The question behind a declining age histogram
A fish catch curve takes numbers of fish at successive ages and asks whether the decline among fully vulnerable older ages can be translated into an estimate of total instantaneous mortality. This is useful because a steeply descending age-frequency line can reflect the loss of fish as cohorts grow older. But a falling chart is not automatically a direct count of fish that died. Sampling gear, which ages can be captured, changes in recruitment, migration and observation error also alter how many individuals appear in a particular age bin. The first responsibility of a reader is therefore to distinguish the biological population from the observation system that produced the histogram.
The label “number of fish” is not enough. Were the fish aged from an independent scientific survey, from commercial landings, or from a subset of a particular fleet? Does a count mean raw aged individuals, fish per standardized unit of effort, or estimates raised to total landed catch? Were discards observed? Were all ages sampled in the same season and area? A catch curve usually summarizes a narrow observational domain. When translating the graph for readers, begin with that domain and the chosen age range. If these details disappear, the slope can be mistaken for an exact death rate applying to every fish in the stock.
First principles: why the relationship is exponential
The simplest survival assumption says that the number remaining after a time interval is the starting number multiplied by exp(-Z times that interval), where Z is the instantaneous total mortality rate. If the interval is measured in years, Z is conventionally expressed per year. An instantaneous rate is a hazard in a mathematical survival model, not itself the percentage of fish dying over a whole year. At Z = 0.50 per year, expected one-year survival is exp(-0.50), approximately 0.607. Thus about 60.7 percent survive and 39.3 percent die from the combined causes represented by Z, under the model.
The exponential form comes from a proportional-loss mechanism: a small interval removes approximately the current number of fish multiplied by the mortality rate and the interval length. There are fewer fish remaining to be lost in later intervals, so numbers curve down rather than following a constant absolute decline. Taking natural logarithms transforms the exponential relationship into a straight-line expression: natural log abundance equals an intercept minus Z multiplied by age or elapsed time. The slope on a natural-log graph is therefore negative Z. The model is simple enough to calculate by hand, but its assumptions about time, exposure and sampling must accompany the formula.
What goes on each axis
The horizontal axis ordinarily shows age, often in whole years. The vertical axis shows the natural logarithm of fish numbers, numbers per consistent effort or another justified proportional abundance measure. The intercept corresponds to the fitted level when age is zero, though the line may be fitted only to older ages and its extrapolated intercept may have no direct biological interpretation. Each displayed point must correspond to a particular age and defined observation unit. A plot using raw counts on a linear vertical scale has a different expected shape from one using logarithms.
The base of the logarithm is an easy source of translation errors. If the vertical axis uses natural logarithms, the straight-line slope is -Z in inverse years. If the plotted transformation is log base ten, its slope is -Z divided by ln(10). To recover Z from a base-ten slope, multiply the magnitude by approximately 2.302585. A line descending by 0.20 base-ten units per year implies Z near 0.461 per year, not 0.20 per year. Graph captions should state the transformation explicitly; a summary that quotes the slope without identifying its base can produce a numerically wrong mortality rate.
Why the youngest ages usually do not belong
Fish enter a gear’s catch gradually. Very young fish may pass through a mesh, live in inaccessible habitat or behave in ways that make capture unlikely. Age composition may therefore rise from younger classes toward an age of full vulnerability. Including this ascending limb in a linear mortality fit violates the assumption that observed numbers decline mainly because fully selected fish are being lost. The rising counts may primarily reflect increasing catch probability, not growing population abundance. Analysts commonly fit the descending limb beginning at an age judged fully recruited to the sampling system.
That age boundary is not a universal biological constant. It depends on species, gear, area, season and potentially regulations. A mesh change can move the apparent boundary even if actual age-specific survival remains unchanged. The appropriate fit should be justified with selectivity evidence, comparison of gears or explicit sensitivity analysis. When a report simply states “ages five through nine were used,” ask why age five was selected and whether the same conclusion follows if age six is the starting point. An attractive straight line alone cannot establish that the fitted segment represents equal vulnerability across ages.
A worked four-age dataset
Take an illustrative, consistently standardized series with 1,000 fish at age four, 607 at age five, 368 at age six and 223 at age seven. These are deliberately constructed teaching numbers, not observations of a named fishery. Their natural logarithms are approximately 6.908, 6.409, 5.908 and 5.407. Every extra year produces a drop of about 0.50 on the natural-log scale. A regression through those four values therefore estimates a slope near -0.50, giving an apparent Z of about 0.50 per year if the ages are fully vulnerable and the cohort and sampling assumptions are reasonable.
