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A method for simulating demographic stochasticity
Ecological Modelling~ 54 (1991) 133-136
133
Elsevier Science Publishers B.V., Amsterdam
Short Communication
A method for simulating demographic stochasticity H. Re§it Ak~akaya Department of Ecology and Evolution, State University of New York, Stony Brook, NY 11794, USA
Life-history parameters (e.g. survivorship) of a population or cohort are usually expressed as probabilities, such as the probability that an individual will survive to the next year. If the population size is large, the observed frequency of survivors will approximate this parameter, although each individual can either survive or die. For small population sizes, the fact that there is a finite, integer number of individuals introduces a variation to the parameter estimates. This variation is called demographic stochasticity. The underlying value of the survival probability can itself vary as a function of environmental conditions. This variation caused by uncertainties in environmental factors is called environmental stochasticity. Environmental stochasticity is usually modeled by randomly varying the parameters in a deterministic model, such as the Leslie matrix model of age-structured populations. When all parameters (survivorships and fecundities) of a Leslie model vary randomly with given variances and covariances, the analytical solution of the model is difficult, and hence simulations are used (Boyce, 1977; Ginzburg et al., 1982). Demographic stochasticity can be modeled as a birth-and-death process (Bartlett, 1960; MacArthur and Wilson, 1967; Richter-Dyn and Goel, 1972; Goodman, 1987), but this formulation does not incorporate age-structure. Shaffer (1983; Shaffer and Samson 1985) developed a model that simulates the dynamics of an age-structured population under both environmental and demographic stochasticity. He simulated environmental stochasticity through variation in population parameters, and demographic stochasticity through the explicit simulation of individual organisms. His model followed t Present address: Applied Biomathematics, 100 North Country Road, Setauket, NY 11733, U.S.A. 0304-3800/91/$03.50
© 1991 - Elsevier Science Publishers B.V.
134
rt.R. AKqAKAYA
each individual separately, at each time step deciding on its survival and reproduction according to the probabilities given by population parameters. This method requires very long simulations, especially when the population size is large. Since environmental and demographic stochasticities as defined above and as simulated by Shaffer are independent, an alternative method of simulating both types of stochasticity is to make a simulation using the stochastic Leslie matrix approach and sample the number of survivors from a binomial distribution. This approach is computationally much faster, and easier to implement on a computer. The results of this study demonstrate that this approach approximates the simulation of each individual very well. In the proposed method, the variance of each parameter will have two components where the environmental component is simply the variance of vital rates under environmental variation. For the demographic component, consider a population of x-year old individuals. If the number of individuals in this population at time t is k and probability of survival from age x to x + 1 is p, the expected number of (x + 1)-year-old individuals at time t + 1 will be pk. Since each of these individuals has an independent probability p of surviving to age x + 1, there is a probability pk that all k individuals will survive to become x + 1 years old (in which case the observed survivorship will be one). Similarly there is a probability (1 - p ) k that none of them will survive (and the observed survivorship will be zero). Clearly this process will result in a binomial distribution of the number of x + 1 year old individuals. In order to demonstrate that this result holds in an age-structured population, I made a simulation similar to Shaffer's. I used data on the population of Dali's sheep (Ovis dalli dalli) reported by Simmons et al. (1984). I started each iteration of the simulation with the same age structure, and decided on the survival and reproduction of each individual according to age-specific vital rates. I recorded the number of individuals and the observed survivals and fecundities for each age class after this one year transition. After 1000 iterations of this transition, the distribution of number of survivors was very closely approximated by a binomial distribution. For example, in the initial age distribution there were ten 7-year-old individuals. After one year of simulation the average number of 8-year-olds in this cohort was about 8.2, as expected from the survivorship rate of 0.82 for this age class, but the observed survivorships ranged from 0.4 (four individuals out of 10) to 1.0, and the distribution followed the expected binomial distribution. The two distributions were virtually identical (goodness of fit test, G = 3.84 with 10 d.f.; P > 0.90) (Fig. 1). In simulating the growth of populations, it is necessary to take into account the combined effects of environmental and demographic stochasticity. Environmental stochasticity can be simulated by sampling the mean value of survivorship from a distribution with observed mean and variance,
135
SIMULATING DEMOGRAPHIC STOCHASTICITY 0.4
0.35
0.3
>. ¢J z bJ "~ CY
0.25
0.2
~J
0.15
0,1
0.05
0 0.3 177]
0.4
OBSERVED
0.5
0.6
0.7
SURVIVORSHIP ~ EXPECTED
0.8
0.9
1
(Binomial)
Fig. 1. Distribution of survivorships of 7-year-olds. Bars at left show the observed distribution after 1000 iterations; bars at right show the expected binomial distribution with parameters p = 0.82 and k -- 10.
