Frequently Asked Questions:
RAMAS Age

Note: Page numbers refer only to single-user manuals in tan binders; the site license manuals have different paginations.

Q. How do I print the graph of a result?

(1) Use cursor keys and press enter to view of the results as a graph

(2) If you have not yet printed any graphs:
      (a) press Escape (you will see a numerical table of the results)
      (b) press F7 (the program will display the name of a printer)
      (c) if your printer is not this, press F8 (you'll see a list of printers)
      (d) select a printer by typing its number
      (e) the program will now display the graph again (if not, select and view the graph, as in 1 above)
(3) Press F7 and wait.

Note: you need to do the steps in (2) if you are printing for the first time, or if you have changed your printer.

Q. How do I estimate fecundities?

        The age-specific fecundity values in screen 2 are those that go into the first row of a Leslie matrix. A common mistake is to use "maternities" (i.e., estimates of number of offspring per individual, such as clutch size, litter size, belly counts, etc.) directly as estimates of fecundity. To understand this problem, first read the last part of the section "How to estimate vital rates" in Chapter 3 of the User Manual (pages 31-32 in single-user binders).

        In a Leslie matrix, fecundity values incorporate two kinds of mortality over the time step. Some of the mothers that were alive during the last census die before reproducing, and some of the babies that are born die before they can be counted in the next census. The elements of the top row of the Leslie matrix must take all this into account. They are those values that, when multiplied by the population's current abundance vector, yield the number of newborns that live to be censused at the next time step.

        If you use a statistical approach to estimate the fecundity vector (using a multiple regression of the number in the zeroth age class in each year on the numbers in all the other age classes in the previous years), then you don't have to worry much about this particular mistake. If you can census all age classes, we strongly recommend this method. However, this may not always be possible. In some cases, younger age classes may be more difficult to census because of their smaller size. In other cases, the census method may work only for the breeding population (e.g., only territorial owls respond to calls by surveyors; juvenile salmon disperse to the ocean and cannot be censused before they return to the rivers to breed). In such cases, survival rates and fecundities may need to be estimated separately. Survival rates, for example, can be estimated by marking a large number of individuals and following each of them as they age. Another common method for estimating survival rates is a mark-recapture study.

        Fecundities may be based on measures such as number of chicks fledged per nest, average litter size, etc. If you have such measures (which we call maternities), you must modify these values to use in a Leslie matrix. How you do this depends on the scheduling of census in relation to mortality and reproduction, and on the definition of age of an individual. For more information, see discussions by Jenkins (1988), Caswell (1989, pages 8-15), Burgman et al. (1993, pages 127-129 and 140-142), and Akçakaya et al (1996).

Q. How do I use the sex ratio?

        When making population projections, RAMAS Age multiplies the fecundities in screen 2 with the sex ratio parameter specified in screen 1. See the discussion in section "2. Initial abundance, ..." of Chapter 7 (pages 76-77 in single-user binders), and in the "Population projection" section in Chapter 8 (pages 92-93). See also the matrix in the section on "Summary statistics" in Chapter 8 (page 100).

        If you are modeling only the female population, set sex ratio to 1.0. In this case the fecundities should be in terms of "number of daughters per female" (but also see the discussion on fecundity above), and the initial abundances should be the number of females in each age class. If you are modeling both sexes, the sex ratio should be the proportion of females (i.e., 0.5 for equal number of males and females), the fecundities should be in terms of "offspring (of both sexes) per female", and the initial abundances should be the number of all individuals (male and female) in each age class.

Q. How do I estimate parameters for density dependence?

        Density dependence in RAMAS Age determines the number of individuals entering the population (recruits; denoted by Z in the equations in the manual) as a function of total reproductive effort (E in the equations in the manual). Before attempting to use density dependence in any model, carefully read Chapter 4 on density dependence, section "3. Density dependence" in Chapter 7 (pages 78-82 in single-user binders), and the section on "Population projection" in Chapter 8 (pages 92-94).

        Note especially that most studies on density dependence are made with a scalar model in mind, whereas the implementation in RAMAS Age is based on an age-structured model (see section on "Interaction with age structure" in Chapter 4, pages 52-54). Parameters based on a scalar model have different meanings from those used in RAMAS Age. For example in screen 3, the parameter K for the logistic density dependence in RAMAS Age is not carrying capacity or the equilibrium population size, unless you have a model with a single age class (i.e., a scalar model). A common mistake is to use parameters based on scalar models directly in an age-structured model.