The calculation can be checked from count ratios without specialist software. The ratio 1,000 divided by 607 is approximately 1.647; taking its natural log gives about 0.499. The ratio 607 divided by 368 gives about 1.649, or a natural log near 0.500. The final adjacent pair is similarly consistent. These checks make it harder to accidentally use log base ten, reverse the sign or confuse a five-year interval with a one-year one. The numerical agreement, however, is a feature of this invented example. Real observations are uneven, and a tidy fitted line does not demonstrate that selectivity and recruitment assumptions are true.
Mortality has two main components
Standard fisheries mortality accounting decomposes total instantaneous mortality into natural mortality and fishing mortality: Z = M + F. Natural mortality M includes the non-fishing hazards represented by the assessment model, such as predation and disease. Fishing mortality F is the hazard associated with fishing. These rates share inverse-time units and can be added in the stated mathematical framework. Catch curves ordinarily estimate an apparent Z; they do not separate M and F by themselves. An independent assumption or estimate of M is needed before the fishing component can be inferred by subtraction.
Suppose Z is estimated at 0.55 per year and other biological evidence supports M near 0.20 per year. The implied F is 0.35 per year. A careless report might announce that “35 percent of fish were caught,” but this is not what F means. In a simple competing-hazards model, the fraction lost to fishing during a year is F/Z times [one minus exp(-Z)], approximately 0.35/0.55 times 0.423, or around 0.269. That is a modelled fishing-attributable loss fraction, not necessarily the fraction retained in landings. Discards, delayed death and unobserved catch all require additional treatment.
Natural mortality is not a free number
Estimating M is often one of the most uncertain steps. Longevity, growth, maturity, predation, environmental conditions and independent survival information may all contribute to a defensible assumption. Yet natural mortality can vary by age, region or time. Borrowing a generic value merely because it appeared in another report risks presenting an untested biological assumption as a measurement. If M is larger, the inferred fishing component for a fixed Z is smaller; if M is smaller, the inferred F rises. This relationship must be carried into the uncertainty statement.
For instance, at Z = 0.55 per year, using M = 0.15 produces F = 0.40, while using M = 0.30 produces F = 0.25. Those are materially different conclusions from the same age-frequency line. If uncertainty in Z also ranges from 0.45 to 0.65, it is misleading to report F = 0.35 with unsupported two-decimal precision. Extreme combinations yield illustrative F values between 0.15 and 0.50, but this range is not itself a formal confidence interval. Correlation and likelihood matter. A negative F from subtracting incompatible estimates indicates an identification problem, not a fishery creating additional fish.
One cohort or several birth years
An age-frequency plot sampled in one calendar year compares different birth cohorts. The six-year-olds were born earlier than the four-year-olds. If one birth year produced unusually many recruits, that cohort can distort the slope even when mortality remained broadly stable. A cohort-following design instead traces fish from the same birth year across successive sampling years. Those two designs do not have identical assumptions, although both can generate a declining line after logarithms. A reader should not infer that a snapshot literally followed the same group of fish through time.
Consider a very strong cohort entering the selected ages. In the first sampling year it may raise age-five counts, flattening one part of the apparent catch curve. In the next year the same cohort enters age six and creates a different bump. A comparison of age-by-year data can reveal diagonal movement of a cohort feature across the table. By contrast, a persistent excess at the same age every year may indicate a selectivity or ageing problem. An unusual point is information requiring investigation, not automatically an outlier to be deleted until the line looks smooth.
Catch records and sampling weights
Fish from different fleets and ports may have unequal chances of being sampled. A survey can age several hundred individuals from one landing and a few individuals from another even though the second landing represents much greater fishing effort. Raw aged counts then reflect the sampling design as well as the fishery. If a report raises measured age frequencies to landed catch using documented weights, the resulting age numbers may differ. Uniformly multiplying all ages by one constant changes only the fitted intercept. Age-specific weighting changes the shape and therefore can change estimated Z.
