Additional implications

In total, the invasibility of the di↵erent habitats in this landscape has been compared using three di↵erent approaches; first, by comparing the initial recruitment levels (Miller 2006), then by evaluating non-spatial demographic models which incorporate the entire lifecycle (Chapter 2), and finally by evaluating those same demographic models in a spatial context by pairing them with dispersal models (Chapters 5 and 6). In each case, the approaches were based on largely the same data set, and in each case projections of these models produced di↵erent conclusions. This serves as an excellent example of how the choice of the specific method used to analyse and interpret a phenomenon can significantly impact the outcome, and highlights the importance of choosing a method which accurately reflects the suite of mechanisms responsible for the observed process. In order to alleviate some of this

risk, I used a bottom-up approach to develop the simulations, similar to the application pattern-oriented modelling (Grimm et al. 2005; Railsback & Johnson 2011), where I first identified those mechanisms which appeared to be primarily driving the dynamics of the system (in this case demography and dispersal), and then constructed and validated sub- models of those processes before combining them in the final application. In an e↵ort to minimise the e↵ect of my own bias and allow for the influence of di↵erent mechanisms to be present (or not) in the the demography sub-model, I began by constructing a minimal model framework that included those mechanisms without explicitly defining their strength or interdependencies a priori. Instead, these aspects of the mechanisms were modelled as latent variables, which allowed the strength and direction of the mechanisms to be dictated by data collected from the field. For example, I developed a generic model framework to describe the survival of juveniles amongst the di↵erent ages of H. lepidulum which contained parameters to describe the strength and direction of density dependence. These parameters and their uncertainty were estimated for each unique combination of age and habitat, producing individualised measures of density dependence for each life stage in each habitat. This allowed the response to be flexible and predominately driven by the data, and the result was that the strength, direction, and variability of the density dependence was unique for each transition. This type of approach can be extremely useful in cases such as this where a variety of mechanisms may or may not be having an e↵ect on the final outcome, but omitting them from the model (and removing the potential for their influence) is not desirable. This is especially useful for applications such as this, where I wanted to apply the same model form to produce di↵erent transition probabilities between ages and habitats, and the e↵ect of a particular mechanism might only be relevant in a subset of these situations. In addition, properly capturing and incorporating the range of responses provides a much more robust and realistic approach than would otherwise be possible.

The approach of defining the strength and direction of these mechanisms as latent vari- ables and estimating their values and associated uncertainty using data collected from the seed sowing trials was made more accessible through the use of a hierarchical Bayesian approach to parameterise these models (Clark 2005; Latimer et al. 2009). This approach increases the fidelity of parameter estimation by not constraining it to a standard dis- tribution which is determined a priori, but instead produces an empirical distribution, the characteristics of which (i.e. magnitude, frequency, and direction of deviation from the mean) are determined by the iterative fitting of the model to the data during the parameterisation process (Clark 2003). Incorporating this type of parameter estimation into the simulations helps to remove some of the opportunity for researchers to inject bias into their models (i.e. constraining a parameter to adhere to a predetermined dis- tribution); bias which on the surface may seem small, but propagated throughout the simulation can have a significant e↵ect (Finley et al. 2011; Halstead et al. 2012). The hierarchical Bayesian approach to modelling has proven quite useful in this (and many

other) applications, and while it is not the best answer for every situation, this disserta- tion serves as a good example that ecological researchers should at least be aware that a range of approaches exists for activities from model construction to parameterisation, and highlights the importance of selecting an appropriate method for their analysis. The more clearly we are able to formulate and quantify models of ecological processes, the more useful they will tend to be.

As previously mentioned, the models used to project spread dynamics in Chapters 5 and 6 are the result of a bottom-up construction approach to model building. In such applications, the top-level phenomenon of interest (in this case, spread dynamics) are described by synthesising a suite of sub-models that describe lower-level processes. My intention with this approach was to begin by construct the sub-models which represent relatively straight forward mechanistic processes, and their collective behaviour and inter- action when combined should then reflect the top-level phenomenon. In this dissertation, the spread of H. lepidulum was deconstructed into multiple sub-levels; the overall spread model was the result of a combination of demographic and dispersal processes, while the demographic processes were further deconstructed into models reflecting transitions be- tween life stages. In some cases these transitions were deconstructed even further, as in where total plant level seed production was deconstructed into independent models repre- senting the number of flowers per plant, and the number of seed per flower. One important aspect of this approach is that the combination of these mechanistic sub-processes may not be a simply additive process, and interactions between the sub-processes may result in unexpected responses. This was certainly the case in Chapter 5, where the combination of the demographic and spatial components resulted in unanticipated spatial population dynamics.

The bottom-up approach is in contrast to the more common phenomenological (or top- down) approaches to modelling specific phenomenon, where mathematical functions are used to describe the patterns of the observed phenomenon, without regard to the underly- ing processes. Phenomenological approaches are generally simpler and faster to construct, however the bottom-up approach will generally lead to more robust solutions that can be applied outside the conditions for which they were developed, as long as the appro- priate underlying mechanisms are incorporated (Grimm et al. 2005). This can be critical for simulation applications, such as in this dissertation. In this case I have not used a completely mechanistic approach, but have used a hybrid approach, developing phe- nomenological models for some of the sub-models. In fact this is often the case (especially in ecology) where reducing the problem to their most basic first principles is not usually a feasible option. Even so, these this type of bottom-up approach still helps to provide a deeper understanding of the complex phenomena by requiring the deconstruction and identification of the underlying processes that shape them. Just as with the hierarchical Bayesian approach to modelling, the approach of bottom-up modelling is not intended to be a ‘one-size-fits-all’ type of solution that can be applied in all situations. Again, how-

ever, it is important to stress the importance that it be identified as a potential option in the tool set available to the ecologist, as it can certainly be a useful approach if applied in the right settings.

Lastly, an important outcome from this analysis is the recognition that the classification of landscape heterogeneity into finer scales than the binary classification of suitable or unsuitable produces a much more complex dynamic, that is often necessary for under- standing of invasion dynamics. Reducing the landscape to this binary classification may be the only option where data is limited, but the result is a very coarse approximation of the actual dynamics of the system. It is clear from this and other studies that the suit- ability of di↵erent habitats within a landscape is most appropriately measured along a gradient, and that simplifying the landscape to a binary response fundamentally changes not only the representation of the environment, but also the interactions of population processes that interact with that environment (Meekins & McCarthy 2002; Melbourne et al. 2007; Chytr`y et al. 2008). For example, improving the fidelity of this classification has been show to have implications on population stability (Hector et al. 2010; Oliver et al. 2010), life history (Harris et al. 2011), and metapopulation dynamics (Holle & Simberlo↵ 2005; Warren et al. 2011). While refined classification of habitat suitability is becoming more common (Pitt et al. 2009; Fitzpatrick et al. 2012), it is still important and relevant to reiterate how improving the fidelity of this classification can translate to improved understanding of the dynamics under study. Just as with the hierarchical Bayesian and bottom-up approaches described earlier, it is important to make sure these choices and their implications are taken into consideration when planning data collection or analyses.