Introduction to Models

Abstract

Models are essential tools in landscape ecology, as they are in many scientific disciplines. Spatial models, in particular, play a prominent role in evaluating the consequences of landscape heterogeneity for ecological dynamics. Because we refer to models throughout this book—and because we are aware that many students have had little training in modeling or systems ecology—the first part of this chapter presents an elementary set of concepts, terms, and caveats for students to understand what models are, why they are used, and how models are constructed and evaluated. We also define what we mean by a spatial model and indicate the circumstances where spatial models will be most useful. The second part of this chapter introduces neutral landscape models (NLMs) and illustrates the utility of simple models for understanding landscape heterogeneity and testing hypotheses linking pattern with process. There are many excellent texts that address modeling issues in greater depth. Students interested in the modeling process are referred to the recommended readings at the end of the chapter.

Appendices

Appendix: Classification of Models

Models may be described or classified in various ways, and it is helpful to understand some commonly used terms. We review the terms often used to describe ecological models; similar distinctions are also presented by Grant et al. (1997).

Deterministic vs. stochastic. A model is deterministic if the outcome is always the same once inputs, parameters, and variables have been specified. In other words, deterministic models have no uncertainty or variability, producing identical results for repeated simulations of a particular set of conditions. However, if the model contains an element of uncertainty (chance), such that repeated simulations produce somewhat different results, then the model is regarded as stochastic. In practice, the heart of a stochastic simulation is the selection of random numbers from a suitable generator. For example, suppose that periodic movements of an organism are being simulated within a specified time interval. It may be likely that the organism will move, but it is not certain when this event will occur. One solution is to represent the movement event as a probability, say 0.75, and the probability of not moving as (1.0–0.75) = 0.25. Selection of a random number between 0.0 and 1.0 is done to “decide” randomly if movement occurs during a specific time interval. If the simulation is repeated, the time-dependent pattern of movement will be different, although the statistics of many movement events will be quite similar. Inclusion of stochastic events within a model produces variable responses across repeated simulations – a result that is quite similar to our experience of repeated experiments.

Analytical vs. simulation. These terms refer to two broad categories of models that either have a closed form mathematical solution (an analytical model) or lack a closed form solution and therefore must rely on computer methods (a simulation model) to obtain model solutions. For analytical models, mathematical analysis reveals general solutions that apply to a broad class of model behaviors. For instance, the equation that describes exponential growth in a population is an example of an analytical model (Table 1), as are many of the model formulations used in population ecology (May 1973; Hastings 1996).

In contrast, the complexity of most simulation models means that these general solutions may be difficult or impossible to obtain. In these cases, model developers rely on computer methods for system solution. Simulation is the use of a model to mimic, step by step, the behavior of the system we are studying (Grant et al. 1997). Thus, simulation models are often composed of a series of complex mathematical and logical operations that represent the structure (state) and behavior (change of state) of the system of interest. Many ecological models, especially those used in ecosystem and landscape ecology, are simulation models.

Dynamic vs. static. Dynamic models represent systems or phenomena that change through time, whereas static models describe relationships that are constant (or at equilibrium) and often lack a temporal dimension. For example, a model that uses soil characteristics to predict vegetation type depicts a relationship that remains the same through time. A model that predicts vegetation changes through time as a function of disturbance and succession would be a dynamic model. Simulation models are dynamic.

Continuous vs. discrete time. If the model is dynamic, then change with time may be represented in many different ways. If differential equations are used (and numerical methods available for the solution) then change with time can be estimated at arbitrarily small time steps. Often models are written with discrete time steps or intervals. For instance, models of insects may follow transitions between life stages; vegetation succession may look at annual changes, etc. Models with discrete time steps evaluate current conditions and then “jump” forward to the next time while assuming that condition remains static between time steps. Time steps may be constant (i.e., a solution every week, month, or year) or event-driven, resulting in irregular intervals between events. For example, disturbance models (e.g., hurricane or fire effects on vegetation) may be represented as a discrete time-step, event-driven model.

