Density dependent Markov population processes: models and methodology

A feature exhibited by all populations is demographic stochasticity. This is randomness that arises from chance events inherent in birth and death; finding a partner, successfully reproducing, the number of offspring produced and age at death are just some examples of events governed by chance in population dynamics. Additionally, the study of the probability of extinction and expected time to extinction can only be undertaken in a stochastic framework. I provide models and accompanying methodology for enabling the use of stochastic models for real biological modelling. In particular, we will consider a class of continuous-time Markov chains with a particular form of rates, known as (asymptotically) density-dependent Markov pop- ulation processes. These are continuous-time Markov chains that satisfy the properties that the changes in state (in each dimension) are of magnitude unity, and the transition rates take the form (at least asymptotically in N) qN(i, i + l) = Nf ( i N , l), l 6= 0, for a suitable scaling parameter N and function f. These processes have been widely studied in the applied probability literature, and have appeared extensively in the theoretical literature in a variety of contexts. Results of Kurtz and Barbour have played a major part in establishing this ubiquity. Kurtz established a functional law of large numbers, and a functional central limit law, for such processes, consequently establishing unique deterministic and Gaussian diffusion approximations, respectively. I utilise these results to firstly establish deterministic and diffusion approximations for models of metapopulations in a dynamic landscape, before establishing a method for incorporating initial state uncertainty, and a method of parameter estimation, for general (asymptotically) density-dependent Markov population processes. As habitat loss and fragmentation continues around us, populations are being forced to occupy patches of subdivided habitat. With the addition of the numerous species that naturally occupy landscapes of this type, such as wood roaches in fallen logs, fish on coral reefs and parasites on hosts, metapopulations are rapidly becoming the pre-eminent paradigm for ecological modelling. A key component affecting metapopulation fluctuations is often the dynamics of the landscape itself. We look at models for metapopulations that account for dynamic landscape. I show, for each model, that a suitably scaled version converges, uniformly in probability over finite time intervals, to a deterministic model. Additionally, I establish bivariate normal approximations to their (quasi-)stationary distributions. This allows us to consider the effects of habitat dynamics on metapopulation dynamics and provides an effective means for modelling metapopulations inhabiting dynamic landscapes. I find that habitat dynamics can have a significant impact upon metapopulation dynamics, reducing the expected number of occupied patches and increasing the variability in patch occupancy. Additionally, I find that mainland-type patches help alleviate these affects, and thus argue for the introduction of mainland-type patches as a conservation management action. Ecologists and conservation biologists have become interested in the optimal way to manage and conserve such metapopulations. I investigate an extension of the previous models, that allows a number of the patches to be protected from disturbance, and we use this model to answer the question - how can we optimally trade-off returns from protection versus creation? I find that when habitat dynamics are slow in comparison to patch occupancy dynamics it is optimal to create new patches, while under the reverse conditions it is optimal to protect existing patches before creating new patches. When the time scales are similar it is difficult to establish a robust rule of thumb. However, a deterministic approximate solution is presented, which provides a reasonable rule of thumb, in particular when the number of patches is large. Next we will consider initial state uncertainty in population modelling. Most modelling in ecology assumes that the initial state of the system being studied is known. In reality, this initial state is rarely known to such a high degree of precision. I investigate under what conditions ignoring uncertainty in the initial state matters, and additionally provide an extension to the technique of Kurtz, allowing us to establish explicit expressions for the mean and variance of the process, incorporating an initial distribution, at any point in time. Whilst continuous-time Markov chains are ubiquitous in the theoretical literature, their application has been limited, partly due to a lack of clear statistical procedures for fitting the models to data. I address these statistical limitations by providing straightforward methods for parameter estimation, so enabling this rich family of models to be used in real biological modelling. A general approach, applicable to any finite-state continuous-time Markovian model, is presented, and this is specialised to a computationally more efficient method applicable to (asymptotically) density-dependent Markov population processes. This method is found to be consistently highly accurate under certain (weak) conditions.