Science, Irving K. Barber Faculty of (Okanagan)
Computer Science, Mathematics, Physics and Statistics, Department of (Okanagan)
DSpace
UBCO
Wyse, Sarah Kathleen
Master of Science - MSc
University of British Columbia
It is known that human activity is driving extreme weather patterns and extreme events impact climate change mitigation behaviour (Demski et al., 2016; IPCC, 2021). However, few models allow for human behaviours and the climate to dynamically interact. The models presented in this thesis are an initial step to filling this gap. These models contain three components: a social model, a climate model, and a means to connect the two into a social-climate model. First, I study a social model to determine the conditions under which a committed minority can influence the greater population and overturn a pre-established social convention. I formulate a new tool, opinion response functions, that analytically computes the steady states of my social model and allows for a study of the bifurcation structure as the committed minority size changes. I find that a committed minority and memory bank length play a vital role in causing a population to tip and shifting public opinion. Second, I modify an existing climate model to include climatic variability. With this very simple climate model, I obtain results that are comparable with more complex models. Lastly, I connect my social and climate models in a feedback loop. My results demonstrate that the social-climate feedback loop may be important in accurately predicting future temperatures, in contrast to the standard approach where human behaviour is considered fixed. Additionally, I find that a sufficient number of extreme events, occurring over a sufficiently short time span, is needed to shift the population into action (i.e., cause the social convention to be overturned through a tipping event) and the time at which the tipping event occurs has a large impact on future temperatures.
The Role of Committed Minorities inClimate Change ActionQualitative Insights from a Social-Climate ModelbySarah Kathleen WyseB.Sc. Hons., The University of British Columbia, 2021A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE COLLEGE OF GRADUATE STUDIES(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)June 2024© Sarah Kathleen Wyse, 2024The following individuals certify that they have read, and recommendto the College of Graduate Studies for acceptance, a thesis/dissertation en-titled:The Role of Committed Minorities in Climate Change Actionsubmitted by Sarah Kathleen Wyse in partial fulfilment of the require-ments of the degree of Master of ScienceDr. Eric Foxall, Irving K. Barber Faculty of ScienceSupervisorDr. Rebecca Tyson, Irving K. Barber Faculty of ScienceCo-SupervisorDr. Bert Baumgaertner, University of IdahoSupervisory Committee MemberDr. John R.J. Thompson, Irving K. Barber Faculty of ScienceSupervisory Committee MemberDr. John Janmaat, Irving K. Barber Faculty of Arts and Social SciencesUniversity ExamineriiAbstractIt is known that human activity is driving extreme weather patterns andextreme events impact climate change mitigation behaviour (Demski et al.,2016; IPCC, 2021). However, few models allow for human behaviours andthe climate to dynamically interact. The models presented in this thesis arean initial step to filling this gap. These models contain three components: asocial model, a climate model, and a means to connect the two into a social-climate model. First, I study a social model to determine the conditionsunder which a committed minority can influence the greater populationand overturn a pre-established social convention. I formulate a new tool,opinion response functions, that analytically computes the steady statesof my social model and allows for a study of the bifurcation structure asthe committed minority size changes. I find that a committed minorityand memory bank length play a vital role in causing a population to tipand shifting public opinion. Second, I modify an existing climate model toinclude climatic variability. With this very simple climate model, I obtainresults that are comparable with more complex models. Lastly, I connect mysocial and climate models in a feedback loop. My results demonstrate thatthe social-climate feedback loop may be important in accurately predictingfuture temperatures, in contrast to the standard approach where humanbehaviour is considered fixed. Additionally, I find that a sufficient numberof extreme events, occurring over a sufficiently short time span, is neededto shift the population into action (i.e., cause the social convention to beoverturned through a tipping event) and the time at which the tipping eventoccurs has a large impact on future temperatures.iiiLay SummaryIt is well-established