PhD Thesis. The Australian National University, Canberra.

** Scaling Up Population Dynamics: Spatial Variance and the Scale Transition for Stream Biofilm and Grazers**

Supervisors: Dr Peter Chesson (Australian National University & University of California, Davis), Dr Julian Ash (Australian National University), Prof. P. Sam Lake (Monash University).

Examiners: Prof. Scott Cooper (University of California, Santa Barbara), Prof. Hugh Possingham (University of Queensland), Prof. Mary Power (University of California, Berkeley). What they said about my thesis.

**Abstract**

I examine how spatial variation at small scales modifies population dynamics at regional scales in a two-level trophic system (stream biofilm and grazers). The approach is organised around Chesson's general theory of the scale transition and combines theory, field studies and model fitting. The scale transition refers to the qualitative and quantitative changes that take place in dynamics when the view shifts from one scale to another, such as in scaling up from local to regional dynamics. The scale transition can be understood as an interaction between 1) nonlinearity in population dynamics at the local scale and 2) variation over the whole population. The magnitude of the scale transition is determined by the degree of nonlinearity and the amount of spatial variance. Thus, how spatial variation modifies population dynamics can be gauged from field data by focusing empirical attention on these two elements.

I first derive a basic model for the dynamics of biofilm at the scale of the ecological neighbourhood. The basic model is a balance equation with general functions for biofilm growth and the effect of foraging by grazers. The basic model is restricted to the dynamics of biofilm over short time scales to match the temporal scale achievable in a field experiment.

From this model of dynamics at the scale of the ecological neighbourhood, I derive a model for the regional dynamics of biofilm using scale transition theory. The regional model includes spatial variation in three key components of the biofilm grazer system: biofilm biomass, photosynthetic rate, and grazer biomass. The regional model reveals that the scale transition (i.e. the difference between the local and regional scale models) is determined by four terms: 1) the interaction of nonlinearity in biofilm growth and the spatial variance of biofilm biomass, 2) the interaction of nonlinearity in grazer foraging and the spatial variance of biofilm biomass, 3) the spatial covariance of photosynthetic rate and biofilm biomass, and 4) the spatial covariance of grazer biomass and biofilm biomass. An analysis of the scale transition, and hence the importance of spatial variation to dynamics, amounts to a consideration of the relative magnitude of these four terms along with two further considerations: 1) the scale at which the system is closed, and 2) the scale of the ecological neighbourhood for biofilm growth and for grazer foraging.

To examine nonlinearities at the scale of the ecological neighbourhood, I derive specific models for the growth and foraging functions of the basic model, including alternatives for each function. From consideration of more detailed models of biofilm growth as a function of light intensity and as a function of nutrient concentration, I derive a general nonlinear model of biofilm growth that retains the essential features of resource-limited growth. I derive two linear models of biofilm growth, as alternative models to test the hypothesis of nonlinearity. I also present alternative models for the removal of biofilm by grazers, including both linear and nonlinear models. I formulate six hypotheses that are derived from these models of local biofilm dynamics. 1) Biofilm growth is limited by grazers. 2) Biofilm growth is nonlinear and depends on biofilm biomass. 3) Biofilm growth rate and carrying capacity depend on solar radiation. 4) Biofilm carrying capacity is limited by flow velocity. 5) Biofilm growth rate depends on temperature. 6) Biofilm removal rate by grazers is nonlinear and depends on biofilm biomass.

Analysis of the scale transition required empirical evaluation in two main areas: 1) the six process-based hypotheses, including the nature of nonlinearities and 2) scales of spatial variance and covariance. To achieve these goals simultaneously, I conducted a field-experiment with an hierarchical design spanning 6 spatial scales. The experiment was repeated in two years (1997, 1998). The study area consisted of two catchments, the Bimberamala and Yadboro Rivers, part of the Clyde River system, New South Wales, Australia. The design included the following spatial scales: catchments, 1.6 km reaches within catchments, 10 m sites within reaches, 1 m by 1 m blocks within sites, rocks within blocks, and samples within rocks. Sites were stratified into five classes of solar radiation determined from horizon and riparian-canopy profiles. At twenty sites, I excluded grazers from small arenas on the stream bed using a high voltage electric pulse and observed the dynamics of biofilm growth with and without grazers.

