Program LIFETIME: explanations and documentation

The R-script LIFETIME estimates a population's lifetime from the population's demographic parameters.

 

Introduction and installation

The R-script does not require any previous knowledge of R, but presupposes that R has been installed on your computer. A point-by-point instruction follows here:

·    R is an open and free programming language and environment. It can be downloaded from http://www.r-project.org. Follow the installation instructions at that site to install the package.

·    After you have installed and started R, the lifetime script can be loaded in one of two ways:

·     write load(url("http://www.evol.no/hanno/12/lifetime.rtx")) directly in your R pane (this requires your computer to be online); or

·     use your browser to navigate to http://www.evol.no/hanno/12/lifetime.rtx and save this file to your harddisk; later, write load("...") in your R pane, where "..." specifies the file location [for example, load("c:/aliens/lifetime.rtx"); this requires your computer to be online only when dowloading the file for the first time, whereupon it can be loaded locally from your computer].

·    Now you can run the script by writing lifetime(...), where "..." represents the parameters, which are explained in detail below. Example:
   lifetime(N0=200, lambda=1.02, demvar=0.75, envvar=0.1, C=10)
Not all parameters are required (see below for details), e.g.:
   lifetime(N0=50, lambda=0.96, envvar=0.01)
Parameter names can be omitted if they are provided in the precise order N0/lambda/demvar/envvar/C:
   lifetime(1000, 1.05, 0.9)
is thus identical to
   lifetime(N0=1000, lambda=1.05, demvar=0.9)
Some more parameters can be used if desired (r, K, ndis, pdis, theta, varK, rho), although this may not often be needed.

Please note that this R-script is not part of any R package. Therefore, no R help will be available for this function. Please refer to this site instead.

 

Parameters

The function lifetime has the following parameters:

N0            Current population size (N0). This figure constitutes the basis for estimates of future population sizes.

·    Definition: The number of individuals that currently make up the population.

·    Significance: The larger N0, the larger is the expected population lifetime (see fig. 1).

·    Default: none. The parameter must be specified with a number ≥ 1.

lambda  The annual multiplicative population growth rate (λ). If r is provided, lambda can be omitted.

·    Definition: λ = Nt+1 / Nt, where t is time (year).

·    Significance: The larger λ, the larger is the expected population lifetime. The parameter has a huge effect on the result (see fig. 2).

·    Default: none. Either lambda or r should be specified. Otherwise, lambda is assumed to be 1 (no increase, no reduction), which may be far from correct.

demvar  Demographic variance (). This is a measure of the extent of demographic stochasticity.

·    Definition: Demographic variance is the variance in fitness w (number of surviving offspring per individual) in the population, i.e. , where μ is the population's mean fitness.

·    Significance: The larger , the smaller is the expected population lifetime. At large population sizes, the parameter has minor effects on the result (see fig. 3).

·    Default: Exact knowledge of alien species' demographic variance will rarely be available from their novel environments. If the parameter is omitted, it is set to 0.5. This is a realistic assumption for several vertebrates, but can be strongly misleading for other taxa. It should be attempted to specifiy a realistic value at least for the larger taxon to which the species belongs.

envvar  Environmental variance (). This is a measure of the extent of environmental stochasticity.

·    Definition: , where   is the variance of the multiplicative population growth rate.

·    Significance: The larger , the smaller is the expected population lifetime. The parameter can have major effects on the result (see fig. 4).

·    Default: The estimation of environmental variance requires several years of population estimates for the species in question. This information will often be unavailable for alien species. If the parameter is omitted, it will be set to 0.05. This is a realistic assumption for several vertebrates, but can be strongly misleading for other taxa. It should be attempted to specifiy a realistic value at least for the larger taxon to which the species belongs. If environmental variance is mainly caused by relatively rare "disasters" characterised by unusually high mortality, an alternative way of specifying environmental variance is by using the parameters pdis and ndis (see below).

C               Quasi-extinction threshold (C). The population size at which the species is regarded extinct. Under most circumstances, it is realistic to assume that this threshold is larger than zero.

·    Definition: The largest number of individuals that is sufficiently small to effectively prevent reproduction (e.g., because the inability to find potential mates).

·    Significance: The larger C, the smaller is the expected population lifetime. It is sufficient to provide the correct order of magnitude (see fig. 5).

·    Default: If the parameter is omitted, it will be set to 10. Potentially, it can be both lower (down to 1, e.g. in species with vegetative reproduction) and larger (e.g. under strong Allee effects).

r               The intrinsic population growth rate (r). If lambda is specified, r will be ignored.

·    Definition: r = lnλ.

·    Significance: The larger r, the larger is the expected population lifetime. The parameter has a huge effect on the result (cf. fig. 2).

·    Default: none. See lambda.

K               Carrying capacity (K).

·    Definition: The population size at which density regulation balances the growth rate.

·    Significance: The effect of K depends on the growth rate. If λ > 1 (r > 0), the larger K, the larger is the expected population lifetime. If K is assumed to be much larger than N0, the precise value of K has negligible effects on the result (see fig. 6). If, however, λ < 1 (r < 0), the larger K, the smaller is the expected population lifetime.

·    Default: Exact knowledge of alien species' carrying capacity will rarely be available from their novel environments. If the parameter is omitted, it is set to 100 times N0. For most purposes, this is a sufficient approximation. Please note that a negative population trend can be caused by K < N0, and not only by λ < 1 (r < 0). The script can handle both cases. However, please avoid specifying K < N0 and λ < 1 at the same time (this would result in meaningless estimates).

ndis       Number of disasters (ndis). If ndis and pdis are provided, envvar is ignored.

·    Definition: The mean or expected number of "disaster years" within a 50-year period, where disaster year is defined as a year with unusually high mortality that is caused by an environmental factor common to the entire or large parts of the population.