that human activity is a driving force for climatechange and that extreme weather events influence human behaviour. How-ever, few models allow for human behaviours and the climate to dynamicallyinteract. The models presented in this thesis address this gap and serveas an initial framework to expand on current models by using a dynamicsocial-climate feedback loop. First, I study a social model to determine theconditions under which a committed minority can overturn a pre-establishedsocial convention. Second, I modify a previous climate model to include cli-matic variability. Lastly, I formulate a social-climate feedback loop to studythe interplay between human behaviour and the climate. I find that a com-mitted minority plays a vital role in shifting public opinion towards climateaction and that the time at which the climate inaction social convention isoverturned has a large impact on future temperatures.ivPrefaceThe content in Chapter 2 is in preparation for submission to Journal ofMathematical Sociology.The code used in this thesis is run using Python 3.9.18 (Spyder 5.5.1) andis available at https://github.com/sarahwyse/SarahWyseMScThesis.git.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xiiChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . 11.1 Social Behaviours . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Climate Systems . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Social-Climate Interactions . . . . . . . . . . . . . . . . . . . 41.4 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 6Chapter 2: Social Models . . . . . . . . . . . . . . . . . . . . . . 82.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.1 General Framework . . . . . . . . . . . . . . . . . . . 92.2.2 Agent-Based Model . . . . . . . . . . . . . . . . . . . 112.2.3 Ordinary Differential Equation Models . . . . . . . . . 122.2.4 Opinion Response Functions . . . . . . . . . . . . . . 162.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.1 Agent-Based Model . . . . . . . . . . . . . . . . . . . 202.3.2 Ordinary Differential Equation Models . . . . . . . . . 21viTABLE OF CONTENTS2.3.3 Opinion Response Functions . . . . . . . . . . . . . . 232.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Chapter 3: Climate Model . . . . . . . . . . . . . . . . . . . . . 363.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2.1 Ordinary Differential Equation Models . . . . . . . . . 383.2.2 Stochastic Differential Equation Model . . . . . . . . . 403.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.3.1 Ordinary Differential Equation Model . . . . . . . . . 403.3.2 Stochastic Differential Equation Model . . . . . . . . . 413.3.3 Comparison to IPCC Projections . . . . . . . . . . . . 423.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Chapter 4: Social-Climate Model . . . . . . . . . . . . . . . . . 464.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Chapter 5: Conclusions . . . . . . . . . . . . . . . . . . . . . . . 59Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62viiList of TablesTable 2.1 Key terms used in our social model. . . . . . . . . . . 10Table 3.1 Variables and parameters used in the climate models. . 39Table 3.2 Projected CO2 emissions for each IPCC scenario. . . . 40Table 4.1 Count data from attribution studies on extreme weatherevents. Extreme events are categorized into three cat-egories: human influence found, human influence notfound, and inconclusive. Data from Pidcock and Mc-Sweeney (2022). . . . . . . . . . . . . . . . . . . . . . . 49viiiList of FiguresFigure 2.1 Compartmental diagram for the uncommitted com-partments of the M = 1 ODE model. The com-partments that speak opinion A are A and CM (notshown) and the compartment that speaks opinion Bis B. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Figure 2.2 Compartmental diagram for the uncommitted com-partments of the M = 2 ODE model. The compart-ments that speak only opinion A are AA and CM (notshown). The compartment that speaks only opinionB is BB. Compartments AB and BA speak bothopinions equally often. . . . . . . . . . . . . . . . . . 13Figure 2.3 Compartmental diagram for the uncommitted com-partments of the M = 2 with combined undecidedcompartments ODE model. Compartments AA andCM speak only opinion A, compartment BB speaksonly opinion B, and compartment U speaks opinionsA and B equally often. . . . . . . . . . . . . . . . . . 14Figure 2.4 Compartmental diagram for the uncommitted com-partments of the M = 3 ODE model. Compart-ments AAA,AAB,ABA,BAA, and CM speak opin-ionA and compartmentsBBB,BBA,BAB, andABBspeak opinion B. . . . . . . . . . . . . . . . . . . . . . 