Deterministic models predict an exact outcome, given certain initial conditions. Examples: logistic and exponential growth models discussed previously in lab.
Stochastic models predict variable outcomes based on probabilities of occurrence. For example, growth rate (lambda) is no longer fixed, but is a random variable based on some probability distribution.
Demographic stochasticity - variation in demographic rates among individuals.
Demographic stochasticity often arises because of variation in sampling of individuals. An extremely skewed sex ratio also is an example of an unusual demographic event. This stochasticity occurs even in a constant environment.
In relatively small populations, demographic stochasticity can be an important factor influencing extinction probability.
As population size increases, random demographic differences get averaged over larger and larger number of independent individuals. Impact of demographic stochasticity decreases substantially when N>100.
Environmental stochasticity - usually refers to unpredictable events (e.g., changes in weather, food supply, or populations of competitors, predators, etc.) that affect all members of a population.
...therefore, mean values of r, mx, and Sx change randomly.
Large population sizes do less to buffer against extinction in this case, in contrast to demographic stochasticity.
Natural catastrophes - extreme cases of environmental uncertainty, such as hurricanes and large fires. Catastrophes are usually short in duration but widespread in their impact.
Genetic degradation – resulting from the founder effect, genetic drift, and/or inbreeding. As a general rule, genetic degradation and demographic stochasticity are important factors affecting the viability of only very small populations (e.g., <50 individuals). For example, inbreeding may result in a short-term decrease in vital rates (i.e., survival and birth rates) due to loss of genetic variation following population constriction.
Dennis et al. 1991 – developed a stochastic model of exponential growth based on the theory of age- or stage-structured population analysis (i.e., Wiener-drift model). The model is based on the assumption of a stable but variable mean instantaneous rate of change. Variance (the stochastic factor) and the mean instantaneous rate of change are calculated using maximum-likelihood-estimation (MLE) techniques. The model can be used to predict the likelihood of population persistence or conditional mean time to extinction.
STOCHMVP – a program developed by Dr. E. O. Garton at the University of Idaho. The program is based on the Dennis et al. (1991) stochastic-exponential model but also incorporates metapopulation theory. We will use program Stochmvp to estimate minimum viable population (MVP) size for a metapopulation and subpopulations.
INMAT2AC – a program written by Dr. Scott Mills at the University of Montana. It is a stochastic exponential model based on age-structured Leslie matrices. We will use program Inmat2ac to explore 2 questions:
does demographic stochasticity and population growth rate affect a small population's likelihood of going extinct?
- does environmental stochasticity influence the likelihood of larger populations going extinct?
VORTEX – a software package that can be used to evaluate a population's potential for going extinct as influenced by stochastic events.
PVA: evaluation of data and models to determine the likelihood that a population will persist for some arbitrarily chosen time into the future (Boyce 1992).
Minimum Viable Population: a threshold number of individuals that will ensure, with some level of probability, that a population will persist in a viable state for a given interval of time (Koford et al. 1994).
Metapopulation: basic demographic unit composed of a set of populations in different habitat patches linked by movement of individuals (Koford et al. 1994).
Quasi-extinction (management threshold): the population size (>0) where political and/or biological influences (e.g., Allee effect) become important. In other words, populations may be in trouble long before they go extinct. In modelling exercises, we may wish to use this "threshold size" to determine the probability of a population going quasi-extinct (and thus invoking a management and, possibly, political response).
Note: P(persistence) = 1 - P(extinction)
...all populations experience environmental stochasticity; therefore, all populations eventually go extinct (Pulliam and Dunning 1994). In other words, extinction is best described as a probabilistic phenomenon.
However, time to extinction or probability of extinction is affected by:
Population size and number -- the larger the number of populations and the larger the size of each population, the lower the probability of extinction.
Stochastic effects: demographic, environmental (including natural catastrophes), and genetic factors, which affect temporal variation in vital rates.
Mills and Smouse (1994) cautioned that "it is not genetic or environmental variation per se that are important, but rather how this variation manifests itself as variation in vital rates, and population numbers."
Population growth rates -- populations with higher growth rates (e.g., lambda) have a higher probability of growing away from extinction.