·    Significance: The larger ndis, the smaller is the expected population lifetime.

·    Default: none. The figure must be ≥ 1 (one disaster per 50 years) and < 50 (one disaster each year). Disasters, defined as episodes during which an unusually large proportion of the population dies through a common environmental effect (such as frosts, fires, droughts), are an alternative way of specifying environmental variance. It is based on the assumption that environmental variance is in its entirety caused by such disasters. Note that the population growth rate (lambda or r) must be specified for an average non-disaster year.

pdis       Extent of disasters (pdis). If pdis and ndis are provided, envvar is ignored.

·    Definition: The proportion of the population dying in a disaster year over and above normal mortality.

·    Signficance: The larger pdis, the smaller is the expected population lifetime.

·    Default: none. The value must be a fraction larger than 0 (no additional individuals dies during a disaster year than during a normal year) and less than 1 (all individuals die during a disaster year). See ndis for further explanations.

 

Further parameters are available, although they may rarely be needed. The ones that are implemented thus far are:

·    quiet (suppresses messages and warnings if it is TRUE; defaults to FALSE),

·    theta (the value used for θ in theta-logistic population models; the default is theta=1, which assumes logistic population dynamics; let theta=0 for density dependence that follows the Gompertz law),

·    varK (the variance of carrying capacity on a logarithmic scale, ; defaults to 0),

·    rho (the environmental correlation ρ between growth rate and environmental noise; defaults to 0),

·    kontroll (a list of control parameters that can have the following entries:
tol = number specifying the relative error tolerance during integration;
sub = maximum number of subdivisions during integration;
kor = number specifying the weighting of estimation error during integration;
det = logical variable indicating whether detailed error messages should be displayed;
tra = logical variable indicating whether parameters should be transformed;
def and int are reserved for internal control routines).

 

Output

The output of the function is a number, representing the population's median lifetime in years. Some additional information is displayed on the screen only (provided you didn't specify quiet=TRUE), viz. conversions of the result into different equivalent quantities (extinction risk, expected lifetime and median lifetime).

 

Estimation may take several minutes (especially when the population size is high). If you get unexpectedly high or low estimates, this may be due to convergence problems during estimation. If you suspect that this is the case, or if the program produces error messages, this may be a bug, and you are welcome to send me the input parameters that produced them (although I cannot guarantee that I will be able to help you).

 

Illustrations

The following figures are meant to illustrate how sensitive the estimate of expected lifetime is to variation in the different parameters. (For the time being, the figures are in Norwegian.)

 

 

Figure 1: Expected lifetime increases with increasing population size (shown for three different population growth rates, λ = 0.98, 1.00, 1.02; other parameters: demvar = 0.5, milvar = 0.05, C = 10, K = 100 * N0). Red lines indicate the threshold values of criterion A. Note that the x axis is on logarithmic scale (base of 10, i.e. 9109).

 

 

Figure 2: Expected lifetime increases with increasing population growth rate λ (shown for three different population sizes, N0 = 20, 100, 20000; other parameters: demvar = 0.5, milvar = 0.05, C = 10, K = 100 * N0). Red lines indicate the threshold values of criterion A. Note that the y axis is on logarithmic scale (base of 10, i.e. 6one million).

 

 

Figure 3: Expected lifetime decreases with increasing demographic variance (shown for three different population growth rates, λ = 0.98, 1.00, 1.02; other parameters: N0 = 100, milvar = 0.05, C = 10, K = 10 000). Red lines indicate the threshold values of criterion A.

 

 

Figure 4: Expected lifetime decreases with increasing environmental variance (shown for three different population growth rates, λ = 0.98, 1.00, 1.02; other parameters: N0 = 100, demvar = 0.5, C = 10, K = 10 000). Red lines indicate the threshold values of criterion A. Note that the y axis is on logarithmic scale (base of 10, i.e. 6one million).

 

 

Figure 5: Expected lifetime decreases with increasing quasi-extinction threshold (shown for three different population sizes, N0 = 100, 500, 2500; other parameters: lambda = 1.01, demvar = 0.5, milvar = 0.05, K = 100 * N0). Red lines indicate the threshold values of criterion A.

 

 

Figure 6: Expected lifetime increases with increasing carrying capacity (shown for three different population sizes, N0 = 50, 500, 5000; other parameters: lambda = 1.01, demvar = 0.5, milvar = 0.05, C = 10). If carrying capacity can be assumed to be at least 100 times population size (arrow heads), its precise value is of negligible significance. The dots mark situations where K = N0. Red lines indicate the threshold values of criterion A. Note that the x axis is on logarithmic scale (base of 10, i.e. 8100 millions).

 

Documentation and definitions

A population's expected lifetime is the mean estimated time to extinction (averaged over the probability distribition of times to extinction), taking demographic and environmental stochasticity into account. It is estimated using the following equations:

,(1)

where

,(2)

,(3)

.(4)

See Leigh (J Theor Biol 90:213, 1981) for the derivations of these equations. For their application, see Lande, Engen & Sæther (Stochastic population dynamics in ecology and conservation, Oxford 2003, s. 38–40).

 

If environmental variance can be assumed to due to the occurrence of disaster years alone, growth rate and environmental variance can be estimated using the following equations (based on a 50-year period and with n := ndis; p := pdis; λ := mean growth rate in a non-disaster year):

,(5)

.(6)

Derivation of equation 6 (where μ is the arithmetic mean annual growth rate, i.e. ):

 

About the program

The R-script LIFETIME has been written by Hanno Sandvik at the Centre for Biodiversity Dynamics (CBD), Norwegian University of Science and Technology (NTNU).

Its present version number is 1.6 (April 2017).

In case of questions or comments, please contact Hanno Sandvik.