Modelo De Leslie

Modeling Population Dynamics
Gary C. White INTRODUCTION Population modeling is a tool used by wildlife managers. At its simplest, population modeling is a book keeping system to keep track of the four components of population change: births, deaths, immigration, and emigration. In mathematical symbolism, a population model can be expressed as Nt+1 = Nt + Bt - Dt + It - Et . Population size (N)at time t+1 is equal to the population size at time t plus births (B) minus deaths (D) plus immigrants (I) minus emigrants (E). The simplicity of the relationship belies its usefulness. If you were managing a business, you would be interested in knowing how much money was being paid out of the business, relative to how much money was coming in. The net difference represents your “profit”. The sameis true for managing a wildlife population. By knowing the net change in the size of the wildlife population from one year to the next, you as a wildlife manager know whether the population is “profitable” (i.e., gaining in size), or headed towards bankruptcy (i.e., becoming smaller). Two examples will demonstrate typical uses of population models in big game management. In November 1995, votersin Colorado passed an amendment to the state’s constitution that outlawed hunting black bears during spring. Wildlife managers were asked if the bear population would increase because of no spring harvest. A population model can be used to answer this question. The second example of the use of a model is to determine which of two management schemes (e.g., no antlerless harvest versus significantantlerless harvest) will result in

Modeling Population Dynamics – Draft May 6, 1998 the largest buck harvest in a mule deer population. Both of these examples require specific information to answer the questions posed. The models in both cases are important in conceptualizing the problem, and in defining exactly what information is needed to answer the question. The main use of the model in theprevious examples is to predict the result of some management action; such as stopping spring bear hunting, or implementing an antlerless harvest. Another use of population models is to conceptualize the dynamics of a population in a rigorous

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mathematical notation. Such a conceptual model allows biologists to think more clearly about the dynamics of a population. A third use of models isto test hypotheses about population dynamics from observed data. For example, biologists might entertain competing models, one based on density independence, the other with density dependence. Which of these models best explains our observations of a mule deer population? Computational tools are needed to build population models on a computer. Spreadsheets are a natural vehicle to build populationmodels. They were developed for modeling the functioning of a business (i.e., keeping track of income and outflow) and hence are particularly suited for modeling population dynamics. Modern spreadsheets include a natural language to specify a population model, graphics to display the results, random number functions to model stochasticity, and optimization functions useful in optimizing someattribute of the model’s output (e.g., annual harvest), or in estimating parameter values from data. Complex models can be built (Lubow et al. 1997) and then made available via the WWW (see http://www.cnr.colostate.edu/~bruce/software.htm for these spreadsheet models of linked-sex harvest strategies.

Modeling Population Dynamics – Draft May 6, 1998

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A major advantage of using spreadsheets todevelop the model is that you fully understand the mechanics of the model you’ve developed. You know what relations are assumed in the model because you put them in the model. As a user of another’s model, spreadsheet models allow you to dissect the model to see what relations were programmed. Most of the earlier models of population dynamics were built with pre-defined models, such as ONEPOP...