The six hypotheses were tested using data from the experiment in 1997 (a series of spates during the experiment in 1998 prevented model fitting and hypothesis testing for that year). The results were as follows. 1) Biofilm growth was limited by grazers in both the Bimberamala and Yadboro catchments. Removal of grazers caused biofilm biomass to more than double within 25 days. There was no difference in grazer impact between the two catchments. 2) By fitting alternative models of biofilm growth to the data, I found that biofilm growth was nonlinear and dependent on biofilm biomass. The fitted parameters suggested moderate competition for resources within the biofilm. 3) Biofilm growth rate and carrying capacity were not influenced by solar radiation at the site scale, as measured by horizon profiles, or the rock scale, as measured next to each rock. At the rock scale, solar radiation was measured with microsensors installed on the stream bed and recorded with electronic dataloggers. 4) Biofilm carrying capacity was not limited by flow velocity, possibly because flow velocities during the experiment were low. 5) Biofilm growth rate was positively related to temperature differences among sites (monitored with electronic dataloggers) and was described by an exponential, Arrhenius, relationship. 6) The rate of biofilm removal by grazers was nonlinear and dependent on biofilm biomass. A model with linear density dependence provided the best description of the data compared to alternative models, although a "type III like" model performed similarly.

I used variance components estimation to obtain the variances and covariances that determine the scale transition in the regional model and to identify important spatial scales of variation in 1997 and 1998. There were five findings. 1) Variation at the smallest scales (rock, sample) always made up a substantial proportion (20-50 percent) of the total variation for biofilm and grazers. 2) The pattern of variation among scales differed between biofilm and grazers. Biofilm was most variable at small scales (sample, rock, site), whereas grazers were variable at large scales (catchment, reach) and small scales (rock) but not intermediate scales. 3) The pure-spatial structure differed between times for biofilm. Biofilm dynamics were highly variable in both space and time at smaller spatial scales and over all temporal scales. Grazers also displayed large spatio-temporal variation over small space and time scales. However, in contrast to biofilm, the pure-spatial structure of grazers changed little on any time scale, despite drying and flooding episodes encountered during the study. 4) Flow velocity was variable on all scales, whereas depth, temperature and solar radiation had more distinct domains of variation. 5) The association in space of biofilm biomass and photosynthetic rate, and biofilm biomass and grazer biomass was weak. The covariance components for these variables were small and few significant components of covariance were observed.

To examine how spatial variation could affect dynamics, I combined the considerations of nonlinearity and scales of spatial variation. Using the regional model, I examined the scale transition for the rate of change of biofilm biomass and the short-term equilibrium biomass of biofilm. There were five predictions from the regional model. 1) Spatial variation will lead to a large negative scale transition. At the regional scale, the instantaneous rate of change in biofilm biomass will be reduced by up to 118 percent and the equilibrium biomass by up to 32 percent by spatial variation at small scales. 2) The first term of the scale transition (interaction of nonlinearity in biofilm growth and spatial variance of biofilm biomass) will have the strongest influence on the scale transition, followed by the second term (interaction of nonlinearity in grazer foraging and spatial variance of biofilm biomass), with the third and fourth terms (covariance terms) making only a small contribution. 3) The magnitude of the scale transition will increase if grazers perceive biofilm biomass as homogeneous at larger spatial scales. That is, the more different biofilm and grazers are in their scaling, the greater the effect of spatial variation at small scales on biofilm dynamics at the regional scale. 4) Reducing the scale of closure will reduce the magnitude of the scale transition. 5) The magnitude of the scale transition will be greater for a system with high-mobility grazers than for low-mobility grazers.