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Avoiding Mistakes in Population Modeling:
Demographic Structure

List of common mistakes in population modeling

No demographic structure (scalar models for age-structured populations)

Scalar models (with no age or stage structure within populations) are often used in cases where the only available data are a time series of population size estimates (these methods are also known as "count-based models" and "diffusion approximation"). However, if the population being modeled has age structure, a scalar model of this population may overestimate the variability in the population size, and hence overestimate the risks faced by the population (Holmes 2004).

An analysis of scalar and structured models for a set of populations has indicated a precautionary bias (overestimated risks of decline) by scalar models, and a set of simulations has indicated that the bias increases as a function of the generation time of the species (Dunham et al. 2006). Correcting the bias (e.g., based on the species' generation time) seems difficult if not impossible, because the bias is not a simple function of generation time, and because any deviation of the initial age structure from the stable age structure adds uncertainty (Dunham et al. 2006).

Future developments in the analysis of time series data for building stochastic scalar models may address this issue. Until then, if a population is known to have age structure, but age-structured data are not available to build a matrix model, the results of scalar models of this population should be viewed as preliminary (and likely precautionary).

Bias in fecundity estimation

Fecundity should be based on total productivity of individuals over the time step of the model. Fecundity calculations based on a single nesting attempt underestimate fecundity. See Anders & Marshall (2005) for methods of estimating season-long productivity in landbirds.

Note that in addition to productivity or fertility (number of fledglings per pair per time step), the fecundity values in a matrix model must also incorporate survival rate, sex ratio, and proportion breeding.

Fecundity vs. maternity

A common mistake is to use a measure of maternity ( m(x), e.g., number of eggs per female; number of fledglings per nest, etc.) in the stage matrix, instead of fecundities, F(x). In a matrix model, the fecundities must incorporate two types of survival:
(a) survival of breeders (from census to next breeding)
(b) survival of newborns (from birth to next census)
Most matrix models assume either post-breeding or pre-breeding census, and thus either (a) or (b) is assumed to be 1.0 (but never both). In using maternities or life table data to estimate the stage matrix, the exact definition of m(x) is very important; check the source of the information in order to use the correct index. For more information, see Akçakaya (2000a), Caswell (2001) and Akçakaya et al. (1999).

Not incorporating proportion breeding

In a matrix model, the fecundities must incorporate the proportion of individuals breeding in an age class or stage. Fecundities have units of "per individual" or "per female" in that age class or stage (not "per breeder"). If, for example, two-year old breeders and three-year old breeders have the same fecundity, but only half of two-year old individuals and all of three-year old individuals breed, then fecundity of the two-year-old age class will be half of the fecundity of the three-year-old age class.

The fecundities must include the proportion of breeding individuals, regardless of the type of sex structure or the mating system. In RAMAS Metapop, when females and males are modeled in separate stages (i.e., when the model includes sex structure), the proportion of breeders is entered in the Stages dialog. However, the parameter in the Stages dialog is used only to determine the limiting sex. If less than 100% of individuals breed, then this must be reflected in the stage matrix, as described above (even if the model includes sex structure, and the proportion of breeders have been entered in the Stages dialog).

Not incorporating sex ratio

Sex ratio is the proportion of females in a population or in an age class. Sex ratio at birth is the proportion of daughters among the offspring. In a matrix model, the sex ratio must be incorporated into the estimation of fecundities.

If you are modeling only the female population, the fecundities should be in terms of "number of daughters per female", and the initial abundances should be the number of females in each age class.

If you are modeling all individuals, the fecundities should be in terms of "offspring per individual", and the initial abundances should be the number of all individuals in each age class.

If you are modeling females and males separately, fecundities are entered in two different elements of the stage matrix, as "number of daughters per female" and "number of sons per female" (assuming a monogamous or a polygynous mating system). For more information, see "Modeling sex structure" in the RAMAS Metapop help file.

Extra stage

Some matrix models include a stage that should not be included in the matrix. One example of this mistake is including a "seed" stage in a plant model, even though seeds do not need to "wait" for a whole time step (e.g., for 1 year) before they germinate. An example of a similar mistake in a model of an animal population might be an "egg" stage.

