Mathematics (Faculty of)

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Now showing 1 - 9 of 9

Coupled models of structured contagion processes in human-environment systems
(University of Waterloo, 2021-09-15) Jentsch, Peter Carl; Bauch, Chris; Anand, Madhur
Models of infectious processes are a common feature in the landscape of applied mathematics. It is rare that these processes are isolated from other significant dynamics in nature, and therefore we can incorporate some of the complexity inherent in real systems by coupling infections to major features of the ecosystems they inhabit. Infectious processes can take many forms, but in this thesis we consider three: the COVID-19 pandemic, the invasion of eastern North American forests by wood-borne pests, and the outbreak cycles of an endemic forest pest. The first chapter covers a model of Sars-CoV-2 in a structured population, coupled with a replicator equation representing sentiment towards the use of non-pharmaceutical interventions. We use this human-environment model of to compare the efficacy of vulnerable-first and transmission-preventing age structured vaccination strategies. The buildup of natural immunity in a population combined with a low vaccination supply is shown to cause a transmission-preventing vaccination strategy to be more effective. The second chapter considers a spatially structured model of forest pest contagion over an empirically-derived network of forest patches in eastern Canada. Since these pests can frequently be spread long distances by wood transport, we couple this model to the sentiment of local populations towards avoiding firewood transport from outside their area. Three possible countermeasures to the spread of the invasive pest are compared: social incentives, direct interception of infested firewood, and quarantine of patches. The level of effort needed to significantly reduce forest damage with any of these methods is substantial and unlikely to be implemented. The final chapter extends a model of mountain pine beetle (MPB) in western north american pine forests to incorporate tree mortality due to wildfire. We find that wildfire acts as a disturbance that increases the heterogeneity in age structure, and therefore is able to increase the resilience of the forest to outbreaks of MPB. A targeted thinning procedure aimed specifically at increasing heterogeneity in the forest age structure is proposed and shown to be highly effective at reducing the severity of outbreak. The effectiveness of targeted thinning in the manner described further emphasizes the importance of heterogeneity in forest stand structure. Each model tests the importance of indirect protection in preventing the spread of an infectious agent through a specific host population, with respect to key parameters. Models let us use counterfactuals to gain potentially invaluable understanding of these complex human-environment systems.
Detecting and distinguishing transitions in ecological systems: model and data-driven approaches
(University of Waterloo, 2020-01-22) Bury, Thomas; Bauch, Chris; Anand, Madhur
There exists a plethora of systems that have the capacity to undergo sudden transitions that result in a significantly different state or dynamic. Consider the collapse of fisheries, outbreak of disease or transition to a 'Hothouse Earth' to name a few. The common factor among these transitions is mathematical - they are the result of crossing a bifurcation point. This thesis is concerned with the detection and description of these bifurcations from time series data, and the mechanisms that lead to these transitions. We begin in the domain of climate change, where models of the climate system are extremely sophisticated, but those that incorporate social dynamics and its two-way coupling with climate dynamics are lacking. In developing a simple socio-climate model, we show how mechanisms such as social learning, social norms, and perceived mitigation costs play a major role in climate change trajectories. These social effects can strongly determine the predicted peak global temperature anomaly, how quickly human populations respond to a changing climate, and how we can chart optimal pathways to climate change mitigation. However, we also show that if the climate model is subject to a tipping point, the climate can transition to a new state before mitigating behaviour becomes sufficiently widespread to prevent the transition. This motivates a need for early warning signals (EWS) of tipping points. Hence, in the next chapter we focus on the development of EWS in time series data that can be used to detect an upcoming bifurcation. This thesis develops two 'spectral EWS', which are derived from the power spectrum. We show that the peak in the power spectrum provides a more sensitive and conservative EWS when compared to conventional metrics, and the shape of the power spectrum, quantified using AIC weights, provides clues as to the type of approaching bifurcation. We validate these spectral EWS with empirical data from a predator-prey system. Finally we focus on EWS for population extinction, where we study the efficacy of EWS in seasonal environments. We find that conventional EWS prevail under seasonal environments, however asymmetries exist in higher-order metrics such as skewness and kurtosis that could be used to distinguish the driver of extinction. To conclude, nonlinear behaviour arising from social learning and social norms yield bifurcations that have profound impacts on future trajectories of climate change, and bifurcations can be anticipated across a wide range of systems using spectral EWS, that also provide information on the type of bifurcation. The further development of generic and system-specific EWS will play an important role in preserving healthy ecosystem functioning in the Anthropocene.