Spatial variation: effect of individuals moving among populations that have more or less independent dynamics (i.e., subpopulations as part of a metapopulation).
Deterministic effects (e.g., the underlying relationship of growth rate to pop'n size)
Subjective Assessment – "expert" approach
Rules of Thumb Assessments – do not have to guess but data are limited, e.g., IUCN (World Conservation Union) categorization of over 3,000 species/subspecies of vertebrates based on total population size, population trend, level of population fragmentation, and susceptibility to catastrophic crashes. Does not include statement of uncertainty about conclusions.
Analysis of Trends (time-series models) – using trend data and variation in population size to predict viability, e.g., Dennis et al. (1991) and program STOCHMVP. Concerns about trend data: (1) will past trends be representative of the future? (2) how was the trend estimated (e.g., based on 1st and last data points)? Also, if subpopulations are independent, then the metapopulation is less likely to go extinct (i.e., increasing the number of independent subpopulations increases the persistence of the metapopulation). In reality, dynamics of subpopulations probably fall between the 2 extremes of being totally independent and closely correlated.
Demographically Explicit Models – e.g., age- or stage-structured population models (Leslie Matrix). Potentially data hungry, but very useful. Can be used as a tool not only to identify species of concern but to determine what can be done to address impacts and provide a setting for conservation (e.g., sensitivity or elasticity analysis to identify important factors influencing population growth rates).
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"In general, population viability cannot be assessed by focusing on a single patch of suitable habitat and the organisms living in it... Studies of population viability of most organisms will therefore need to consider the importance of factors that link subpopulations within a metapopulation" (Pulliam and Dunning 1994:194). |
What factors negatively impact a population, and how?
Over what time frame are we interested? Is >40 years a realistic projection?
What degree of security are we interested in? (e.g., probability of persistence)
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Minimum Viable Population Program developed by Dr. E. O. Garton. |
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STOCHMVP calculates MVP and probability of persistence based on stochastic-exponential model of population growth (Dennis et al. 1991). Input data consist of population counts over time. The program calculates the mean instantaneous rate of growth and the variance associated with mu. What are the sources of variation and stochasticity that result in variance of mu? |
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STOCHMVP treats subpopulations as completely independent (i.e., no exchange of individuals and no correlation of demographic rates) and assumes they are of equal size. Is this realistic? How would increasing the correlation of subpopulations affect estimates of MVP and P(persistence)? |
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Data can be entered interactively (from the keyboard) or from a raw data file consisting of a title, followed by year of survey and estimated population size (see Chinook.inp). |
Copy the subdirectory k:\wlf\448\stochmvp to the g: or h: drive. Move to the DOS window. Change directories until you are in the subdirectory that contains stochmvp.exe and chinook.inp. Start the program by typing stochmvp.
Type Chinook.inp when the program asks for data or name of the input file. The program will calculate and report the mean instantaneous rate of growth (mu), finite rate of growth (lambda), and variance (sigma squared) associated with mu .
Enter years (t) that you want the population to persist, e.g., 40.
Enter the lower threshold (i.e., management threshold) for N to avoid extinction, e.g., 30.
Enter the desired probability of persistence, e.g., 0.90.
Enter the initial population size (N), e.g., 50.
The program will calculate the MVP necessary to achieve the probability of persistence you specified over your desired time interval. The program cannot just solve for N (MVP size) at your given probability of persistence; it must solve the equation iteratively (i.e., by repeatedly plugging numbers into the equation and comparing the result to your desired probability of persistence). Note: the results of this iteration are for one population (i.e., consisting of no subpopulations). The question of subpopulations is addressed in the next step.
The program will then ask if you want to run the analysis for a different # of populations (i.e., subpopulations). Type "Y" and hit return. Enter the number of subpopulations in the "metapopulation" (e.g., 2). Note: the program will calculate MVP size and probability of persistence for each subpopulation. Because subpopulations are treated as independent, we can calculate the probability of persistence for the initial population size (i.e., the metapopulation) using the following equation:
Pm = 1 - (1 - Ps )n
where,
Pm = probability of persistence for the metapopulation
Ps = probability of persistence for a subpopulation
n = number of subpopulations
1-Ps = probability of extinction for a subpopulation
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What is the effect of increasing the number of independent subpopulations? Run the same analysis but use 5 subpopulations instead of 2. |
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What is the effect of increasing the initial population size? Use the same input data as simulation #1 but increase the initial population size from 50 to 500. |
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What if management actions resulted in a mu = 0.30 and variance=0.55 (i.e., a positive growth trend)?
