Mathematical studies of conservation and extinction in inhomogeneous environments

A fragmented ecosystem contains communities of organisms that live in fragmented habitats. Understanding the way biological processes such as reproduction and dispersal over the fragmented habitats take place constitutes a major challenge in spatial ecology. In this thesis we discuss a number of mathematical models of density-dependent populations in inhomogeneous environments presenting growth, decay and diffusion amongst woodland patches of variable potential for reproductive success. These models include one- and two-dimensional analyses of single population systems in fragmented environments. We investigate and compute effective properties for single patch systems in one dimension, linking ecological features with landscape structure and size. A mathematical analysis of potential impacts on spread rates due to the behaviour of individuals in the population is then developed. For the analysis of the population dispersal between areas of plentiful resources and areas of scarce resources, we introduce a novel development that models individuals hazard sensitivity when outside plentiful regions. This sensitivity is modelled by introducing a term called endrotaxis that generates a dispersal gradient, resulting in realistically low migration between regions of plentiful resources. Numerical methods and semi-analytic results yield maximum patch separations for one and two dimensional systems and show that the velocity of spread depends on inter-patch distances and patch geometries. By introducing Allee effects (i.e., inverse density-dependent responses to the difficulty of finding mates at low density) over the population growth function, we find that dispersal is slowed down when combined with hazard sensitivity. In the final Chapter we sumarise the results of the previous chapters, concluding that the work performed in this thesis complements and enriches the current mathematical models of movement behaviour

Designing strategies for epidemic control in a tree nursery : the case of ash dieback in the UK

Ash dieback is a fungal disease (causal agent Hymenoscyphus fraxineus) infecting Common ash (Fraxinus excelsior) throughout temperate Europe. The disease was first discovered in the UK in 2012 in a nursery in Southern England, in plants which had been imported from the Netherlands. After sampling other recently planted sites across England, more infected trees were found. Tree trade from outside and across the UK may have facilitated the spread of invasive diseases which threaten the sustainability of forestry business, ecological niches and amenity landscapes. Detecting a disease in a nursery at an early stage and knowing how likely it is for the disease to have spread further in the plant trade network, can help control an epidemic. Here, we test two simple sampling rules that 1) inform monitoring strategies to detect a disease at an early stage, and 2) inform the decision of tracking forward the disease after its detection. We apply these expressions to the case of ash dieback in the UK and test them in different scenarios after disease introduction. Our results are useful to inform policy makers’ decisions on monitoring for the control and spread of tree diseases through the nursery trade

Modelling cassava production and pest management under biotic and abiotic constraints.

We summarise modelling studies of the most economically important cassava diseases and arthropods, highlighting research gaps where modelling can contribute to the better management of these in the areas of surveillance, control, and host-pest dynamics understanding the effects of climate change and future challenges in modelling. For over 30 years, experimental and theoretical studies have sought to better understand the epidemiology of cassava diseases and arthropods that affect production and lead to considerable yield loss, to detect and control them more effectively. In this review, we consider the contribution of modelling studies to that understanding. We summarise studies of the most economically important cassava pests, including cassava mosaic disease, cassava brown streak disease, the cassava mealybug, and the cassava green mite. We focus on conceptual models of system dynamics rather than statistical methods. Through our analysis we identified areas where modelling has contributed and areas where modelling can improve and further contribute. Firstly, we identify research challenges in the modelling developed for the surveillance, detection and control of cassava pests, and propose approaches to overcome these. We then look at the contributions that modelling has accomplished in the understanding of the interaction and dynamics of cassava and its' pests, highlighting success stories and areas where improvement is needed. Thirdly, we look at the possibility that novel modelling applications can achieve to provide insights into the impacts and uncertainties of climate change. Finally, we identify research gaps, challenges, and opportunities where modelling can develop and contribute for the management of cassava pests, highlighting the recent advances in understanding molecular mechanisms of plant defence

Early warning of critical transitions in biodiversity from compositional disorder

Global environmental change presents a clear need for improved leading indicators of critical transitions, especially those that can be generated from compositional data and that work in empirical cases. Ecological theory of community dynamics under environmental forcing predicts an early replacement of slowly replicating and weakly competitive “canary” species by slowly replicating but strongly competitive “keystone” species. Further forcing leads to the eventual collapse of the keystone species as they are replaced by weakly competitive but fast‐replicating “weedy” species in a critical transition to a significantly different state. We identify a diagnostic signal of these changes in the coefficients of a correlation between compositional disorder and biodiversity. Compositional disorder measures unpredictability in the composition of a community, while biodiversity measures the amount of species in the community. In a stochastic simulation, sequential correlations over time switch from positive to negative as keystones prevail over canaries, and back to positive with domination of weedy species. The model finds support in empirical tests on multi‐decadal time series of fossil diatom and chironomid communities from lakes in China. The characteristic switch from positive to negative correlation coefficients occurs for both communities up to three decades preceding a critical transition to a sustained alternate state. This signal is robust to unequal time increments that beset the identification of early‐warning signals from other metrics