Emergence and Implications of Conservation Opinion Propagation in Dynamic Coupled Socio-Ecological Systems
(University of Waterloo, 2021-04-22) Thampi, Vivek; Bauch, Chris; Anand, Madhur
Human behaviour is rarely a static phenomenon. In life, individuals are presented with choices that define the trajectories they will experience days, weeks or months later. As an example consider farmer decision-making and orchard dynamics. If an avocado orchard is well taken care of, a bountiful harvest can lead to a lower price of avocados that will more easily attract grocers to stock the product. Alternatively, if the orchard is not properly cared for, avocado prices can surge (due to their low supply) and become a greater risk for grocers, causing them to seek other suppliers. If a particular 'care' routine is developed by the farmer, this can have a significant impact on the long-term trajectories of orchard dynamics. From this simple example, it is clear how dynamic human behaviour can interact with environmental system dynamics. This motivates the potential value of capturing this interaction in mathematical modelling. In this thesis, we develop two different coupled human-environment system (CHES) models that incorporate a dynamic feedback loop mechanism to link human impact and environmental system responses and vice versa. The first model is developed using a game-theoretic approach to describe dynamics of opinion spread. The model is then coupled to a previously established coral reef ecosystem model. We investigate the effects of key factors such as social learning, social norms, and exploitation rate on the trajectories predicted by the model. We discover stable regimes that are made possible by the presence of human coupling and we identify the potentially harmful role of social norms. In the second model, we utilize a similar game-theoretic approach to couple a dynamic human component to a previously established grassland model of the Southeastern Australian grasslands. The aim of this model is to determine conditions that suppress invasive exotic grasses, in the presence of human feedback that determines how strongly the local population mitigates its own pollution. Finally, we conduct a systematic review of the CHES modelling literature between May 2009 and April 2019 using the Web of Knowledge and PUBMED databases. Results reveal an increasing trend in the number of mathematical models using a CHES approach. Results also show that these models utilize a wide range of techniques of varying complexity. In general, most work focuses on agricultural systems. We postulate that application to other environmental systems is relatively unexplored and can be implemented using techniques similar to those of the models developed in this thesis, or via adaptations of other modelling techniques from different fields of research. We suggest that including dynamic human behaviour is necessary in order to improve existing environmental policies and improve the predictive power of mathematical modelling techniques in environmental systems research.
The forest transition and ecological thresholds: resilience, recovery, and predictions
(University of Waterloo, 2018-01-19) Gooding-Townsend, Robert; Bauch, Chris; Anand, Madhur
A central topic in modeling land use change is to understand the forest transition from deforestation to net reforestation. Agricultural land use change is the main driver of this phenomenon; classically, agricultural land expands considerably to feed a growing population, and then declines as efficiency gains are realized, marginal farmland is abandoned, and rural populations move to cities. As a result, existing models have focused on the socioeconomic and demographic factors that drive agricultural intensification. However, in doing so, these models often neglect the role of ecological feedback effects and thresholds. These ecological thresholds can cause rapid shifts in ecosystems, such as forest collapse, based on small changes in parameters, and are very difficult to predict. The existence of these thresholds implies that agricultural expansion carries a risk of forest collapse. We aim to use realistic models to assess the risk of collapse in forest cover, dependence on key parameters, and strategies to avoid it. To address the risk of forest collapse, we develop and analyze a differential equation model that incorporates both agricultural intensification and ecological thresholds. We use parameter values from the literature to adapt this model to boreal and tropical forests. We analyze the model with bifurcation diagrams, simulations of key resilience metrics, and fitted time series of real-world data for China, Costa Rica, and Vietnam. Our analysis shows that there is a risk of forest collapse, and that the system is particularly sensitive to agricultural parameters. We find that regardless of the mechanism by which collapse occurs, there is a critical value of 20-25% forest cover. In scenarios of interest (i.e. forest transitions), initial deforestation would result in collapse if left unchecked. We estimate model parameters at multiple points along historical time series, which allows us to infer the risk of collapse and identify historical patterns. This shows that forest transitions can be caused by more varied parameter patterns than classically assumed in the literature; in particular, rates of land conversion and agricultural abandonment rate may remain elevated, instead of declining after intensification. The agricultural abandonment rate is a key advance predictor of collapse at long time horizons, but at the brink of a crisis forest collapse can best be avoided by reducing the forest conversion rate. We argue that ecological threshold effects should be acknowledged in forest transition models not only for ecological accuracy but also to ensure prudent forest management, particularly in the face of emerging risks such as climate change.