as proposed by Ginzburg et al. (1982) and Shaffer and Samson (1985). After obtaining the mean survivorship for a particular time period, the number of survivors may be sampled from a binomial distribution with the given mean and the given initial number of individuals. This method is used to simulate demographic stochasticity in a model of age-structured populations (Ferson and Ak~akaya, 1990) and a model for spatially structured metapopulations (Ak~akaya and Ferson, 1990). Note that the binomial distribution is based on the presence of two states. In the case of survivorships, these are survival or death, since a given individual either survives or dies in a given time period. For fecundities, the two states may be assumed to be producing one offspring or none. If the fecundity of an individual can be greater than one, the fecundities in the population will not be distributed binomially. In these cases more information is needed on the individual (phenotypic) variation in fecundity to approximate the distribution caused by demographic stochasticity, which in some cases may be approximated by a Poisson distribution. If average fecundity is high and the variation in fecundity is largely due to environmental fluctuations rather than individual differences, demographic stochasticity
136
H.g.AK~AKAYA
will have a smaller effect on the distribution of fecundities than environmental stochasticity. The results of this study demonstrate that demographic stochasticity can be modeled by r a n d o m variates taken from a binomial distribution. This method makes it possible to combine the effects of environmental stochasticity (random variation in population parameters) with demographic stochasticity, and it is computationally much faster than the alternative m e t h o d of simulating each individual separately, which becomes prohibitively inefficient as the population size increases. REFERENCES Aksakaya, H.R. and Ferson, S., 1990. RAMAS/space user manual: Spatially-structured population models for conservation biology. Exeter Software, New York, 87 pp. Bartlett, M.S., 1960. Stochastic Population Models in Ecology and Epidemiology. Methuen, London, 90 pp. Boyce, M.S., 1977. Population growth with stochastic fluctuations in the life table. Theor. Popul. Biol., 12: 366-373. Ferson, S. and Ak~akaya, H.R., 1990. RAMAS/age user manual: Modeling fluctuations in age-structured populations. Exeter Software, New York, 143 pp. Ginzburg, L.R., Slobodkin, L.B., Johnson, K. and Bindman, A.G., 1982. Quasiextinction probabilities as a measure of impact on population growth. Risk Analysis 2: 171-181. Goodman, D., 1987. The demography of chance extinction. In: M.E. Soul6 (Editor), Viable Populations for Conservation. Cambridge University Press, Cambridge, pp. 11-34. MacArthur, R.H. and Wilson, E.O., 1967. The Theory of Island Biogeography. Princeton University Press, Princeton, NJ, 203 pp. Richter-Dyn, N. and Goel, N.S., 1972. On the extinction of a colonizing species. Theor. Popul. Biol., 3: 406-433. Shaffer, M.L., 1983. Determining minimum viable population sizes for the grizzly bear. Int. Conf. Bear Res. Manage., 5: 133-139. Shaffer, M.L. and Samson, F.B., 1985. Population size and extinction: a note on determining critical population sizes. Am. Nat., 125: 144-152. Simmons, N,M., Bayer, M.B. and Sinkey, L.O., 1984. Demography of Dalrs sheep in the Mackenzie mountains, Northwest Territories. J. Wildl. Manage., 48: 156-162.