        For example, the parameter K of the logistic function in screen 3 is the equilibrium number of offspring entering the first age class (recruits at equilibrium, Z*). The other parameter of logistic function (r) is not the population growth rate; rather, it relates to the proportion of potential offspring that enter the population at low population densities; approximately 1+r of the potential offspring become recruits. The function assumes that this proportion decreases as density increases, and becomes zero if E(K+K/r). See the figure in Chapter 4 (page 40). Note that even though it seems that 1+r (the maximum proportion of potential offspring that become recruits) should be less than or equal to 1.0, this is not always the case. Actually, in logistic function r cannot be negative, and r=0 is equivalent to density independence. Thus for this particular function, 1+r must be more than 1.0; i.e., at low densities the number of recruits must exceed the number of potential offspring. This is what is called "recruitment inflation" on page 39. The meaning of "recruitment inflation" is that fecundities increase above the values entered in screen 2 when abundance is low. This makes sense when the fecundity values you entered in screen 2 are estimated for a population close to its carrying capacity. In such a case, you would expect fecundities to be higher than the specified values when the abundance falls below the carrying capacity. Thus "recruitment inflation" would be exactly what you want.

        Below we discuss a simple numerical example for such a case. This example refers to a special case in which you make the following three assumptions: (1) fecundities are measured in a population close to its carrying capacity, (2) you have an estimate for number of recruits at equilibrium (Z*), and (3) you have an estimate of how much higher fecundities would be when population has a low enough density to escape the overcrowding effects of density dependence (but it is not influenced by any Allee effects).

        For a numerical example, assume that you estimated (or guessed) that the number of recruits at equilibrium (Z*) is 650, and when the population density is low, fecundities should be about 40% higher than the values in screen 2 (which are measured in a population close to its carrying capacity). In such a case, parameters of the three density dependence functions would be estimated as follows.

Logistic:  The parameter r would be 0.4 (because 1+r is assumed to be 1.4, as the recruitment at low densities is 40% higher than the recruitment at equilibrium). The parameter K would be 650, which is the number of potential offspring (E*) or number of recruits at equilibrium (Z*). The two are the same number at equilibrium.

Ricker:  The proportion of potential offspring that enter the population at low population densities (which was 1+r for logistic) is equal to the parameter alpha. For the above example, alpha = 1.4.

        The equilibrium number of potential offspring or equilibrium number of recruits (Z*) in the Ricker function is equal to Z* = ln( alpha ) / beta (natural logarithm of alpha divided by beta), so beta can be estimated as beta = ln (alpha) / Z*, where Z* is the equilibrium number of recruits at equilibrium. For the above example, beta = 0.00052.

Beverton-Holt:  In this case, the proportion of potential offspring that enter the population at low population densities (which was 1+r for logistic) is equal to 1/k (note that the parameter k of the Beverton-Holt function has nothing to do with the parameter K of the logistic). For the above example, k = 1/1.4 = 0.714.

        The equilibrium number of potential offspring or equilibrium number of recruits (Z*) in the Beverton-Holt function is equal to Z* = (1-k)/rho, where rho is the other parameter for this function. So, rho can be estimated as rho = (1-k) / Z*, where Z* is the number of recruits at equilibrium. For the above example, rho = (1-0.714) / 650 = 0.00044.

        Note that the number of recruits at equilibrium (Z*) is not the carrying capacity or the total equilibrium abundance. If you want to use the density dependence functions by specifying a carrying capacity or ceiling for the total abundance, you might consider using RAMAS Metapop, in which all density dependence functions have the same set of parameters, including the carrying capacity or ceiling (Akcakaya 1994, pp.43-61).

Q. How do I estimate the coefficient of variation (CV)?

        The coefficient of variation (CV) parameters in screen 4 are proportions, not percentages; a value of 1.0 means that standard deviation is equal to the mean. See section "4. Stochastic parameters ..." in Chapter 7 (pages 82-83 in single-user binders). Remember that these parameters refer to environmental stochasticity, i.e., temporal variation in vital rates. Do not use estimates of measurement error or spatial variation from a single time step. A common mistake is to base the CV estimates on the standard error of mean, rather than on standard deviation of the series of estimates of the vital rate.

Q. Which distributions should I use?

        If you guessed the values of coefficients of variation in screen 4, the results may be biased because of truncation. Selecting a lognormal distribution instead of a normal distribution may be helpful. Read the section "4. Stochastic parameters ..." in Chapter 7 (pages 83-84 in single-user binders).

        For example, if zero-year old survival rate is 0.1, with a standard deviation of 0.1, the CV for zero-year old survival rate would be 1.0. If you specify a normal distribution, about 16% of the distribution would be negative, and truncated to 0.0. This truncation would make the realized values of zero-year old survival rates higher than 0.1, and their CV lower than 1.0, resulting in an underestimation of extinction risks. In such a case (low survival rate or fecundity with a high CV), the use of a lognormal distribution avoids the bias resulting from truncation.

        A similar problem may occur if you have a survival rate close to 1.0 with a high CV. For example, if you have adult survival rate of 0.8 with a CV of 0.25 (i.e., a standard deviation of 0.2), then about 16% of the distribution would be above one, and truncated to 1.0. This would make the realized values lower than 0.8, and the CV lower than 0.25, with unpredictable effect on extinction risks. At this time there is no solution to such a case (lognormal would not work either). In RAMAS Metapop, the lognormal distribution is modified to handle such cases as well, by sampling the mortality rates (1-S) instead of survival rates (S) from a lognormal distribution (Akcakaya 1994, pp.66-69).

