g TAN per gram of feed, and , so the two agree exactly for a 24 per cent protein feed.

Step 3 — nitrate production. Complete nitrification conserves nitrogen, so every milligram of TAN becomes a milligram of nitrate nitrogen. Converting to the nitrate ion that test kits report uses the molar mass ratio:

Formula: R_NO3 = R_TAN ⁢ 62.004 / 14.007 ⁢ (1 − ε)

where is the fraction of nitrogen you physically export as harvested plant tissue. The ratio evaluates to 4.4266.

Step 4 — steady-state nitrate. Replacing a fraction of a volume every days produces a sawtooth. At steady state the nitrate mass removed by one change equals the mass produced during one interval, which fixes the peak concentration reached immediately before each change:

Formula: C_peak = C_tap + (R_NO3 ⁢ t) / (r ⁢ V)

Inverting that expression against your chosen ceiling gives the nitrate-limited stock mass directly:

Formula: M_NO3 = ((C_max − C_tap) ⁢ r ⁢ V) / (t ⁢ 4.4266 ⁢ (1 − ε) ⁢ φ ⁢ P ⁢ 0.092)

Step 5 — biofilter capacity. Nitrifying media are conventionally rated by the daily feed ration they can support per unit of media volume. SRAC Publication 453 sizes moving-bed reactors at 8 to 16 kg of feed per cubic metre of media per day, the lowest of the modern engineered designs it reviews; this calculator adopts the conservative end, 8 g of feed per litre of media per day, so

Formula: M_filter = (8 ⁢ V_media) / φ

Step 6 — un-ionised ammonia. Total ammonia partitions between the ammonium ion and free ammonia gas, and only the latter crosses the gill freely. Emerson and colleagues fitted the dissociation constant in 1975 as a function of absolute temperature:

Formula: p K_a = 0.09018 + 2729.92 / (T + 273.15)

Formula: f = 1 / (1 + 10^pK_a−pH)

Reproducing all 90 cells of SRAC Publication 452's Table 2 from this pair of expressions gives a median disagreement of 0.6 per cent; 71 per cent of cells agree within 1 per cent and every cell agrees within 4.4 per cent, the largest gaps sitting in the 32 °C column at the edge of the table. At pH 7.0 and 20 °C the equations give 0.394 per cent against the table's 0.40, and at pH 8.0 and 26 °C they give 5.74 per cent against 5.71. The chronic ceiling on total ammonia nitrogen, and the time the tank would take to reach it if nitrification stopped, follow immediately:

Formula: C_TAN,max = 0.02 / f, t_fail = (C_TAN,max ⁢ V) / R_TAN

The 0.02 mg/L NH₃-N figure is the bottom of the 0.02–0.07 mg/L range at which SRAC Publication 452 reports slowed growth and tissue damage in warmwater species.

Worked example: twenty tetras against one adult goldfish

Take a 55 US gallon tank, which is 208.2 litres, held at 26 °C and pH 7.4, with 25 per cent of the volume changed every 7 days, 3 litres of biological media, a 40 mg/L nitrate ceiling, nitrate-free tap water and no plant export.

Scenario A: twenty elongated-to-fusiform community fish of 6 cm adult length, fed 2 per cent of body mass daily on a 45 per cent protein flake. Each fish weighs 0.01303 × 6³ = 2.81 g, so the stock is 56.3 g. Daily feed is 1.126 g, giving 1.126 × 0.45 × 0.092 = 0.0466 g, or 46.6 mg, of TAN each day, which becomes 46.6 × 4.4266 = 206.3 mg of nitrate per day. Steady-state peak nitrate is 206.3 × 7 ÷ (0.25 × 208.2) = 27.7 mg/L, comfortably under the 40 mg/L ceiling. Inverting for the limit gives a nitrate-limited stock of 81.1 g, or 28 fish. The biofilter, rated at 3 × 8 = 24 g of feed per day, would support 1 200 g of fish, so it is nowhere near binding. At pH 7.4 and 26 °C the un-ionised fraction is 1.505 per cent, the TAN ceiling is 1.33 mg/L, and if the filter died the tank would reach it in about 5.9 days. The inch rule would have allowed 55 ÷ 2.36 = 23 fish, so here it lands in roughly the right neighbourhood by accident.

Scenario B: one short-bodied 20 cm common goldfish in the same tank, fed the same 2 per cent ration on a 35 per cent protein pellet. Its mass is 0.01770 × 20³ = 141.6 g, so the single fish outweighs twenty tetras two and a half times over. Daily feed is 2.832 g, producing 91.2 mg of TAN and 403.7 mg of nitrate per day, and the steady-state peak is 403.7 × 7 ÷ 52.05 = 54.3 mg/L. One goldfish alone breaches a 40 mg/L ceiling. The nitrate-limited stock mass is 104.3 g, so the permitted count rounds to zero, and holding this fish at 40 mg/L would require a 34 per cent weekly change instead of 25 per cent. The inch rule, meanwhile, sees a 7.9 inch fish in a 55 gallon tank and licenses six of them. That is the gap between counting inches and counting nitrogen.