A robust interpretation records the number aged, the method for selecting specimens, the stratification variables and how data were raised or standardized. “Per haul,” “per hour,” “per trip” and “entire fishery catch” refer to different denominators. The effect of seasonal sampling is particularly important for migratory fish whose age distribution shifts among habitats. Avoid combining counts from incomparable observational processes and calling the result a population survival curve. Independent survey data can support the stock-wide picture, but even an independent survey has its own selectivity and spatial footprint. The assumption linking observed age frequencies to abundance needs evidence.
The sparse upper tail and plus groups
Old fish may be rare in samples, so upper-age counts can be small or zero. Taking the natural logarithm of zero is undefined. Adding one to every age count is a tempting shortcut but has a proportionally large effect on sparse groups, and may alter the estimated slope. A count-likelihood model, explicit treatment of sparse ages, or careful reporting of insufficient data may be more defensible than applying a hidden pseudocount. Removing rare old ages after examining the desired result can also bias the fit. A sensitivity analysis should explain how sparse classes were handled.
A plus group, for example “age ten and above,” sums several ages into one category. It is not an ordinary one-year bin with an exact age of ten. Using the aggregate as if it were an age-ten point violates the equal-age interpretation of a simple catch curve. The analyst can omit the plus group from the straight-line fit, or use a model representing its grouped composition explicitly. The graph caption must identify where precise ages end. Otherwise the largest aggregation may be mistaken for a surprising biological increase in survival at the oldest age.
Gear differences change the interpretation
A trawl, a gillnet and a hook-and-line fishery may encounter different sizes, behaviors and habitats of the same species. Their age compositions can diverge even while the stock shares a common underlying survival history. Gear retention and encounter probability are not interchangeable with mortality. A catch curve from one fleet can appear steep because older fish are less available in that fleet’s fishing grounds, not because those fish died at a faster rate. Comparing the raw slopes from different gear types without checking selection is therefore risky.
The repair is to specify the gear and examine selectivity curves or independent calibration where available. If gear selectivity can be estimated, the analyst can identify which ages might be fully vulnerable and compare like-for-like strata. If selectivity is unknown, uncertainty remains and should be stated. This issue is explained separately in eduKate’s fishing-gear selectivity guide. A catch-curve slope is a conditional summary of sampled age composition. It is not a universal mortality constant that every fishing method must display identically.
Reading a neat line with scientific restraint
A regression may fit the selected log counts with a high R-squared and small numerical standard error. That shows the points align well under the fitted mathematical model. It does not establish that the ages were fully selected, the cohorts were equally strong, fish did not migrate or ageing error was absent. Statistical precision within a model is different from certainty that the model describes the right mechanism. A small opportunistic sample can produce an excellent-looking line by chance; a larger representative survey may produce noisier points because it captures more real variation.
A strong report checks both mathematical fit and evidence for assumptions. Present counts and their denominators, the fit interval, the logarithm base, uncertainty in ageing, sensitivity to the first fully selected age and the biological basis for M. If the result is used in management discussion, distinguish estimating Z from judging whether F exceeds a stock-specific reference point. A catch curve is a useful diagnostic and teaching tool precisely because the equation is transparent. Its simplicity should encourage verification, not license overconfident claims about fishery sustainability.
Calculation by adjacent age ratios
For a sufficiently comparable pair of successive fully selected ages, the simple exponential approximation gives Z approximately equal to the natural logarithm of the younger-age count divided by the older-age count, when ages differ by one year. If ages six and seven have counts of 400 and 240, respectively, the ratio is 400/240 = 1.6667, and ln(1.6667) is about 0.511. This corresponds to an apparent instantaneous total mortality rate of roughly 0.511 per year. The survival associated with that model is exp(-0.511), approximately 0.600, and the annual fraction dying is approximately 0.400. The sign is correct because the ratio is greater than one.
The pairwise calculation is a transparent check, not an automatic substitute for fitting several ages. A single point can be unusually high because a cohort was strong, or unusually low because a sample was incomplete. With three or more ages, examine the pattern of log counts, fit an appropriate model, and inspect the residuals. If all adjacent ratios imply nearly the same Z, the model may describe the chosen segment reasonably well. If ratios swing strongly, investigate cohort strength, selectivity, ageing and measurement issues before reporting a single slope. A four-point classroom exercise can demonstrate the mathematics but cannot independently prove its biological assumptions.
