1. Introduction to Models Lecture 8 February 22, 2005

2. Models help us to generate or test hypotheses. To formally organize ideas or data. To provide a framework for making comparisons. Identify areas of understanding Identify range of variability Identify sensitive parameters ??? Input Output Model Why use any models?

3. ??? Input Output Model Why use any models? To interpolate or extrapolate understanding, often across scales. Management applications - make predictions or test different management scenarios. To explore scenarios where experiments are not easily conducted.

4. ??? Input Output Model Why use landscape models? Spatial and temporal constraints on landscape studies Experiments on large areas are difficult. Even more difficult to replicate experiments or even "sample" and analyze replicates. Many large-scale processes operate slowly, so landscapes also change slowly.

5. Operationally, useful to think of three general types of landscape models Neutral Models Landscape change models Land cover classes, ecosystem types, or habitats Influenced by natural or anthropogenic processes Includes landscape process models Individual-based models Three Model Types

6. Neutral landscape models generate raster maps in which complex habitat structures are generated with analytical algorithms. Thus, they are neutral to the biological and physical processes that shape real landscape patterns. Neutral Landscape Models

7. What is the value of neutral models? Statistical: How do structural properties of landscapes deviate from theoretical spatial distributions? Modeling: How are ecological processes affected by landscape pattern? Neutral models DO NOT represent actual landscapes!!!!! Neutral Landscape Models

8. Random mapsHierarchical mapsFractal maps Neutral landscape models may be generated by random, hierarchical, or fractal algorithms. Neutral Landscape Models

9. Simple random maps One class Multiple classes p = 0.4 p = 0.6 p = 0.8 Neutral Landscape Models

10. As an increasingly large proportion of the landscape is occupied, the occupied cells coalesce into larger patches. Once p = 0.5928 (0.41 for the 8-neighbor rule), the largest cluster will span the map edge-to-edge. Important since all landscape metrics covary with p. Percolation Theory and Neutral Models

11. Example: Neighborhood Rules examined using Neutral Landscape Models

12. General insights 1.Threshold effects occur as nonlinear relationships between patterns or processes and p. 2.Neutral landscape models are very important for calibrating and understanding different measures of landscape pattern - what is the expected range? 3.Concepts from Neutral Models can be applied to Landscape Change Models - What happens if I turn on/off process X? 4.Specific results of neutral models do not necessarily apply to any actual landscapes, but the insights of the models do apply. Neutral Landscape Models

13. 1.Landscape change models simulate pattern change or state change in a landscape. 2.Most landscape models are different ways of conceptualizing the interactions between three general areas: abiotic template, biotic interactions, disturbances. 3.Depending on needs, a model may need to include processes operating within any of these three areas. 4.All landscape change models include some processes. 5.Questions and scales determine which processes to include. Landscape Change Models: Introduction

14. Markov models: To predict the state of the system at time t+1, you only need to know the state of the system at time t and the probability of transition. (first-order) Landscape Change Models: Simple Markov Models

15. early seral mid- seral closed mid- seral open late- seral open late- seral closed succession low intensity fire high intensity fire thinning Ponderosa pine forest

16. Markov Models: Requires a Transition Probability Matrix (TPM) TPM may be derived from landscape data collected at two time points. TPM may be derived from expert opinion. Landscape Change Models: Simple Markov Models Transition Matrix 1850-1910 Harvard Forest Dioramas

17. The transition matrix is invoked on a cell-by-cell basis. The resulting projected landscape is a stochastic outcome of the transition probabilities. Landscape Change Models: Projecting Markov Models

18. Historical Influences: If the transition probabilities depend on more than the immediately prior state, then the system retains a “memory” of antecedent conditions If so, the dynamics are not first-order. Landscape Change Models: Problems with Simple Models Cell xy time-2 Cell xy time-1 Cell xy t p BG ~ f(time-1, time-2) pBGpBG

19. History (time lags): The transition probabilities become conditional based on the age of the site (transitions occur only after the site has been in a certain state for some time). Time lags are particularly important for disturbance events. Landscape Change Models: Problems with Simple Models Time Old FieldPine ForestFire Old-field succession p = 0 p = 0.2p = 0.8 Transition probabilities

20. History (antecedent events): The transition probabilities become conditional based on whether a particular antecedent event occurred. Landscape Change Models: Problems with Simple Models

21. Spatial dependencies - Covariates: The transition probabilities become conditional probabilities based on some ancillary information about the covariates. Land Cover Classes Soil type Landscape Change Models: Problems with Simple Models

22. Spatial dependencies - Neighborhood Effects: The transition probabilities become conditional probabilities based on the state of neighboring cells surrounding the focal cell. Cellular automata models are best suited to model neighborhood effects (vs. Markov models). Probability of green occupying cell = 3/8 = 0.375 Probability of blue occupying cell = 0.125 Probability of magenta occupying cell = 0.125 Probability of red occupying cell = 0.25 Probability of navy occupying cell = 0.125 Focal cell

23. Landscape Change Models: Problems with Simple Models Nonstationarity: The transition matrix varies over time (i.e., the probabilities are not constant) – which implies that the rules governing landscape change are changing over time.

24. Nonstationarity Solution: Calculate new transition matrices for each time period of interest, or calculate transitions as functions of time. Harvard Forest Dioramas Transition Matrix 1740-1850 Transition Matrix 1850-1910 Landscape Change Models: Problems with Simple Models

25. Disturbance: Disturbances are a special case in modeling, because they are an integration of all the special cases affecting transition probabilities. Disturbances (e.g., fires) may be physically constrained (spatial covariates), may spread contagiously (neighborhood effects), may be lagged in time (time lags), and may change through time (nonstationarity), and may be stochastic. InsectsFire Wind Landscape Change Models: Problems with Simple Models

26. Cellular automata models: "systems of cells interacting in a simple way but displaying complex overall behavior" (Phipps 1992) System of cell networks or grids Cells interact with neighborhood Each cell adopts one of m (m may be infinite) possible states Transition rules for each state can be simple, deterministic, or stochastic. Transition rules ~ f(abiotic constraints, biotic interactions, disturbances) Cellular Models