Q. How do I specify correlations?

        The correlation values in screen 4 should be estimated from time series of survival rates, fecundities and migrations; they refer to correlations among these variables over time, not over individuals. If you don't have such data, you should think what kind of assumption you are implicitly making. In most cases, an assumption of complete correlations (all values of the correlation matrix in screen 4 equal to one) may be more reasonable than an assumption of no (zero) correlation (e.g., because a "good" year for adults is more likely to be "good" than "bad" for zero-year olds). The two assumptions result in different extinction risks: complete correlation results in a higher extinction risk than no correlation. Also see section "4. Stochastic parameters ..." in Chapter 7 (pages 84-85 in single-user binders).

Q. Should I use demographic stochasticity?

        YES! ... unless one of the following applies to your model:

• you are modeling densities (such as number of animals per km² ) instead of absolute numbers of individuals,
• you think that the coefficient of variation estimates you are using incorporate the effects of demographic variability, because of the small sample sizes used in their estimation (and the model never predicts abundances in any age class lower than the sample sizes you used in estimating CVs),
• you have a very slow computer and your model never predicts abundances (for any age class, at any time step, in any replication) of less than 100.

Q. How many replications should I use?

        Use the maximum number of replications (500 in the last version), unless you are running a test simulation or making a demonstration. A common mistake is to use too few replications. With 500 replications, the risk curves have 95% confidence interval of about ± 0.04 (based on Kolmogorov-Smirnov test; see Sokal and Rohlf, page 721). With 50 replications, the confidence interval is ± 0.13, i.e., if a model predicts zero risk of extinction, the actual risk may be as high as 13%! If you think you need to use more than 500 replications, consider using RAMAS Stage or RAMAS Metapop.

Q. How many time steps (duration) should I use?

        This depends on the reliability of your data, the longevity of the species you are modeling, the duration of one time step (often 1 year, but may be any time interval), and the questions you are addressing. Don't attempt to predict population dynamics for longer time horizons than can be justified by the available data. In very few cases can use of 50-100 year simulations make sense. RAMAS/age can predict only 50 time steps. If you have a compelling reason to model time horizons longer than 50 years, you can either model multiple-year time steps (see below), or use RAMAS Stage or RAMAS Metapop.

Q. How many age classes should I use?

        The maximum number of age classes in RAMAS Age is 18 (age 0 to age 17). For species that live longer than 18 years, there is usually limited information on survival rate and fecundity for older ages, or the available information suggests that these demographic rates do not change much with age after a certain age. In this case, you can use the last age class as a composite age class. Specifying a non-zero survival rate for the last age class describes a life history with infinite number of age classes (of course, the number of survivors at older ages included in the composite age class quickly decline to less than 1.0 for any reasonable survival rate). The sample file HAWK is such an example (see the section "Extensions to Leslie matrix" and the box on pages 21-22).

        Another way to get around the limitation is to define age classes with time intervals of more than 1 year (for example, the standard practice in human demography is to define 5-year intervals). This can be done when the demographic rates do not change very much from one year to the next, and it is especially useful since in most cases the available data will not allow the estimation of more than twenty or so survival rates with any statistical precision. The GORILLA sample file, for example, defines an "age" as a 5-year interval. This also makes it possible to model time horizons longer than 50 years; in this example 50 time steps correspond to 250 years.

        The third solution is to use RAMAS Stage, and define as many age classes (or stages) as the memory of your computer allows.

Q. How do I interpret trajectory summary (screens 5 and 7)?

        Carefully! The average line is often not representative for two reasons: (1) we know that the population will not behave like the mean trajectory in the future (just as none of the replicated trajectories displayed during the simulation looked like it), and (2) in most cases the population abundances have a skewed distribution, so an arithmetic average doesn't make sense. For the same reason, the ± one standard deviation bars usually don't make sense (they might even go below the minima or above the maxima!). The remaining information on this screen (maxima and minima) are more effectively presented in the risk screens. So, if it doesn't make sense, why is there such a screen in the first place, you might ask. We might say that it is used to demonstrate interesting deterministic dynamics such as cycles and chaos (which is true), but the real reason is user demand: everyone asks for it!

References

Akçakaya, H.R. 1994.RAMAS Metapop: Viability Analysis for Stage-structured Metapopulations (version 1.0). Applied Biomathematics, Setauket, New York.

Akçakaya, H.R., M.A. Burgman, and L. Ginzburg. 1996. Applied Population Ecology: principles and computer exercises with RAMAS EcoLab.

Burgman, M.A., S. Ferson and H. R. Akçakaya. 1993. Risk Assessment in Conservation Biology. Chapman and Hall, London.

Caswell, H. 1989. Matrix Population Models: Construction, Analysis, and Interpretation. Sinauer Associates, Sunderland, Massachusetts.

Jenkins, S.H. 1988. Use and abuse of demographic models of population growth. Bulletin of the Ecological Society of America 69: 201-202.

Sokal, R.R. and F.J. Rohlf. 1981. Biometry (second edition). W.H. Freeman and Company, New York.

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