Introduction to models for wildlife management
All models are wrong.... some are useful - G.E. Box
Models Versus Full Reality
We begin the discussion of fish and wildlife modeling with a reality check.
Our fundamental assumption is that there is no “true” model that generates
biological data. Rather, we believe that truth in biological sciences has
essentially infinite dimension. Indeed, biological systems are complex
with many small effects, interactions, individual heterogeneity, and
environmental covariates. For example, a list of unrelated, short-term, small-scale events that could have caused a bird to be in a location where it was
counted by a biologist might include: random foraging movement, frightened by
potential predator, interaction with conspecific in adjacent territory, warming
itself in the sun, and many others. These also could be the events that led to
that bird being preyed upon. When such small effects are combined across
individual birds within a population, they would no doubt contribute to
population-level processes and our estimation thereof. Thinking back to
statistics class, recall that the number of independent variables cannot exceed
the number of data points. That is because each datum conveys a certain amount
of information and the model (the independent variables) cannot represent more
information than is contained in the data set. Given that we usually do not have
infinite data sets, full reality (i.e., the truth) cannot be revealed. This
means that biological models cannot represent full truth and are, by definition,
wrong. Now you are probably asking yourself, then what good are models? To
address that, we first must define a model.
When defining models, it is often useful to think about caricatures, such as the one of a famous Georgian below.
Caricatures do not accurately portray their subjects. Rather, they exaggerate some features and ignore others and only include those features necessary to convey the identity of the subject. Similarly, good (or useful) biological models attempt to approximate reality and include only those features that are necessary to predict the likely outcomes of biological processes and that are supportable by the data. This latter constraint has led to the development and use of a variety of methods for modeling fish and wildlife populations, ranging from simple models with few parameters to very complex models requiring a very large number of parameters. We believe that it is important for biologists to consider data (information) requirements and availability when building population models because the use of inappropriate (e.g., over-parameterized) models is (1) not scientifically sound and (2) indefensible. However in practice, this constraint is often ignored as evidenced by the widespread use of such programs as Vortex to evaluate population viability.
Population models are used for a variety of reasons ranging from the theoretical--to explore how populations might work-- to the applied-- to predict how a system behaves to identify the best management action. Here, we focus on the applied aspects of population modeling and consider single population models.
Single population models are by far the most widely use in fish and wildlife management. They are generally used to evaluate the effect of some management action (e.g., harvest rates, bag and size limits) on population sizes or to evaluate the change in population persistence (e.g., quasi extinction, time to extinction) in response to some management action, such as purchasing conservation lands and setting up reserves. Single population models range from simple population growth models-- to stage based (Leslie Matrix ) models-- to complex Individual Based Models (IBM). The data requirements for each type of model range from minimal to large, respectively. Example models are located in population modeling.xls.
Population growth rate models
The simplest form of a single population
model is the population growth rate model that requires few parameters: a growth
rate, r, for a density independent model and r and carrying
capacity, K, for the density dependent form. These parameters are usually
obtained by fitting models to time series data, such as population monitoring
data. The population growth rate represents the combined effects of birth (b),
death (d), immigration (i) and emigration such that:
r = b + i - d - e
The use of these simple models requires several assumptions. First, all individuals in the population have the same reproductive potential (e.g., males and females) and the same probability of mortality. Generations are non-overlapping or offspring reach maturity within a single time-step. Fisheries folks have gotten around this assumption by only considering the sexually mature (recruits) in the models. Finally, the growth rate is not subject to temporal variability.
Two forms of population growth rate models are the density independent and density dependent. The density independent form assumes that there are no mechanisms that can change the population rate of growth in response to changing population densities. Thus, sustainable harvest rates are independent of population size. The exponential growth model is a density independent growth model with the form:
Nt = N0ert
where the the population size at time t is a function of the initial population size N0 and the growth rate, r. Biologists often decompose the population rate of growth into birth (F, fecundity) and death components such that:
Nt+1 = Nt + r*Nt
Nt+1 = F*Nt + S*Nt
where survival S is 1 minus death. Fecundity and survival also are often expressed in terms of:
F + S = l = r + 1
where l > 1 indicates F > (1-S) and population growth, l = 1 indicates F = (1 - S) and no population growth, and l < 1 indicates F < (1- S) and no population growth.
Density independence assumptions is likely not met for most populations. An alternative population growth rate model is the density dependent model. One density dependent model includes the logistic growth model:
where K is the carrying capacity. Other density dependent population growth rate models include the Ricker stock recruit model. In contrast to the density independent model, sustainable harvest rates for the density dependent model depend on population size.
Age or Stage – based models
Stage or age-based models differ substantially from population growth rate models. Here, ages groups (cohorts) or stages (young, juvenile, adult) are modeled separately. Below is a graphical representation of a simple 3 stage population model. Here, the model begins by creating young at time t, the number of young is a function of the number of adults at time t and fecundity. The number of juveniles in time (t +1) is a function of survival and young at time (t), whereas the number of adults at time (t+1) is a function of the number of juveniles at time (t) and survival plus the number of adults at time t and survival.
In stage based models, factors such as survival, growth, fecundity can vary among stages. Stage based models, also referred to as Leslie matrix models, can be expressed in matrix formulation as:
where Ni are the stage (i) specific abundance at time t and (t + 1), Si are age specific survival probabilities, and Fi are the stage specific specific fecundity. In this notation, the survival of the last stage k is usually assumed zero. The population in t+1 is calculated by multiplying the population vector for t with the transition (Leslie) matrix and the population size then is the sum of the individual stage specific population sizes.
The assumptions of the stage based models are not nearly as restrictive as the population rate models. One big assumption is that all individuals in a stage have the same rates such as fecundity, growth, and survival. That is, stage based models cannot take into account individual variation.
Individual based models (IBM)
Of the models considered here, IBMs are the most complicated and data-intensive. Like stage based models, they include factors such as growth, survival, fecundity, plus other factors such as movement and feeding. These factors, however, are allowed to vary among individuals and in more complicated versions of IBMs, progeny can inherit characteristics from parents. IBMs are generally useful for more theoretical applications. For example, evaluating the change in the frequency of specific traits through time as environments vary. They also have been used to examine •spatially explicit phenomena, such as habitat use patterns. However, some applications have been developed for natural resource management, such as ATLAS that uses IBMs for multiple species to guide management of the Everglades. Though these are the most detailed models (i.e., they attempt to approximate truth as closely as possible), they tend to be over-parameterized and hence are generally poor at predicting future population size.