A while back I wrote about the potentially misplaced preoccupation with statistical power in species distribution models. Our attempts at drawing out some relationships between our deer distribution data and descriptors of land cover is proving taxing – the relationships evident at a more coarse spatial resolution (e.g. county level) than we are considering aren’t found in our stand-level data. As a result we moving toward taking a modelling approach that is driven less by our empirical data and more by inferences based on multiple information sources. Particularly I’m drawn toward emphasising an approach I first encountered in my undergraduate landscape ecology class taught by George Perry – ‘Pattern-Oriented Modelling‘.
A prime example of the POM approach is its use to model the spread of rabies through central Europe. The rabies virus has been observed to spread in a wave-like manner, carried by foxes. Grimm et al. (1996) describe how they developed a cellular automate-type model that considers cells (of fox territory) to be in either a healthy, infected or empty state. Through an iterative model development process, their model was gradually refined (i.e. its assumptions and parameters modified) by comparing model results with empirical patterns.
The idea underpinning this iterative POM approach is
“… if we decide to use a pattern for model construction because we believe this pattern contains information about essential structures and processes, we have to provide a model structure which in principle allows the pattern observed to emerge Whether it does emerge depends on the hypotheses we have built into the model.”
This approach has been found particularly useful for the development of ‘bottom-up’ agent-based models. Often understanding of the fine-scale processes driving broad-scale system dynamics and patterns is poor, making it difficult to both structure and parameterise mechanistic models. However, whilst the logical fallacy of affirming the consequent remains, if a model of low-level interactions is able to reproduce higher-level patterns, we can be confident that our model is a better representation of the system mechanics than one that doesn’t. Furthermore, the more patterns at different scales that the model reproduces, the mode confident we can be in it. Thus, in POM
“multiple patterns observed in real systems at different hierarchical levels and scales are used systematically to optimize model complexity and to reduce uncertainty.”Grimm et al. (2005)
Grimm and Berger outline the general protocol of a pattern-oriented modelling approach (whilst reminding us that there is no standard recipe for model development):
- Formulate the question or problem
- Assemble hypotheses about essential processes and structures
- Assemble (observed) patterns
- Choose state variables, parameters and structures
- Construct the model
- Analyse, test and revise the model
- Use patterns for parameterisation
- Search for independent predictions
Several iterations of this process will be required to refine the model. In initial iterations, steps 2 and 4 may need to be largely inferential if the state of knowledge about the system is poor. However, by moving iteratively back through these steps, and in particular exploiting steps 6 and 7 to inform us about model performance relative to system behaviour, we can improve our knowledge about the system whilst simultaneously ensuring our model recreates observed patterns. For example, during the development of the landscape fire-succession model in my PhD, I compared the landscape-level model results of different sets of (unknown) flammability probabilities (parameters) of each vegetation type required by the model with empirically observed wildfire regime behaviour. By modifying parameters for individual vegetation types I was able to reproduce the appropriate wildfire frequency-area distribution for Mediterranean-type environments that had previously been found (I’m currently writing this up for publications – more soon).
But what does this all have to do with our model of the relationship between deer browse and timber harvest in Michigan’s Upper Pensinsula? Well, right now I think we’re at steps 2,3 and 4 (all at the same time). As our deer and land cover relationships are weak at the stand-level (which is the level we are considering so that we can integrate the model with an economic module), I am currently developing hypotheses (i.e. assumptions) about the structure of the system from previous research on different specific aspects of similar systems. Furthermore, we’re continuing to look for spatial patterns in both vegetation and deer distribution so that we can compare the results of our hypothetical model.
For example, one thing I’m struggling with right now is is how to establish the probability of which individual trees (or saplings) will be removed from a stand due to a given level of deer browse (which in turn is dependent upon a deer density). This is not something that has been explicitly studied (and would be very difficult to study at the landscape level). Therefore we need to parameterise this process in order for the model to function. We should be able to do this by comparing several different parameterisations to empirically observed patterns such as spatial configuration of forest types classified by age class or age/species distributions at the stand-level. That’s the idea anyway – we’ll see how it goes over the next months…
In the meantime, next week I head back to the study area for the first stage of our seedling experiment. We’re planting seedlings now across a gradient of browse and site conditions with the intention of returning in the spring to see what has been browsed and count deer pellets. This should improve our understanding of the link between pellet counts and browse pressure and provide us with some more empirical patterns which we can use in our ongoing model development.