Global land use and the future of sustainable consumption: projections of a coupled social-land use model
(University of Waterloo, 2020-09-10) Pal, Saptarshi; Bauch, Chris; Anand, Madhur
Throughout history, agricultural land use activities have shaped the environment and its inhabitants. Humans have sought to maintain their well-being for centuries by finding revolutionary ways to produce and gather food. But challenges that are new to us--both in nature and intensity--face the human race and its planet as we move further into the 21st century. Global food demand is currently at its peak and is projected to grow in the near future due to growing global population and global affluence. Subsequently, pressure on the agricultural system is expected to increase to meet the elevated demand. Agricultural expansion, a possible pathway for meeting increasing demand, has been shown to bring with it severe impacts on climate, environment, and ecosystems. While several land use mitigation strategies have been thoroughly explored in the literature, they have mostly been centered around policy regarding land use management and investment in agricultural technology. In most of the models, future dietary patterns of populations are assumed to behave independently from land use change. The immense potential of sustainable consumption in land use mitigation, stimulated by desires to avoid agricultural expansion into sensitive ecosystems, has only been explored through scenario constructions. Little effort has been spent on understanding how sustainable consumption might begin evolving in a population as a behaviour in response to land use dynamics. This thesis introduces a minimal mathematical model based on evolutionary game theory that couples human behavioural dynamics with land-use system dynamics. This coupled human-environment model helps in gaining a better understanding of the evolution of sustainable diets within populations while making global land use projections till 2100 under multiple future socio-economic scenarios. Results in this thesis highlight the direct impact of social processes on global agricultural land use and underline the barriers and drivers of human consumption behaviour. The model framework lays the foundation for further developments in complex coupled human-land system models that focus on gaining deeper insights into system dynamics and possible future outcomes of interventions.
Modeling human-coupled common pool resource systems with techniques in evolutionary game theory and reinforcement learning
(University of Waterloo, 2021-05-26) Farahbakhsh, Isaiah; Bauch, Chris; Anand, Madhur
Shared resource extraction among profit-seeking individuals involves a tension between individual benefit and the collective well-being represented by the persistence of the resource. In these systems, the decisions of rational agents have been modeled from a game theoretic, and more recently, a reinforcement learning approach. Within game theoretic models, the mechanisms used for learning dynamics are often assumed, and the influence of the type of learning dynamics are not systematically compared under identical models. Models using reinforcement learning techniques are a relatively recent addition to this field, and the literature on multi-agent systems with spatial structure is very sparse. This thesis presents two common pool resource models, each using one of these two different approaches. In the second chapter, an evolutionary common pool resource game is simulated on a social network with payoff functions that depend on the state of the resource. Model predictions under two types of learning, best response and imitation dynamics are compared and it is shown that best response dynamics lead to an increase in sustainability of the system, the persistence of cooperation while decreasing inequality and debt. Given the strikingly different outcomes for best response versus imitation dynamics for common-pool resource systems, our results suggest that modellers should choose strategy update rules that best represent decision-making in their study systems. In the third chapter, an analogous model to the one above is presented, however it uses reinforcement learning techniques to inform the agents' harvesting decisions. Here, the harvesting strategies of the agents are learned, rather than prescribed a priori, and the payoff function is the weighted sum of a profit goal and a social conforming goal. Preliminary results show that an increased cost of harvesting has a positive effect on the resource level and sustainability of the system, however, a high cost parameter brings the system to an unprofitable state where agents harvest above the analytically derived optimal level. Additionally, the effect of the weight of the conforming goal shows contradictory outcomes, which are highly dependent on the profitability of the system. These different outcomes are posited to be due to strong social conformity amplifying existing trends in the social dynamics. Results from both chapters demonstrate the profound effect human learning models can have on common-pool resource systems, as well as the potential for sustainable outcomes to emerge among a non-hierarchical system of self-interested agents.
Modelling resilience and sustainability of complex human-environment systems in agriculture and ecology
(University of Waterloo, 2020-06-01) Fair, Kathyrn; Bauch, Chris; Anand, Madhur
As we move further into the Anthropocene, numerous challenges to sustainable development present themselves. Questions abound: How do we feed a growing population? What steps must we take to conserve ecologically valuable ecosystems? How can we create the greatest improvements in global food security and equality? The increasing impacts of climate change on Earth’s systems only serve to heighten the importance of, and difficulty in, answering these questions. Given the complexity of the systems -- trade networks, ecosystems, etc. -- to which these questions pertain, it is crucial that we gain a comprehensive understanding of their dynamics before taking action. Without this, any changes to these complex human-environment systems could have unintended and potentially calamitous effects. As such, the value of modelling techniques for exploring the dynamics and potential futures of these complex systems is high. This thesis uses models to examine the behaviour and possible future trajectories of 3 such systems. We begin by delving into the temporal evolution of the global wheat trade network using a dynamic network model. A preferential attachment mechanism is found to provide a good fit to the empirical network, based on several key metrics. Our modelled trade network is quite fragile to shocks. However, as it grows towards 2050, its resilience to attacks will increase. Next, we implement a spatially-explicit agent-based model for the forest-grassland mosaics of Southern Brazil. These ecologically valuable systems are fragile, with simulated mosaics persisting only over a narrow range of conditions. Mosaics may cease to exist in scenarios where climate change impacts greatly reduce fire-mediated recruitment thresholds. When climate change effects are less severe mosaics that do not disappear exhibit substantial alterations to their spatial structure. Finally, we explore the dynamics of a human metapopulation linked through a trade network. Centrality to the network is key to obtaining high food per capita, and differences in centrality may result in inequalities between patches. Inequalities and issues of food security can also arise when patch-level behaviours differ. Larger and more regular network structures facilitate more equal patch-level outcomes and higher levels of food security. However, when patch-level import behaviours are heterogeneous, the best course of action is to first modify these behaviours before adjusting the network topology. Across all 3 projects, modelled systems display complex behaviours. This emphasizes the necessity of further development of models for complex human-environment systems that can provide a more complete understanding of system dynamics and potential futures. The insights gained from these models can be used to inform policies for facilitating positive outcomes in real-world complex systems.
