Comparison of model structure

The primary di↵erence in utilising the complete model structure (which incorporates dispersal) as opposed to the closed structure (which does not incorporate dispersal) is that incorporating dispersal allows the local demographic processes (that occur within individual cells) to be influenced by populations in locations through the exchange of seed. This creates an interlinkage between the individual cell-level populations, and the landscape-level dynamics are more representative of a single spatially connected popula- tion as opposed to the aggregate behaviour of multiple independent replicates. At the same time, applying the demographic models at the local (individual cell) scale ensures an accurate representation of the density dependent processes, as this is the scale used when fitting the models which describe this phenomenon (Chapter 2). Comparing the results of simulations using the two approaches allows me to examine how well we can apply these models which reflect the local demographic processes to project population dynamics at a larger scale. While in the end, it appears that both approaches tended to produce very similar projections of overall population size (Figure 5.2), a close compari- son of the occupancy and density of individual cells reveals that they arrive at these end

Figure 5.4: The projected performance of simulated populations in the di↵erent habitats (using the complete model structure), measured in terms of total population size, occupancy rate of cells by adults, and mean density of occupied cells. These simulations were all initiated with 40 adult individuals distributed among the centre four cells and run for 100 timesteps.

points via distinctly di↵erent paths. In general, omitting dispersal with the closed model simulations results in a strategy that utilises a relatively small number of cells occupied at high densities (Figure 5.1 and Figure 5.3). The relatively high rate of local extinction in the absence of dispersal creates a condition of highly localized populations (Kean & Barlow 2004). Conversely, incorporating dispersal (using the complete model structure) results in a greater proportion of cells containing H. lepidulum at lower densities, cre- ating sparse population conditions (Kean & Barlow 2004). Depending on the tolerance of the existing community for H. lepidulum, these two di↵erent dynamics may result in significantly di↵erent outcomes; the sparse distribution is more likely to be successfully assimilated into the existing community, while the localized invasions are more likely to displace other species, at least at smaller scales.

The omission of dispersal in the closed model structure results in seeds remaining within their cell of origin, which produces a localised e↵ect of propagule pressure compared to the open model structure, in which a portion of the seed would disseminate, and the propagule pressure originating from one cell would be dispersed among a number of neighbours. This localisation of propagule pressure e↵ectively amplifies the trajectory achieved by the local population through stochastic variation in the simulation; if a local population starts out poorly, the result is compounded by a decrease in the number of reproductive individuals, which reduces replacement of dying individuals, creating a negative feedback loop. The converse is also true, where local populations which are successful early on produce more reproductive individuals, which produce more propagules, and so on creating a positive feedback. However, while increased seed supply does result in increased recruitment, the relationship is not linear, and the proportion of seed that result in a viable recruit decreases as the seed input to a cell increases (Duncan et al. 2009). Permitting the dispersal of seed between cells by using the complete model structure achieves two things; first, it disseminates the seed more broadly, helping to mitigate the e↵ects of intra-specific competition, and e↵ectively increases the per-seed recruitment rate (Howe & Smallwood 1982; Hil Res Lambers et al. 2002; Harms et al. 2000). Secondly, the dissemination of seed between cells reduces the impact of the feedback loops, which tend to reinforce the initial population trajectory. This dissemination of seed essentially works to ‘spread the wealth’ of a locally successful population, and allows the diminishing return of increasing propagule pressure in successful cells to be redispersed to locations with potentially lower competition from conspecifics, where they would have a higher probability of surviving. However, the use of absorbing boundaries probably also reduces the relative success of the complete model structure observed in figure ˜refChapter05-closed-v-open-popSize; the use of reflective boundaries or a toroidal surface would likely result in higher di↵erentiation between the two approaches. This interaction between cells (via dispersal) adds a measure of resiliency against local extinction through this addition of propagules, or can even provide an avenue for potential recolonisation after a local extinction occurs. This is analogous to the rescue e↵ect in metapopulation dynamics (Brown & Kodric-Brown 1977),

but in a spatially continuous landscape. This e↵ect of dispersal increasing occupancy at the landscape level is seen on all but one of the habitats (scrub) simulated. These results help to reveal exactly how incorporating dispersal into models of invasion can have a significant e↵ect on the resiliency and extent of invasive populations, two attributes that can substantially influence the e↵ectiveness of control e↵orts.

If utilised properly, demographic models such as this can serve as a useful tool when attempting to predict invasion success (Parker 2000; Caswell et al. 2003). However, this comparison highlights a few of the potential caveats that should be acknowledged when using projections from these types of models to decide on management actions. First, it is clear from this comparison that the overall assessment of invasion success may be influenced by the model structure. The incorporation of explicitly spatial processes (such as dispersal) with demographic models may be necessary to obtain accurate projections (Jongejans et al. 2008). Accounting for these spatial interactions provides a better under- standing of the inherent resiliency in the population; this is exemplified in the dynamics of the occupancy rates when the population is initialised with low numbers, and both model structures start with a high rate of local extinction, but the simulations of the complete model structure are in most cases able to rebound and end up with positive growth by the end of the simulation period (Figure 5.3). The use of a non-spatial demographic projec- tion may therefore lead to an underestimation of the resiliency of these populations, and in turn an underestimation of the level of intervention required to successfully combat the invasive population. Secondly, the e↵ect of including a spatial component to these population projections can vary depending on the local environmental conditions and ex- isting community structure (e.g. Brown et al. 2008; Harris et al. 2011). The di↵erentiation between the two model structures is dependent on what habitat is simulated, particularly when examining the simulations that were initialised at lower densities (Figure 5.3). This is exemplified in the tussock habitat simulations, where the population represented using the complete model structure appears to be self-sufficient, while that of the closed model appears headed for extinction. This interaction between model structure and landscape heterogeneity can be an important aspect to consider when projecting population growth. When applied properly, demographic projections can be an invaluable tool to evaluate the potential e↵ectiveness of control e↵orts (Davis et al. 2006; Brown et al. 2008; Jiao et al. 2009). However, these findings suggest that the e↵ectiveness of removal e↵orts may be underestimated using a traditional closed model approach. However, this investigation does not directly investigate the minimum population size required for persistence; such information would presumably also be useful when evaluating the e↵ectiveness of control e↵orts.

In this example, the results show that the choice of model structure had a substantial impact on not only the final predicted state of the invasion, but also the underlying mechanisms and dynamics by which the invasive population reached that state. It was therefore prudent in this application to utilise the additional complexity a↵orded by the

open/complete model structure. However, this is not necessarily the case in all applica- tions, and persons should strive for parsimony, and find the simplest model that addresses the question at hand. In addition, the costs in terms of additional data collection and analysis required for these types of analyses must be carefully weighed. The application described in this chapter is meant as an example to illustrate the potential for model structure to influence the outcomes, not to infer that additional complexity is always better.