For an example of this common mistake, consider the following model. Flowering plants produce 50 seeds/plant per year, of which 20% germinate to become "seedling" in the next time step. There is no seed bank, so any seed that does not germinate by the next census dies. We also assume that the census is pre-reproductive. In this case, the first row of the correct matrix would have the following form (ignore the other rows).

Correct Seedling Sapling Flowering plant
Seedling 0 0 10.0
Sapling S1 S3 0
Flowering plant S2 S4 S5

In this case, seeds are not included in the model, because in a pre-reproductive census, seeds would not be counted. If this matrix were parameterized according to post-productive census, it would be correct to name the first stage"seed", but in that case, the fecundity value must include the survival rate to the flowering stage.

However, some models incorrectly include a "seed" stage:

WRONG! Seed Seedling Sapling Flowering plant
Seed 0 0 0 50
Seedling 0.2 0 0 0
Sapling 0 S1 S3 0
Flowering plant 0 S2 S4 S5

This model is wrong (as a model of the same species as described above), because it introduces an extra time step between seed production and germination. It would also be wrong under post-productive census. For an example of this mistake, see Werner and Caswell (1977), and the discussion of the corrected model in Caswell (2001, p. 60-62, figs 4.3, 4.4).

As mentioned above, a similar mistake that is common in models of animal populations is including stages such as "egg" and "hatchling". This is a mistake if an egg does not take one time step to hatch. In general, the life cycle of the organism, in conjuction with the time step (projection interval) of the model determine the stages. Just because a phase in the life cycle of a species is important does not mean that that phase will always be represented with a stage in the matrix. In most cases, if the projection interval is longer than the time individuals spend in that phase of their life cycles, that phase will not be represented as a separate stage in the matrix model.

Case with seed bank: Now consider a similar model of a species with seed bank.Flowering plants produce 50 seeds/plant per year, of which 20% germinate to become "seedling" in the next time step. 30% of the seeds remain in the seed bank, of which only 5% germinate in the following year, and the rest die (i.e., no seed remains in the seed bank longer than a year). In this case, the first 2 rows of the correct matrix would have the following form:

Correct Seed bank Seedling Sapling Flowering plant
Seed bank 0 0 0 15.0
Seedling 0.05 0 0 10.0
Sampling 0 S1 S3 0
Flowering plant 0 S2 S4 S5

Fecundity of first age class or stage

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.

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.

Bias in survival rate estimation

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 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).

Uncertainty in survival rate estimation

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:

  1. 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.
  2. 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).

Using survivorship instead of survival rate

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).

Survival rates in the diagonal

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.

Ignoring constraints

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.

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.

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.

Too many (or too few) age classes or stages

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).

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).

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).

Modeling only females

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.

The stage matrix of such a model may look like the figure on the left. In this example, there are 3 age classes of females, followed by 3 age classes of males. There are 2 sets of age-specific fecundities. One row of fecundities (highlighted yellow) represents female offspring (daughters per female), and the other row (highlighted blue) represents male offspring (sons per female). The upper left quadrant also includes female survival rates; this quadrant is also the 3x3 matrix of the female-only version of this model. The lower right quadrant includes male survival rates. To complete this model, you must also specify the mating system, and the degree of polygamy (see the program manual or the help file for details).

Ignoring genetics

Most population models do not incorporate genetic factors (such as inbreeding depression). In many cases, this is justified, because data on genetic effects are rarely available, and incorporating such effects usually makes the model too complex for the available data. However, genetic factors may be important, for some populations. Inbreeding depression, for example, may strongly affect populations of naturally outbreeding diploid species, if they are not rapidly declining, if they exhibit large variations in population size, and if the simulated time horizon is long (Brook et al. 2002).

If you do not have sufficient data to model inbreeding depression, consider estimating risk of decline (rather than total extinction). If you do have sufficient data, read about modeling inbreeding depression in the program manual or the help file (RAMAS Metapop/GIS version 4 or later)..

List of common mistakes in population modeling

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