Spatial and Temporal Discounting in a Social-Climate Model
(University of Waterloo, 2024-01-25) Cameron, Mackenzie; Bauch, Chris; Anand, Madhur
This thesis analyzes how individuals' devaluation of distant impacts of climate change affects mitigation behaviours and projected climate conditions. To approach this question, spatial and temporal discounting is applied to a coupled social-climate model. This model represents a two-way feedback between human decision-making, social norms, and human behaviour with changes in the climate. This is achieved through coupling an evolutionary game theoretic model of opinion dynamics and a simple Earth System Model. The results showed that shifting from current-looking to future-looking behaviours (preferring lower discounting scenarios) and considering multiple locations and population groups, supports a higher proportion of the population choosing mitigation strategies. This shift produces a pathway to reducing temperature anomalies and carbon dioxide emissions. However, the approach to a better state of the climate is best achieved by targeting both discounting and social behaviours rather than just one or the other. These results highlight the benefits of including human behaviour in climate models and the need for a more multifaceted approach to mitigating the negative effects of climate change.
A spatially explicit modelling approach for predicting and managing the effects of coral reef stressors
(University of Waterloo, 2022-09-09) Milne, Russell; Bauch, Chris; Anand, Madhur
Coral reefs represent simultaneously one of the most beloved and one of the most endangered ecosystems in the world. Millions of people visit coral reefs every year for tourism purposes, and millions of people living in areas adjacent to reefs have reef fish as a key part of their diet, but both of these important services provided by reefs are under threat by anthropogenic stressors. These include overfishing, which is known to cause regime shifts to an equilibrium dominated by macroalgae instead of coral; nutrient loading, which facilitates greater nutrient uptake by macroalgae, allowing them to overgrow coral; sedimentation, in which coral and algae alike become smothered by particulate matter, leading to food web disruption; and many others such as ocean warming and acidification due to climate change. Outbreaks of crown-of-thorns starfish (CoTS), a fast-acting coral predator, are also projected to become more serious in the future, as more CoTS larvae survive at elevated nutrient concentrations. Due to the large number of reef organisms (including many coral species, macroalgae, and CoTS) that reproduce by dispersing their larvae into the ocean, the fact that sedimentation can be caused by soil erosion many miles inland, and the spatial variation in fishing pressure and nutrient input that arises from different human land use patterns, these reef stressors are explicitly spatial in nature. As the large spatial scales that coral reef dynamics take place on make comprehensive field studies expensive in money, time, and labour, determining optimal strategies for managing these stressors cannot rest on field work alone. In this thesis, we build and parametrize three spatially explicit mathematical coral reef models of intermediate complexity, and use these to produce ecological predictions and conservation recommendations for reefs with high levels of anthropogenic stress. We show that coral and herbivorous fish populations respond to fragmented habitats induced by overfishing in opposite ways, and that spillover from marine protected areas can sustain herbivorous fish populations even in heavily overfished areas. We demonstrate that local economic transitions from fishing to tourism can facilitate larger-scale recovery first of fish and subsequently of coral within approximately 30 years. We show that on reefs experiencing CoTS outbreaks, slight increases in fishing and nutrient loading rates could cause sharp transitions to states with less coral and continuous CoTS presence. We predict how future CoTS outbreaks would affect coral cover on reefs adjacent to two growing cities (Cebu City, Philippines and Jeddah, Saudi Arabia), and evaluate four strategies for CoTS management in these cities. We evaluate the resilience of reef fish in four different functional groups to sedimentation caused by deforestation, as well as the robustness of reef fisheries to decline in fish stock stemming from deforestation, and show that flexible harvesting strategies can mitigate this decline. Our work joins novel ecological theory with concrete recommendations for reef ecosystem management, and represents a substantial step forward for understanding marine spatial dynamics.

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