A worked case with a base-ten axis
Suppose the caption explicitly says “log10 fish count,” and the descending fitted slope is -0.22 for each additional year of age. The correct instantaneous Z is 0.22 times ln(10), or approximately 0.507 per year. If M is independently assessed as 0.18 per year, the implied F is 0.327 per year. One-year survival under the total hazard is exp(-0.507), approximately 0.602, leaving a total one-year loss fraction near 39.8 percent. These numbers have different meanings despite being derived from the same plot, and each should be identified by its symbol and unit.
An incorrect summary might say the fish suffered 22 percent annual mortality because it read -0.22 directly from the graph, or 50.7 percent because it equated instantaneous Z with the annual fraction dying. Both conclusions are wrong. The reliable method checks the logarithm base, transforms the gradient, calculates survival, then translates into ordinary language. If the original plot does not disclose its logarithm convention, avoid silently guessing from its appearance. Request its axis definition or infer it from a supplied numerical table and state that inference. Translation cannot be accurate when the underlying quantity is ambiguous.
Comparing two legitimate fit windows
Consider illustrative standardized counts at ages three to eight: 1,800, 1,650, 1,100, 725, 475 and 310. The younger classes may not be fully selected. A rough endpoint-based decline from ages four to eight is ln(1650/310) divided by four, approximately 0.418 per year. The corresponding decline from ages five to eight is ln(1100/310) divided by three, about 0.422. Including age three across the full five-year span gives ln(1800/310) divided by five, or roughly 0.352. These calculations demonstrate why the chosen first age matters, even before using a fitted regression.
An actual assessment would justify the fit window using gear and biological evidence and generally examine regression diagnostics, not use endpoint arithmetic as its only estimator. The early count of 1,800 is lower than the roughly extrapolated abundance implied by the older-age trend, suggesting incomplete age-three selection in this constructed example. If age four is also partly selected, the alternative starting at five may be preferable. A reader should be able to reconstruct all three comparisons from the displayed data and understand why the final choice was made. The correct answer is not merely whichever segment gives the most impressive line or the desired mortality number.
What if the catch curve bends
A descending catch curve may curve downward faster among the oldest ages, or flatten after a certain age. This can happen for several reasons. Older fish may move into deeper habitats, experience changing fishing vulnerability, suffer changing mortality, or include more ageing error because the otolith structures are difficult to interpret. A plus group can distort the upper tail. Different recruitment strengths across the represented birth years may also break a single straight-line pattern. The curve is telling the analyst that constant apparent mortality under equal observation probability may be an oversimplification.
Instead of forcing one global line, inspect whether the bend aligns with changes in gear retention, age-reading protocol or distribution. Consider separate age ranges as sensitivity tests, but do not assign a second mortality regime without biological evidence. If a species changes habitat at maturity, spatial redistribution may explain a feature that otherwise resembles an age-dependent survival change. Where data support more complex analyses, age-structured models can explicitly represent selectivity, natural mortality by age or movement. When they do not, the honest result may be a qualitative description and a recommendation for better age and habitat coverage.
A two-year interval is not a one-year interval
Suppose a cohort proxy falls from 500 at age five to 200 at age seven, with no comparable age-six observation. The elapsed age interval is two years. In the simplified exponential model, Z equals ln(500/200) divided by two, not the full log ratio. Since ln(2.5) is approximately 0.9163, Z is approximately 0.4582 per year. Dividing by the actual time interval preserves the unit of the hazard. Forgetting that denominator would double the estimate and distort every later comparison to mortality reference points.
Similarly, when a graph is based on half-year age steps or months, the slope unit must be converted before describing annual mortality. A rate of 0.06 per month would be 0.72 per year in a constant-instantaneous-rate model, not 6 percent annual loss. If Z is multiplied by time to form the dimensionless exponent, the units cancel. This unit check is a practical safeguard. A well-written technical explanation gives the sampling interval, rate units and transformation so readers do not have to reconstruct them from an unlabeled axis.
What a fishing-attributable fraction really means
If natural and fishing mortality operate as constant competing hazards, total hazard Z equals M plus F. Over one year, total probability of dying under the simplified model is one minus exp(-Z). The fishing-attributable probability is F/Z times that total probability. For example, with M = 0.20 and F = 0.30 per year, Z = 0.50 per year, total death probability is approximately 0.3935, and fishing-attributable probability is 0.30/0.50 times 0.3935, approximately 0.2361. Natural causes account for about 0.1574 of the starting population in this mathematical partition.