Does this new growth "trend" change the estimated MVP and/or the probability of persistence for your initial population size? If so, why? If we now have a positive growth trend (i.e., mu>0), how can our population have a probability of going extinct? |
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A stochastic-exponential growth model written by Dr. Scott Mills of the University of Montana. The model projects population growth and extinction under the simultaneous influence of stochastic variation (due to environment and birth/death schedules) and inbreeding depression. The heart of this program is a Leslie Matrix projection model (female-based model), which is currently set for 10 age classes. The user can project population growth with a deterministic model, or add appropriate levels of stochastic variation. In addition, inbreeding depression can be incorporated. |
Type "INMAT2AC" to start the program (make sure you are in the right subdirectory).
Enter initial population size of females (N0 = 5).
Input vital rates for your population or choose one of the demographic data sets provided:
ungulate population w/ low growth rate ( lambda = 1.05): Enter "u"
felid population w/ moderate growth rate ( lambda = 1.24): Enter "c"
rodent population w/ high growth rate ( lambda = 1.56): Enter "r"
Enter the Coefficient of Variation (moderate CV = 0.33). Remember: this parameter represents environmental uncertainty!
Press Enter to view the Leslie matrix (this should look familiar!)
Input inbreeding costs (for this week, enter "e" for equal).
Input lethal equivalents for survival (for this week, enter "0")
Input lethal equivalents for fecundity (for this week, enter "0")
Input "theta" -- ratio of male effective population size to female effective population size ---> for this week, enter "1.0"
Select stochastic (s) or deterministic (d) population growth. Note: if we set CV=0 but choose "stochastic", what type of stochasticity are we modelling?
The program will project the population over 10 intervals, based on 40 replicates, when the stochastic option is selected. Each replicate represents a population. Is 40 replicates enough? Will we all get the same answers with the stochastic model?
For each set of stochastic runs, the output will be saved to stochrun.out located in your directory (i.e., from wherever you ran Inmat2ac). NOTE: stochrun.out is overwritten each time you run program Inmat, so rename or print the file if you want to save the output. Output is in ASCII format and can be imported into a word-processing program, but select left and right margins of 0.5" to make it fit on a 8.5" page. You may also print directly from the Inmat by using program LISTER (i.e., type "lister stochrun.out" at the DOS prompt).
After completing the simulations, the program will report how many of the 40 replicates (populations) went extinct and how many populations exceeded K, which is arbitrarily set at 100,000 (note: this is an exponential model, so K doesn't influence population growth!).
first 2 lines contain information on input parameters and summary of results (% of 40 populations that went extinct).
results for each of the 10 time intervals (population projections).
time period 10 (the last one) will be a summary of results. Pay special attention to (1) mean and SE of population size (females) for all 40 populations (including populations where N=0), (2) mean and SE of N for persisting populations (i.e., N>0), and (3) # and % of 40 populations that went extinct.
If you get lost in the program or make a mistake, hit "CTR BRK" to end the program, or type "n" when it asks if you want to continue with another simulation.
Boyce, M. S. 1992. Population viability analysis. Ann. Rev. Ecol. and Syst. 23:481-506.
Dennis, B., P. L. Munholland, and J. M. Scott. 1991. Estimation of growth and extinction parameters for endangered species. Ecol. Monogr. 61:115-143.
Koford, R. R., J. B. Dunning, Jr., C. A. Ribic, and D. M. Finch. 1994. A glossary for avian conservation biology. Wilson Bull. 106:121-137.
Mills, L. S., and P. E. Smouse. 1994. Demographic consequences of inbreeding in remnant populations. Am. Nat. 144:412-431.
Pulliam, H. R., and J. B. Dunning. 1994. Demographic processes: population dynamics on heterogeneous landscapes. Pages 179-205 in G. K. Meffe and C. R. Carroll, eds., Principles of conservation biology. Sinauer Associates, Inc., Sunderland, Mass.
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