133
Elsevier Science Publishers B.V., Amsterdam
Short Communication
A method for simulating demographic stochasticity H. Re§it Ak~akaya Department of Ecology and Evolution, State University of New York, Stony Brook, NY 11794, USA
Life-history parameters (e.g. survivorship) of a population or cohort are usually expressed as probabilities, such as the probability that an individual will survive to the next year. If the population size is large, the observed frequency of survivors will approximate this parameter, although each individual can either survive or die. For small population sizes, the fact that there is a finite, integer number of individuals introduces a variation to the parameter estimates. This variation is called demographic stochasticity. The underlying value of the survival probability can itself vary as a function of environmental conditions. This variation caused by uncertainties in environmental factors is called environmental stochasticity. Environmental stochasticity is usually modeled by randomly varying the parameters in a deterministic model, such as the Leslie matrix model of age-structured populations. When all parameters (survivorships and fecundities) of a Leslie model vary randomly with given variances and covariances, the analytical solution of the model is difficult, and hence simulations are used (Boyce, 1977; Ginzburg et al., 1982). Demographic stochasticity can be modeled as a birth-and-death process (Bartlett, 1960; MacArthur and Wilson, 1967; Richter-Dyn and Goel, 1972; Goodman, 1987), but this formulation does not incorporate age-structure. Shaffer (1983; Shaffer and Samson 1985) developed a model that simulates the dynamics of an age-structured population under both environmental and demographic stochasticity. He simulated environmental stochasticity through variation in population parameters, and demographic stochasticity through the explicit simulation of individual organisms. His model followed t Present address: Applied Biomathematics, 100 North Country Road, Setauket, NY 11733, U.S.A. 0304-3800/91/$03.50
© 1991 - Elsevier Science Publishers B.V.
134
rt.R. AKqAKAYA
each individual separately, at each time step deciding on its survival and reproduction according to the probabilities given by population parameters. This method requires very long simulations, especially when the population size is large. Since environmental and demographic stochasticities as defined above and as simulated by Shaffer are independent, an alternative method of simulating both types of stochasticity is to make a simulation using the stochastic Leslie matrix approach and sample the number of survivors from a binomial distribution. This approach is computationally much faster, and easier to implement on a computer. The results of this study demonstrate that this approach approximates the simulation of each individual very well. In the proposed method, the variance of each parameter will have two components where the environmental component is simply the variance of vital rates under environmental variation. For the demographic component, consider a population of x-year old individuals. If the number of individuals in this population at time t is k and probability of survival from age x to x + 1 is p, the expected number of (x + 1)-year-old individuals at time t + 1 will be pk. Since each of these individuals has an independent probability p of surviving to age x + 1, there is a probability pk that all k individuals will survive to become x + 1 years old (in which case the observed survivorship will be one). Similarly there is a probability (1 - p ) k that none of them will survive (and the observed survivorship will be zero). Clearly this process will result in a binomial distribution of the number of x + 1 year old individuals. In order to demonstrate that this result holds in an age-structured population, I made a simulation similar to Shaffer's. I used data on the population of Dali's sheep (Ovis dalli dalli) reported by Simmons et al. (1984). I started each iteration of the simulation with the same age structure, and decided on the survival and reproduction of each individual according to age-specific vital rates. I recorded the number of individuals and the observed survivals and fecundities for each age class after this one year transition. After 1000 iterations of this transition, the distribution of number of survivors was very closely approximated by a binomial distribution. For example, in the initial age distribution there were ten 7-year-old individuals. After one year of simulation the average number of 8-year-olds in this cohort was about 8.2, as expected from the survivorship rate of 0.82 for this age class, but the observed survivorships ranged from 0.4 (four individuals out of 10) to 1.0, and the distribution followed the expected binomial distribution. The two distributions were virtually identical (goodness of fit test, G = 3.84 with 10 d.f.; P > 0.90) (Fig. 1). In simulating the growth of populations, it is necessary to take into account the combined effects of environmental and demographic stochasticity. Environmental stochasticity can be simulated by sampling the mean value of survivorship from a distribution with observed mean and variance,
135
SIMULATING DEMOGRAPHIC STOCHASTICITY 0.4
0.35
0.3
>. ¢J z bJ "~ CY
0.25
0.2
~J
0.15
0,1
0.05
0 0.3 177]
0.4
OBSERVED
0.5
0.6
0.7
SURVIVORSHIP ~ EXPECTED
0.8
0.9
1
(Binomial)
Fig. 1. Distribution of survivorships of 7-year-olds. Bars at left show the observed distribution after 1000 iterations; bars at right show the expected binomial distribution with parameters p = 0.82 and k -- 10.