This is distinct from retained catch fraction, landed biomass and nominal exploitation rates defined under other conventions. Fish can be discarded and subsequently die, not all fishing effects are visible in port records, and age-dependent hazards may violate the constant-rate simplification. If an article converts a modelled instantaneous rate into a percentage, it should show the formula and define the event represented. The word “fishing mortality” is often used precisely in stock assessments, but readers from other disciplines may hear it as a simple count of dead fish. A careful explanation prevents that categorical misunderstanding.
Uncertainty from age-reading and sample design
Age assignments are measurements, not perfect labels attached to fish at birth. Otoliths or other hard structures must be prepared and read, and two trained readers can disagree on annuli. A one-year misclassification shifts an individual between adjacent catch-curve bins and can alter the slope. Validation, double reading and age-error matrices help characterize these effects. A large sample may reduce random counting noise while leaving systematic ageing error untouched. A dense, carefully fitted regression line should not disguise uncertainty in the age categories themselves.
Sampling designs also matter. One hundred fish aged from each of ten representative strata carry different information from a thousand fish collected in one convenient port after an unusual season. The latter can overrepresent a particular fleet, habitat or size composition. Weighting and survey-design variance may be needed. A reader evaluating a mortality estimate should ask whether observations represent the same stock, whether strata and effort were addressed, and whether uncertainty reflects the actual sample design. For a related explanation of the underlying age evidence, read fish otolith annuli and age–length keys.
One method should not silently replace another
Catch-curve mortality estimates are relatively simple and can serve as diagnostics. A virtual population analysis reconstructs cohorts through catch histories while modelling survival and fishing losses, and a modern integrated assessment may simultaneously fit catch, indices, composition and biological parameters. These methods can produce different estimates because their observations, assumptions and identification problems differ. It is not scientifically justified to treat a catch-curve F as equivalent to a stock-wide terminal F from a model that incorporates changing selectivity and recruitment. Compare the quantities, periods and definitions before comparing numbers.
A particularly common mistake is to confuse the age-frequency slope with a regression of total catch over calendar time. A declining catch series can reflect regulation, effort, markets or stock abundance; it is not an age catch curve. Another mistake treats the estimated F as a management status threshold by itself. Status determination uses applicable stock-specific reference points and uncertainty. A catch curve helps generate or assess evidence, but management decisions need a broader causal and institutional context. See the associated eduKate catch-at-age VPA guide for the distinct cohort-reconstruction logic.
A misleading real-world style scenario
Imagine a sampling programme whose trawlers previously worked mainly inshore. In a new season, the programme adds offshore strata containing more large and old fish. The new catch curve becomes flatter, and a superficial comparison announces that mortality declined. Yet the change may originate from expanding spatial coverage rather than from improved survival. The same problem can occur when fleets move after closures, if sample weighting does not restore comparability. The appropriate analysis first examines the changed sampling domain, then compares like-for-like strata across years or uses a statistical model incorporating spatial differences.
Now reverse the scenario. A warming event alters distribution and older fish leave the previous survey footprint. The graph may steepen even if those fish remain alive elsewhere. Neither scenario implies that catch curves are useless. Both show why an age-frequency graph is an observation filtered through habitat, gear and sampling decisions. Independent surveys, tagging or environmental data can help distinguish mortality change from movement. A useful explanation names the rival mechanisms and the evidence that could separate them, instead of committing to a causal story solely because a line has changed direction.
Transfer exercise: estimate Z, F and annual survival
A teaching dataset reports 300 fully selected age-eight fish and 180 age-nine fish, sampled with comparable effort. Assuming constant mortality, recruitment and catchability conditions for this simplified exercise, Z = ln(300/180) = ln(1.6667), about 0.511 per year. If M is independently assumed to be 0.17 per year, F = 0.511 – 0.17 = 0.341 per year. One-year survival is exp(-0.511), approximately 0.600, and total one-year loss is around 0.400. The fishing-attributable loss fraction under constant competing hazards is 0.341/0.511 times 0.400, about 0.267.
The answer must describe all three layers correctly: an apparent instantaneous total mortality estimated from a count ratio, an inferred instantaneous fishing component dependent on M, and a one-year survival probability derived from the exponential model. None is a direct observation of stock-wide sustainable catch. If the gear underselects age-nine fish relative to age-eight fish, the ratio overstates losses. If the age-eight cohort was unusually strong, the cross-sectional comparison can also mislead. The arithmetic is precise only conditional on the assumptions, and the statement of assumptions belongs in the answer.
