Optimality conditions for nonsmooth optimization problems via generalised derivatives

Aquatic plants are integral components of freshwater ecosystems and provide essential ecosystem services. However, when invasive species establish in new aquatic environments, there are few natural checks and balances to inhibit their growth and spread. Overabundant aquatic vegetation can harm aquatic systems if left unchecked and negatively impact on agricultural productivity, social amenity and biodiversity values. Prevention and early intervention are recognised as the most cost effective means to manage invasive species that pose a biosecurity risk. This thesis contributes to the development of effective management strategies for one of the world’s most invasive aquatic plant species, known as alligator weed (Alternanthera philoxeroides (Mart.) Griseb.). It focusses on developing management strategies in an early stage of invasion, in order to achieve extirpation of this species from catchments and waterways. Developing effective detection and surveillance strategies are required for invasive aquatic plants, as a key impediment to achieving extirpation is the ability to detect infestations, so that control strategies can be enacted. This thesis investigates the effectiveness of aerial surveillance for detection of alligator weed at different spatial scales, using high altitude aerial imagery (orthophotos) and unmanned aerial vehicle (UAV) technology. An examination of the growth rate of alligator weed in Victoria, Australia, over a five year period, demonstrates the effective use of orthophotos to detect and monitor large infestations of aquatic alligator weed. The efficacy of unmanned aerial vehicle technology, including the use of automated algorithms, to detect patches of alligator weed growing in waterways is evaluated against current detection techniques. Effective management of invasive aquatic plants targeted for extirpation requires the coupling of effective detection and control efforts to prevent reproduction. To date, development of control strategies for aquatic alligator weed has been limited to evaluating the efficacy of short-term control at a local scale without regard to the effects of management strategies on dispersal of propagules throughout catchments. This thesis determines that viable alligator weed stem fragments are produced following herbicide application, which comprises extirpation efforts. This thesis has gone further than current practice in that it has evaluated the efficacy of current and novel control techniques, in both laboratory and field trials and has developed methods to manage viable fragment production post-herbicide application, to limit dispersal throughout catchments. In this respect, the application of the herbicides glyphosate, metsulfuron-methyl and imazapyr, and their effectiveness when incorporating surfactant systems and plant growth regulators, have been evaluated in field and laboratory studies to optimise control techniques for aquatic alligator weed. Results have shown that our approaches, when used in an early stage of invasion, are capable of eliminating patches of alligator weed in two to three years. Integral to the research is an experiment to determine the effect of herbicide treatments on the production of alligator weed stem fragments and their subsequent viability. Further investigation to determine the usefulness of commercially available plant growth regulators (PGRs) to reduce the number of viable propagules produced by alligator weed post-herbicide application was found to be ineffective. This thesis also evaluates the impact of herbicides and surfactant systems, on all key alligator weed response metrics in aquatic environments including; above ground biomass, below ground biomass and viable stem fragmentation. No previous studies have looked simultaneously at these three important measures for determining the efficacy of a particular control regime, and we have determined that this is essential for effective management of aquatic alligator weed in an early stage of invasion. The thesis has underscored the notion that development of more effective management strategies, based upon experimental trials, will result in an increased likelihood of eradicating invasive aquatic plants that pose a biosecurity risk, and thus move toward the mitigation of the threat that high-risk species pose to aquatic ecosystems. PLEASE NOTE: Portions of the full text have been removed due to copyright restrictions.Doctor of Philosoph

Imprecise Continuous-Time Markov Chains

Continuous-time Markov chains are mathematical models that are used to describe the state-evolution of dynamical systems under stochastic uncertainty, and have found widespread applications in various fields. In order to make these models computationally tractable, they rely on a number of assumptions that may not be realistic for the domain of application; in particular, the ability to provide exact numerical parameter assessments, and the applicability of time-homogeneity and the eponymous Markov property. In this work, we extend these models to imprecise continuous-time Markov chains (ICTMC's), which are a robust generalisation that relaxes these assumptions while remaining computationally tractable. More technically, an ICTMC is a set of "precise" continuous-time finite-state stochastic processes, and rather than computing expected values of functions, we seek to compute lower expectations, which are tight lower bounds on the expectations that correspond to such a set of "precise" models. Note that, in contrast to e.g. Bayesian methods, all the elements of such a set are treated on equal grounds; we do not consider a distribution over this set. The first part of this paper develops a formalism for describing continuous-time finite-state stochastic processes that does not require the aforementioned simplifying assumptions. Next, this formalism is used to characterise ICTMC's and to investigate their properties. The concept of lower expectation is then given an alternative operator-theoretic characterisation, by means of a lower transition operator, and the properties of this operator are investigated as well. Finally, we use this lower transition operator to derive tractable algorithms (with polynomial runtime complexity w.r.t. the maximum numerical error) for computing the lower expectation of functions that depend on the state at any finite number of time points

Directional upper derivatives and the chain rule formula for locally Lipschitz functions on Banach spaces

Motivated by an attempt to find a general chain rule formula for differentiating the composition $f\circ g$ of Lipschitz functions $f$ and $g$ that would be as close as possible to the standard formula $(f\circ g)'(x) = f'(g(x))\circ g'(x)$, we show that this formula holds without any artificial assumptions provided derivatives are replaced by complete derivative assignments. The idea behind these assignments is that the derivative of $f$ at $y$ is understood as defined only in the direction of a suitable ``tangent space'' $U(f,y)$ (and so it exists at every point), but these tangent spaces are chosen in such a way that for any $g$ they contain the range of $g'(x)$ for almost every $x$. Showing the existence of such assignments leads us to a detailed study of derived sets and the ways in which they describe pointwise behavior of Lipschitz functions