A method of determining where to target surveillance efforts in heterogeneous epidemiological systems

The spread of pathogens into new environments poses a considerable threat to human, animal, and plant health, and by extension, human and animal wellbeing, ecosystem function, and agricultural productivity, worldwide. Early detection through effective surveillance is a key strategy to reduce the risk of their establishment. Whilst it is well established that statistical and economic considerations are of vital importance when planning surveillance efforts, it is also important to consider epidemiological characteristics of the pathogen in question—including heterogeneities within the epidemiological system itself. One of the most pronounced realisations of this heterogeneity is seen in the case of vector-borne pathogens, which spread between ‘hosts’ and ‘vectors’—with each group possessing distinct epidemiological characteristics. As a result, an important question when planning surveillance for emerging vector-borne pathogens is where to place sampling resources in order to detect the pathogen as early as possible. We answer this question by developing a statistical function which describes the probability distributions of the prevalences of infection at first detection in both hosts and vectors. We also show how this method can be adapted in order to maximise the probability of early detection of an emerging pathogen within imposed sample size and/or cost constraints, and demonstrate its application using two simple models of vector-borne citrus pathogens. Under the assumption of a linear cost function, we find that sampling costs are generally minimised when either hosts or vectors, but not both, are sampled

Economic indicators over time

This program describes the dynamics of a tree nursery selling trees that undergo 4 growth stages before commercialisation. The trees grow from seeds to medium trees. Then it calculates the value of several economic indicators dependent on demand variability over time

Data from: Variability in commercial demand for tree saplings affects the probability of introducing exotic forest diseases

1. Several devastating forest pathogens are suspected or known to have entered the UK through imported planting material. The nursery industry is a key business of the tree trade network. Variability in demand for trees makes it difficult for nursery owners to predict how many trees to produce in their nursery. When in any given year, the demand for trees is larger than the production, nursery owners buy trees from foreign sources to match market demand. These imports may introduce exotic diseases. 2. We have developed a model of the dynamics of plant production linked to an economic model to quantify the effect of demand variability on the risk of introducing an exotic disease. 3. We find that - when the cost of producing a tree in a UK nursery is considerably smaller than the cost of importing a tree (in the example presented, less than half the importing cost), the risk of introducing an exotic disease is hardly affected by an increase in demand variability. - when the cost of producing a tree in the nursery is smaller than, but not very different from the cost of importing a tree, the risk of importing exotic diseases increases with increasing demand variability. 4. Synthesis and implications. Our results suggest that a balanced management of demand variability and costs can reduce the risk of importing an exotic forest disease according to the management strategy adopted

Detecting regime shifts in artificial ecosystems

Ecosystems are subjected to a range of perturbations that have the potential to induce relatively sharp transitions in states. These can be referred to as regime shifts or critical transitions. They may be driven by perturbations that vary over a wide range of spatial and temporal scales, from responses to deforestation within a small field to responses to the gradual increase of carbon dioxide in the Earth's atmosphere. Here we investigate potential early warning signals that may presage regime shifts in model ecosystems. We hypothesise and model a relationship between biodiversity and community structure that influences ecosystem structure. We argue that Artificial Life methodologies have potential to make substantial contributions to efforts searching to predict large changes in ecosystems and other elements in the Earth system, as there is a recognised limitation in empirical data and ability to conduct experiments in the real-world. Consequently simulation and exploration of the low-level mechanisms that give rise to regime shifts in artificial in-silico ecosystems represents a useful line of enquiry

Effect of varying sampling effort on the mean prevalence at first detection for the HLB model (panels (a) and (b) and the tristeza model (panels (c) and (d).

<p>The estimated prevalence at first detection in hosts is shown in the graphs on the left, and that in vectors is shown in the graphs on the right. The dashed line indicates a host (vertical line) and a vector (horizontal line) sampling effort of 800 samples per 28 days, with the intersection of these dashed lines indicating a theoretical scenario in which a total of 800 hosts and 800 vectors were sampled.</p

Parameter values for different models used in the current study.

<p>Parameter values for different models used in the current study.</p