Fecundity of first age class or stage

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A common mistake is to set the fecundity of the first age class (the upper left corner element of the stage matrix) to zero, when it should be greater than zero. If a model has, for example, annual age classes, and some of the individuals that are born/fledged in one year can breed in the following year (when they are almost, but not quite, 1-year old), then this element should not be zero.

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If at least some individuals can reproduce when they are, say, 11-months old (or earlier), then the upper left corner element of the stage matrix should be greater than zero (in a model with annual age classes). Similarly, if age of first reproduction is 2 years, then there should be only one zero in the first row, not two zeros.

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Bias in survival rate estimation

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Immigration and emigration, if not properly accounted for, can bias survival rate estimates. In order to observe and properly account for dispersing individuals in a mark-recapture study, the area covered for recoveries or resightings should be larger than the area within which individuals were marked (Anders & Marshall 2005). Otherwise, dispersing (emigrating) individuals would not be observed and would be assumed dead, biasing (underestimating) the survival rate. Other solutions include correcting based on estimated or independenly available population trend (Ryu et al. 2016), obtaining known-fate data from radio-tagged individuals (i.e., radiotelemetry data) (Pollock et al. 1989, 1995, 2004; Conroy et al. 1996; Powell et al. 2000; Nasution et al. 2001), and using a combination of band recoveries (dead recoveries) and live recaptures to estimate true survival (Barker & White 2002).

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Uncertainty in survival rate estimation

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Reducing uncertainty in survival rate estimation often means increasing the sample size, i.e., marking or censusing a larger number of individuals. However, in cases where individuals can be selected for marking (esp. in plant population studies), the distribution of marked individuals among stages can also determine the amount of sampling error or measurement uncertainty. In most studies, such sampling is random; i.e., on average a similar proportion of individuals in each stage are marked (for example, when sampling is based on fixed plots). Two alternatives to this design may allow more precise estimates of survival rates:

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The first alternative is to determine the number of individuals to be sampled in each stage, based on how much the stages contribute to the growth rate of the population (Gross 2002). This method requires an "educated guess" of the stage matrix, based on prior studies or preliminary data.

The second alternative is to sample an equal number of individuals from each stage. A simulation study by Munzbergova and Ehrlen (2005) demonstrated that sampling an equal number of individuals per stage overall provides better model predictions than plot-based sampling. It is also better than the method proposed by Gross (2002), unless relatively good estimates of the stage matrix are available before the study.

Using either of these alternatives may be more difficult than a plot-based design because of practical reasons (e.g., ease of relocating individuals) and because they require a priori definition of stages (Munzbergova and Ehrlen 2005).

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Using survivorship instead of survival rate

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Life tables often include survivorship, l(x), instead of survival rate, S(x). Survivorship is the proportion of the original number of individuals in the cohort that are still alive at the beginning of age x (by definition, l(0) = 1.0). For a stage matrix, what is needed is survival rate, which is the probability of surviving from a given age to the next, whereas survivorship is the probability of surviving from birth to a given age. Calculate survival rate using S(x) = l(x+1) / l(x).

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Survival rates in the diagonal

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When entering a Leslie matrix (for an age-structured model), the survival rates should be in the sub-diagonal, not diagonal. The exception is that if the last age class is an composite age class, the last survival rate is in the lower right corner of the matrix.

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Ignoring constraints

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A model must have a number of restrictions or constraints on survival rates and the number of survivors:

1. A survival rate must be between 0 and 1.
2. The sum of all survival transitions from a given stage must be less than 1 in any time step.

3. The sum of the number of survivors from a given stage must be less than or equal to the number of individuals in that stage in the previous time step.

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In RAMAS Metapop, the "Constraints Matrix" (accessed from the Stage matrix dialog) identifies the survival rates (as opposed to fecundities), and thus allows the program to impose these three types of restrictions. If this matrix is all zeros (thus assuming all elements are fecundities), or the "Ignore Constraints" box (in General information dialog) is checked, these three types of checks will not be done, and the model will give erroneous results. This is especially important if the model includes any stochasticity (variability) in survival rates.

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If you are using programs other than RAMAS Metapop/GIS, make sure that the model does impose such constraints on survival rates, as well as constraining fecundities to be non-negative.

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Too many (or too few) age classes or stages

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The number of age classes or stages should be consistent with available data, with what is known about the life history of the species, and with the question being addressed (see Model too complex and Model too simple). If there are too many age classes or stages, the sample size for estimating each survival rate and fecundity will be small, increasing error variance (measurement error). If there are too few age classes or stages, the survival rate and fecundity for each stage may not be uniform with that class (because each stage will include a large proportion of the population and therefore a wide variety of individual demographic traits).

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On the one hand, it is necessary to define a sufficiently large number of stages so that the demographic characteristics of individuals within a given stage are similar. On the other hand, it is necessary to have a sufficiently large number of individuals in each stage so that the transition probabilities can be calculated with reasonable accuracy (also see Uncertainty above). For a discussion of this trade-off, and methods of determining the appropriate number of stages, see Moloney (1986) and Vandermeer (1978).

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For an age-structured model, you can reduce the number of age classes either by pooling data from older age classes and creating a composite age class (see Model too complex), or by defining multi-year age classes. For very long-lived species, 2-year, 5-year, or even 10-year age classes may be defined. If you have, say, 2-year age classes, you have to calculate survival rates, fecundities, etc. over 2 years; and each time step of the simulation will be 2 years (so, for a 50-year simulation, you'd set Duration=25).

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Modeling only females

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Most models of vertebrate populations include only females. In most cases, this is fine, as long as sex ratio is properly incorporated into fecundity estimates. However, there are some cases where it is necessary to model both males and females, for example by developing matrix models with different stages or age classes for males and females. If, for example, the purpose of building a model is to evaluate the consequences of different hunting regimes, and only males (and perhaps only males over a certain age) are hunted, then the model obviously needs to have both male and female age classes (e.g., see Sezen et al. 2004). Regardless of the model objective, if males have higher mortality than females (causing a skewed sex ratio), and the mating system is (or is close to) monogamous, then the number of breeding females (and thus, overall population productivity) may be limited by the availability of males; in such cases, the model should include the males in separate stages.

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The stage matrix of such a model may look like the figure below.