Interpreting the four constraints the calculator reports

• Nitrate-limited count. In almost every home display aquarium this is the number that binds, because nitrate has no sink other than the bucket. If it is uncomfortably low, the levers are larger or more frequent changes, a lower ration, or fewer and smaller fish. Buying a bigger filter does nothing for it.
• Biofilter-limited count. This binds in heavily fed or under-filtered systems and in newly cycled tanks where bacterial populations have not caught up. A wide margin here is not a licence to overstock; it just means ammonia is not your problem today.
• Un-ionised ammonia ceiling and failure window. These describe risk, not capacity. A tank whose failure window is under 24 hours has no tolerance for a power cut, a clogged intake or a medication that stalls the filter. Anything under two days deserves a battery air pump.
• Stocking density in kg/m³. SRAC Publication 453 sizes broodstock and display systems below 15 kg/m³ and ornamental production below 30 kg/m³, with engineered filtration and 5–10 per cent daily water exchange. A home aquarium with weekly changes will normally hit its nitrate ceiling far below those densities, which is exactly why the nitrate constraint usually wins.
• Inch-rule comparison. Shown only as a diagnostic. When the rule is far more permissive than the nitrogen budget, you are looking at a fish whose mass the rule cannot see.

A result of zero permitted fish is a real answer, not an error. It means the plan cannot be held at the nitrate ceiling you chose with the maintenance you described, and the panel will tell you what water change fraction would be needed instead.

Comparison table: how each lever moves the answer

Limitations and assumptions behind this nitrogen-budget model

The model is deliberately narrow, and being explicit about what it ignores matters more than adding spurious multipliers.

• It assumes a fully cycled, aerobic system in steady state. During cycling, after a filter clean or following antibiotic treatment, real ammonia and nitrite will exceed anything modelled here.
• It assumes complete nitrification and no denitrification. Deep substrates, plenums and purpose-built anoxic media do remove some nitrate, which makes the nitrate constraint pessimistic. It assumes no protein skimming either, so marine systems with skimmers will do better than predicted.
• It uses a single body-form coefficient from a cross-species meta-analysis rather than a species-specific relationship. Individual species deviate; a specific weight–length pair from FishBase for your species will always beat the shape default.
• It says nothing about oxygen, territory, aggression, swimming space, temperature compatibility, social group size or footprint. Those constraints frequently bind long before nitrogen does. A tank can pass every number here and still be an unacceptable home for the fish in it.
• The 0.092 protein-to-TAN factor is a production-aquaculture average measured on food species at commercial feeding rates. Ornamental fish fed sparingly, and fry fed heavily, will differ.
• The biofilter rating is drawn from actively aerated engineered media. A low-flow sponge, a clogged cartridge or media rinsed in chlorinated tap water will perform below it, so treat the biofilter-limited count as an upper bound.
• The 0.02 mg/L un-ionised ammonia ceiling is a chronic threshold for warmwater species generally. Sensitive species, fry and invertebrates warrant lower; tilapia and some cyprinids tolerate more. It is not a species-specific toxicity value.

Sources

Every constant on this page was taken from the primary document named below, not from secondary summaries. Where two independent sources cover the same quantity, both are listed so the agreement can be checked.

• Masser, M. P., Rakocy, J., and Losordo, T. M. (1999). Recirculating Aquaculture Tank Production Systems: Management of Recirculating Systems, SRAC Publication No. 452, Southern Regional Aquaculture Center, USDA. Source of the 2.2 lb ammonia nitrogen per 100 lb feed figure, the un-ionised ammonia percentage table (Table 2), the 0.02–0.07 mg/L NH₃-N harm range and the 300 ppm nitrate toxicity threshold. SRAC 452 (PDF)
• Malone, R. (2013). Recirculating Aquaculture Tank Production Systems: A Review of Current Design Practice, SRAC Publication No. 453, Southern Regional Aquaculture Center, USDA. Source of the 8–16 kg feed per m³ of media per day moving-bed biofilter rating and the 15 kg/m³ and 30 kg/m³ display and ornamental stocking densities.
• Emerson, K., Russo, R. C., Lund, R. E., and Thurston, R. V. (1975). "Aqueous Ammonia Equilibrium Calculations: Effect of pH and Temperature", Journal of the Fisheries Research Board of Canada 32(12), 2379–2383. Source of the pK_a temperature relationship used for the un-ionised ammonia fraction.
• United States Environmental Protection Agency (2013). Aquatic Life Ambient Water Quality Criteria for Ammonia — Freshwater 2013, fact sheet EPA 820-F-13-013. Recommends an acute criterion of 17 mg total ammonia nitrogen per litre and a chronic criterion of 1.9 mg TAN/L, both at pH 7 and 20 °C. EPA aquatic life criteria for ammonia
• Froese, R. (2006). "Cube law, condition factor and weight–length relationships: history, meta-analysis and recommendations", Journal of Applied Ichthyology 22(4), 241–253. Source of the weight–length body-shape regression (Eqn 17) used for the shape coefficients. Froese 2006 (PDF)
• Camargo, J. A., Alonso, A., and Salamanca, A. (2005). "Nitrate toxicity to aquatic animals: a review with new data for freshwater invertebrates", Chemosphere 58(9), 1255–1267. Recommends a maximum of 2 mg NO₃-N/L (about 8.9 mg/L as nitrate ion) to protect the most sensitive freshwater species and 20 mg NO₃-N/L for marine animals; the basis for the nitrate ceiling presets.
• Timmons, M. B., and Ebeling, J. M. Recirculating Aquaculture. Source of the R_TAN = feed × protein × 0.092 formulation, cross-checked here against SRAC 452 as described above.

Questions aquarists ask about bioload and stocking