Mechanistic, process-based, empirical models. These three terms are frequently confusing. A “mechanism” is “…the arrangement of parts in an instrument.” When used as an adjective to describe models (i.e., a mechanistic model) the term implies a model with “parts” arranged to explain the “whole.” In the best sense of the term, a “mechanistic” model attempts to represent dynamics in a manner consistent with real-world phenomena (e.g., mass and energy conservation laws, the laws of chemistry, etc.). Although there has been waning support for mechanistic approaches to ecological modeling (Breckling and Muller 1994), the use of “mechanistic” in the strictest sense distinguishes these models from “black box” models which grasp at any formulation which might satisfactorily represent system dynamics. Confusion arises when the term “mechanistic” is loosely applied to distinguish less detailed models from more detailed ones. Often the implication is that mechanistic models are more desirable than less mechanistic(less detailed) models. Unfortunately, the assertion that additional detail produces a more reliable model must be demonstrated on a case-by-case basis (Gardner et al. 1982).

A “process-based” model implies that model components were specifically developed to represent specific ecological processes—e.g., equations for birth, death, growth, photosynthesis, and respiration are used to estimate biomass yields rather than simpler, more direct estimates of yields from the driving variables of temperature, precipitation, and sunlight. Although this concept seems clear, there is no a priori criterion defining formulations which qualify (or conversely do not qualify) as process models. Thus, depending on the level of detail, it is possible to have a “mechanistic process-based” model or an “empirical process-based” model.

An “ empirical” model usually refers to a model with formulations based on simple, or correlative, relationships. This term also implies that model parameters may have been derived from data (the usual case for most ecological models). Regression models (as well as a variety of other statistical models) are typically empirical because the equation was fitted to the data.

The problem of distinguishing between types of model is illustrated by the simulation of diffusive processes based on well-defined theoretical constructs (Okubo 1980). These formulations of diffusion allow simple empirical measurements to define the coefficients estimating diffusive spread. Thus, there is a strong theoretical base along with empirically based parameters. Is such a model considered empirical or theoretical? Should complex formulations always be considered more theoretical or simply harder to parameterize?

The essential quarrel with each of these three terms is that most ecological models are a continuum of parts, processes, and empirical estimations. Separating models into these arbitrary and ill-defined classifications lacks rigor and repeatability. One person’s mechanistic model is the next person’s process-based model, etc. There does not appear to be a compelling reason to use these vague and often confusing terms to distinguish between alternative model formulations.

Table 3.3. Terminology for model components and common procedures.

Further Reading

Ecological Modeling: General References

• Ellner SP, Guckenheimer J (2006) Dynamic models in biology. Princeton University Press, Princeton

• Haefner JW (2005) Modeling biological systems: principles and applications. Springer, New York

• Kot M (2001) Elements of mathematical ecology. Cambridge University Press, Cambridge

Spatial Modeling

• O’Sullivan D, Perry GLW (2013) Spatial simulation: exploring pattern and process. Wiley, Chichester

• Perry GLW, Enright NJ (2007) Contrasting outcomes of spatially implicit and spatially explicit models of vegetation dynamics in a forest-shrubland mosaic. Ecol Model 207:327–338

• Sklar FH, Costanza R (1990) The development of spatial simulation modeling for landscape ecology. In: Turner MG, Gardner RH (eds) Quantitative methods in landscape ecology. Springer, New York, pp 239–288

• Tilman D, Kareiva P (1997) Spatial ecology: the role of space in population dynamics and interspecific interactions, Monographs in population biology. Princeton University Press, Princeton, p 367

Neutral Models

• Gardner RH (2011) Neutral models and the analysis of landscape structure. In: Jopp F, Reuter H, Breckling B (eds) Modelling complex ecological dynamics. Springer, New York, pp 215–229

• Gardner RH, Milne BT, Turner MG, O’Neill RV (1987) Neutral models for the analysis of broad-scale landscape pattern. Landsc Ecol 1:5–18

• Hagen-Zanker A, Lajoie G (2008) Neutral models of landscape change as benchmarks in the assessment of model performance. Landsc Urban Plan 86:284–296

• With KA, King AW (1997) The use and misuse of neutral landscape models in ecology. Oikos 79:219–229