15Figure 2.5 Sample opinion response functions, ΨCM(r), for M =25 and the given CM. We note that ΨCM(r) = Φ(r)when CM = 0 and the tipping point for this modeloccurs at CM = 0.31. An identity line is provided toshow the fixed points of ΨCM(r). . . . . . . . . . . . . 19Figure 2.6 A plot of CM∗, the minimal committed minority sizerequired to overturn a social convention, versus M ,the memory bank length, from the ABM. . . . . . . . 20ixLIST OF FIGURESFigure 2.7 Sample time series for the ODE models with a trivialtipping point. Note that the legends are the samewithin each row. . . . . . . . . . . . . . . . . . . . . . 22Figure 2.8 Sample time series for the ODE models with a non-trivial tipping point. Note that the legends are thesame within each row. . . . . . . . . . . . . . . . . . . 24Figure 2.9 A plot of CM∗, the minimal committed minority sizerequired to overturn a social convention, versus M ,the memory bank length, from the opinion responsefunctions. . . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 2.10 Bifurcation diagrams for various values of M . Notethat all diagrams have the consensus on A opinionequilibrium (A only) for all CM and M = 2∗ is theM = 2 with combined undecided individuals case. . . 30Figure 3.1 Projections of global surface temperature change fromthe ODE model for the five IPCC climate scenarios. . 41Figure 3.2 Projections of global surface temperature change fromthe SDE model for the five IPCC climate scenarios.The solid lines show the average behaviour across10,000 simulations and the windows around each solidline are the 95% confidence intervals. . . . . . . . . . 42Figure 3.3 IPCC projections of global surface temperature changecompared to pre-industrial times. The windows aroundSSP1-2.6 and SSP3-7.0 show the “very likely ranges”.Image from IPCC (2021). . . . . . . . . . . . . . . . . 43Figure 4.1 An outline of our social-climate model feedback loop. 48Figure 4.2 Annual counts of extreme events, decaying cumula-tive sum (left), speaking rate of A, and global surfacetemperature (right) in the minimum final tempera-ture scenario. . . . . . . . . . . . . . . . . . . . . . . . 51Figure 4.3 Annual counts of extreme events, decaying cumula-tive sum (left), speaking rate of A, and global surfacetemperature (right) in the late social model tippingevent scenario. . . . . . . . . . . . . . . . . . . . . . . 52Figure 4.4 Annual counts of extreme events, decaying cumula-tive sum (left), speaking rate of A, and global surfacetemperature (right) in the maximum final tempera-ture scenario. . . . . . . . . . . . . . . . . . . . . . . . 52xLIST OF FIGURESFigure 4.5 Annual counts of extreme events, decaying cumula-tive sum (left), speaking rate of A, and global surfacetemperature (right) in the largest maximum temper-ature scenario. . . . . . . . . . . . . . . . . . . . . . . 53Figure 4.6 Left: A violin plot showing the 2100 temperatureprojections from those simulations wherein the so-cial convention is not overturned. Right: A heat mapshowing the 2100 temperature projections as a func-tion of the time the social convention is overturned.The colour indicates the number of simulations. . . . 54Figure 4.7 Projections of global surface temperature change fromour social-climate model. The solid lines show theaverage behaviour and the shading indicates the 95%confidence intervals. . . . . . . . . . . . . . . . . . . . 54xiAcknowledgementsEric and Rebecca: it’s impossible to put how much I have learned fromboth of you into words. On the academic side, thank you for your unendingsupport, pulling me out of rabbit holes as necessary, and reminding me tolook at the bigger picture. On the personal side, thank you for allowing metime to pursue teaching multiple times in the middle of my degree, and forsharing crafts and games and cats.Bert, Hansi, John, and John: thank you for sharing your modelingknowledge. My research would not be what it is without your expertise.Rollerskating adventure team, mountain feelings time, and Buddy’s bud-dies: thank you for being my rocks (in some cases The Boulder) and keepingme sane. There are few other people I’d rather skate over rocks, sit on rocksto chat, or collect math rocks with. You have all brought so much joy andchaos into my life.MBRG and QEL: I so enjoyed sharing a lab space with all of you andam thankful for all of our conversations and time together.My past, current, and future students: I didn’t think it was possible forme to enjoy math even more, you are so much of the reason that I do whatI do.xiiChapter 1IntroductionClimate change is intensifying at an increasing rate (IPCC, 2021). TheUnited Nations has set a goal to limit global warming to 2◦C above pre-industrial times in the Paris Agreement and hope to go even further to keepthis warming below 1.5◦C (UNFCCC, 2015). At the current warming of1.45◦C (WMO, 2024), it is becoming less and less likely that we will