Transfer exercise: why a zero count is not zero survival
A small research sample has 16 fish aged six, seven fish aged seven and none aged eight. A trainee might claim the fish have total mortality of 100 percent after age seven. That conclusion is unsupported. Zero observed age-eight fish could result from sparse sampling, gear selectivity, location, chance or genuinely low abundance. The log of zero is undefined, so an ordinary log-linear fit cannot simply insert that point. Adding one creates an arbitrary transformed count with disproportionate influence when data are sparse. At the minimum, report that the older age category was not observed in the sample.
A better analysis examines the sampling frame, effort and probability of detecting older ages. Pooling more years or strata may increase information, if biological comparability can be defended. Alternatively, an explicit count observation model may handle zero counts without pretending they are exact zero populations. The exercise teaches a general scientific principle: non-detection is a property of the observation process and does not automatically establish absence. Fishery-age interpretation depends on observation probability as much as on the survival equation.
Transfer exercise: choose a defensible fit range
A sample displays the following relative age counts: age three, 220; age four, 400; age five, 360; age six, 240; age seven, 155. A simplistic attempt to fit all ages would combine an ascending segment from three to four with a declining segment from four onward. The proper first question is whether the sampling gear is fully selective by age four or only by age five. If it is fully selective at age five, the ages five to seven should be considered for a simple descending-limb fit, with uncertainty from having only three points and from possible recruitment differences.
The natural-log endpoint decline from age five to age seven is ln(360/155) divided by two, approximately 0.421 per year. This is a rough check, not a definitive estimate, and the exact fitted gradient across all three points may differ slightly. Including age three would greatly alter interpretation because the age-three count may reflect selection rather than mortality. An answer should give the reasoning behind the fit window, identify missing selectivity evidence and explain how adding data could improve precision. Simply choosing the interval yielding the preferred F fails the exercise.
Frequently asked questions
Does an estimated Z of 0.60 mean 60 percent die annually? No. The corresponding one-year loss fraction is one minus exp(-0.60), approximately 45.1 percent under the exponential model. Does the catch curve separately measure M and F? No. A defensible estimate or assumption for M is needed to infer F from Z. Is a high R-squared proof of a healthy stock? No. It describes how closely selected transformed points follow the fitted line, not whether fishing pressure is sustainable. Why does the log base matter? Because a base-ten slope differs from a natural-log slope by the factor ln(10).
Can the oldest plus group be used as an ordinary age? No, because it aggregates multiple ages unless a more suitable model represents that structure. Are different gears expected to yield the same age curve? Not necessarily; vulnerability and sampled habitat differ. Should one anomalous age be removed? Only with a defensible documented reason based on data quality or model design, not simply to straighten a plot. What is the safest summary? State the sampled system, age range, logarithm convention, apparent total Z, assumptions, uncertainty and the independently justified M used to infer F.
How to communicate the result in one accountable paragraph
A model-ready summary could read: “For fish assessed as fully vulnerable to this survey gear at ages five through nine, the fitted natural-log age-frequency gradient implied an apparent total instantaneous mortality of 0.50 per year. Under an independent assumed natural mortality of 0.20 per year, the corresponding fishing mortality estimate was 0.30 per year. The estimate is sensitive to cohort variation, sampling coverage and age-specific selectivity and should not be read as a direct percentage of fish caught.” This communicates what the graph supports without pretending to have measured all biological causes.
That level of precision is useful for students, report translators and decision makers because they can reconstruct the calculation and challenge assumptions. Keep the raw counts, sampling dates, gear description, fitted ages, log base, standard errors and source of M accessible. When new evidence changes the conclusion, revise the inference rather than erasing the earlier uncertainty. A clear catch-curve explanation is not a shorter formula with stronger adjectives; it is a connected account of the measurement, mathematical transformation, biological model and limits of the claim.
Primary methodological sources
FAO: Methods for Estimating Total, Fishing and Natural Mortalities. FAO: Mortality and Catch-Curve Interpretation. FAO: Virtual Population Analysis and Survival Accounting. These first-party methodological sources support the standard equations and important selectivity limits; numerical exercises in this guide are hypothetical and independently worked.