as proposed by Ginzburg et al. (1982) and Shaffer and Samson (1985). After obtaining the mean survivorship for a particular time period, the number of survivors may be sampled from a binomial distribution with the given mean and the given initial number of individuals. This method is used to simulate demographic stochasticity in a model of age-structured populations (Ferson and Ak~akaya, 1990) and a model for spatially structured metapopulations (Ak~akaya and Ferson, 1990). Note that the binomial distribution is based on the presence of two states. In the case of survivorships, these are survival or death, since a given individual either survives or dies in a given time period. For fecundities, the two states may be assumed to be producing one offspring or none. If the fecundity of an individual can be greater than one, the fecundities in the population will not be distributed binomially. In these cases more information is needed on the individual (phenotypic) variation in fecundity to approximate the distribution caused by demographic stochasticity, which in some cases may be approximated by a Poisson distribution. If average fecundity is high and the variation in fecundity is largely due to environmental fluctuations rather than individual differences, demographic stochasticity
136
H.g.AK~AKAYA
will have a smaller effect on the distribution of fecundities than environmental stochasticity. The results of this study demonstrate that demographic stochasticity can be modeled by r a n d o m variates taken from a binomial distribution. This method makes it possible to combine the effects of environmental stochasticity (random variation in population parameters) with demographic stochasticity, and it is computationally much faster than the alternative m e t h o d of simulating each individual separately, which becomes prohibitively inefficient as the population size increases. REFERENCES Aksakaya, H.R. and Ferson, S., 1990. RAMAS/space user manual: Spatially-structured population models for conservation biology. Exeter Software, New York, 87 pp. Bartlett, M.S., 1960. Stochastic Population Models in Ecology and Epidemiology. Methuen, London, 90 pp. Boyce, M.S., 1977. Population growth with stochastic fluctuations in the life table. Theor. Popul. Biol., 12: 366-373. Ferson, S. and Ak~akaya, H.R., 1990. RAMAS/age user manual: Modeling fluctuations in age-structured populations. Exeter Software, New York, 143 pp. Ginzburg, L.R., Slobodkin, L.B., Johnson, K. and Bindman, A.G., 1982. Quasiextinction probabilities as a measure of impact on population growth. Risk Analysis 2: 171-181. Goodman, D., 1987. The demography of chance extinction. In: M.E. Soul6 (Editor), Viable Populations for Conservation. Cambridge University Press, Cambridge, pp. 11-34. MacArthur, R.H. and Wilson, E.O., 1967. The Theory of Island Biogeography. Princeton University Press, Princeton, NJ, 203 pp. Richter-Dyn, N. and Goel, N.S., 1972. On the extinction of a colonizing species. Theor. Popul. Biol., 3: 406-433. Shaffer, M.L., 1983. Determining minimum viable population sizes for the grizzly bear. Int. Conf. Bear Res. Manage., 5: 133-139. Shaffer, M.L. and Samson, F.B., 1985. Population size and extinction: a note on determining critical population sizes. Am. Nat., 125: 144-152. Simmons, N,M., Bayer, M.B. and Sinkey, L.O., 1984. Demography of Dalrs sheep in the Mackenzie mountains, Northwest Territories. J. Wildl. Manage., 48: 156-162.