ENV MODELING ESM 232
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Date Created: 10/22/15
ESM 232 Model Types January 87 2004 NOTE Read Chapters 1 2 in Modeling in Natural Resource Management 1 Broad classes of models Broadly7 models in environmental science and management can be 1 Conceptual a set of ideas 2 Verbal translate into words7 or 3 Mathematical translate words into equations 11 Example Suppose you work for the California Department of Fish and Game and you are responsible for making a recommendation about the daily bag limit of mallard ducks harvested in Cal ifornia next year One relationship in which you might be interested is how this year s pop ulation and harvest will affect next year s population 1 Draw CONCEPTUAL MODEL 2 VERBAL MODEL of this system might sound something like this7 The population of adult ducks is reduced on a oneto one basis by the harvest of adult ducks The larger is the resulting population7 the larger will be the next year s population ceteris paribus7 but there are likely diminishing returns in this growth process And a MATHEMATICAL MODEL might look like the following m Xt 7 Ht Xltt1gt mt V 1 2 where Yt is the breeding population of mallard ducks in year 257 Ht is the harvest of mallard ducks in year 257 Xt is the population of adult ducks prior to harvest in year 257 and the function fY determines the growth of the population7 where f Y gt 0 and f Y lt 0 Equation 1 re ects the rst part of the verbal statement Equation 2 re ects the second part of the verbal statement We can add complexity to each type of model eg temperature affecting biological productivity V 2 What kind of models should we use The type of model for a particular application should always be driven by the questions or hypotheses that will be analyzed using the model Often times7 developing a mathematical model is unnecessary Simply developing a clear conceptual model and a more re ned verbal model is suf cient for determining the answer to a question However many complex environmental problems cannot be solved with conceptual models alone The process of developing a mathematical model can be extremely helpful in clarifying the thought process It forces the modeler you to think clearly about the important processes and relationship that govern important features of the problem Our focus here will be on the sorts of environmental problems that require some type of mathematical model for their solution And of course different environmental problems require different kinds of mathematical models In the next section we will consider a general classi cation of mathematical models 3 A classi cation of mathematical models Successful model development always begins with a question or hypothesis and the model complexity and scale should match those of the question At that point a number of decisions must be made about the type of mathematical model the researcher will develop Below is a general classi cation of six such decisions that must be confronted in model design Most models in environmental science and management will embody one branch of each dichotomy on the list below For example the duck harvest model above is a dynamic deterministic analytical structural predictive model 1 Static vs Dynamic Does the action occur at one point in time or over time a Static What is the globally averaged surface temperature of Earth b Dynamic In 1991 19000 gallons of the pesticide metam sodium spilled into the Upper Sacramento River What will be the consequences of this spill for aquatic invertebrates over time to Deterministic vs Stochastic Are the important processes random or can they be viewed as non random Note that it is usually a good idea to develop a deterministic model rst and then to add randomness a Deterministic How much carbon is released when one hectare of Amazon rain forest is burned b Stochastic What will be the high water level on Mission Creek in February 03 Empirical vs Analytical ls the model driven by data or are the results general to a class of problems a Empirical Are bacteria counts higher near sewage outfalls ceteris paribus b Analytical lf OPEC is dissolved will carbon emissions decrease q Structural vs Reduced Form also called Mechanistic vs Phenomenological Are the underlying processes modeled explicitly or are we working with equations that con dense them into a simple framework a If we are interested in how much sediment is carried into Lake Powell we could model the interrelationships between geomorphology hydrology and climate Structural or we could for example simply model a relationship between annual precipitation and sediment load reduced form 5 Predictive vs Decision Theoretic Are we trying to predict what will happen eg over time or are we trying to decide what is the best action to take today a Predictive What will be the earth s mean temperature by year from 2004 b Decision Theoretic What should be the price structure for water in Santa Bar bara And further re nements are almost always necessary For example when developing dy namic models we must decide whether to use discrete time increments eg years or continuous time with an in nitesimal time step Similarly for decision theoretic models we must be mindful of the difference between simulation models that simulate the behavior of some system and optimization models that nd the optimal level of a variable of set of variables possibly through time 31 Some Modeling Terminology To explain the following modeling terminology I will refer to two models 1 Decay of 85Kr lf qt is the amount of Krypton remaining at time t and k 2042m10 9 is a constant the rate of decay of 85Kr is g 7 7kg lt3 qlt0gt 7 100 lt4 This problem happens to have an analytical solution for the amount of Krypton remaining at any time qt q0e 2 Fishery management Fish biomass at time t is denoted Xt Biomass grows according to X XH1 X rXt17 ft 7 Ht 5 The pro t from harvesting biomass Ht is 7139 19th 7 0lnXt 7 lnXt 7 The objective of the shery manager is 10 7ft max 7 6 Ht 1 7 And this maximization problem is subject to the constraint that 0 Ht X rXt17 We will use these models to illustrate some key modeling concepts below 0 Model A system of equations that show how parameters and variables of interest are related to one another Models are not meant to exhaustively represent all elements of a system The trick is to balance the complexity of reality with the principle of parsimony the simpler the better 0 Analytical Solution A mathematical solution to the objective Analytical solu tions are more general than numerical solutions or empirical results see below