achieveeither goal. The current release of carbon is unprecedented (Zeebe et al.,2016) and human behaviour is driving climate change (IPCC, 2021). Fur-ther, climate change is causing increased frequency and severity of extremeweather patterns (IPCC, 2021). Clearly, climate action is necessary, but it isnot always clear what will shift a population into action and there are manyinteracting factors (von Borgstede et al., 2013; Brulle et al., 2012). In thefollowing sections, we explain some key factors related to social behaviour,climate systems, and interactions between the two, that can influence theprogression of climate change. We discuss the complementary roles of sim-ple/toy and complex/realistic models in generating insights. We then givean outline of the chapters to come.1.1 Social BehavioursPeople have a large influence on each other that is often underestimated(Bohns et al., 2013; Flynn and Lake, 2008). This influence can take manyforms, e.g., propagating social norms (Melnyk et al., 2021; Morris et al.,2015), asking for a favour (Flynn and Lake, 2008), or even encouragingunethical behaviour (Bohns et al., 2013). Social norms define the expectedbehaviours of people participating in society (McDonald and Crandall, 2015)and have immense influence on human behaviours such as participation bywomen in paid labour (Jayachandran, 2021), willingness to vaccinate (Grau-pensperger et al., 2021), decisions on which products to purchase and con-sume (Melnyk et al., 2021), and level of prejudice (McDonald and Crandall,2015). Studies that attempt to leverage social norms to elicit or stop a par-ticular behaviour are mixed: some studies find that marketing campaignsrelated to social norms are effective (Agostinelli et al., 1995; White et al.,11.1. SOCIAL BEHAVIOURS2019), some show no significant change in behaviour (Russell et al., 2005),and others observe a boomerang effect in which an undesired behaviour isincreased (Perkins and Berkowitz, 1986). Further studies have found thatmarketing campaigns that directly state whether a behaviour is sociallyapproved or disapproved eliminates this boomerang effect (Schultz et al.,2007).An individual’s ability to influence others is expanding through the useof social media (Confetto et al., 2023). Mass media can promote health-related preventative behaviours (Hong and Kim, 2019), such as changingbehavioural intentions to not consume sugary drinks (Boles et al., 2014),and increase sustainability behaviours (Confetto et al., 2023). On the otherhand, social media can spread undesired behaviours such as compulsive buy-ing (Maccarrone-Eaglen and Schofield, 2022), smoking (Roby et al., 2020),and sharing misinformation (An et al., 2023). Due to the massive amountof data that can be obtained from social media, it can be difficult to analyzethe data and study online opinion dynamics (Willaert et al., 2020). Re-searchers have proposed creating social media observatories to fill this gap(Willaert et al., 2020).Misinformation is the holding and spreading of false beliefs (Kuklinskiet al., 2000; O’Connor and Weatherall, 2019). The more often an individualencounters evidence supporting their1 false belief, the stronger that beliefbecomes (Kuklinski et al., 2000). We note that while the “evidence” mightnot actually support the belief, the individual may perceive the evidenceas supportive due to effects such as confirmation bias (Allahverdyan andGalstyan, 2014; Lord et al., 1979; Rabin and Schrag, 1999). It has beenfound that misinformation tends to spread faster than factual information(O’Connor andWeatherall, 2019) and can be extremely hard to correct (Jeritand Zhao, 2020; Zeng and Brennen, 2023). The internet, in particular, allowsfor the mass spread of unverified statements which can be further propagatedthrough echo chambers and polarization (Del Vicario et al., 2016). Factchecking can used to suppress misinformation, but might not be as usefulin non-democratic or non-Western societies (Zeng and Brennen, 2023) andthere is no consensus on methods used to effectively counter misinformation(Jerit and Zhao, 2020).Echo chambers form when like-minded people gather and echo eachother’s beliefs (Baumgaertner, 2014; Nguyen, 2018). Echo chambers canresult in amplified confidence in one’s beliefs (Baumgaertner, 2014) and a1We use the words “their” and “they” as gender-neutral pronouns that can be eithersingular or plural, based on the context.21.2. CLIMATE SYSTEMSdecreased exposure to opinion diversity (Kitchens et al., 2020; Ranalli andMalcom, 2023). While echo chambers may form for positive reasons suchas promoting climate action, one study argues that all echo chambers areinherently harmful due to the lack of opinion diversity and limited abilityto strengthen the reasons behind one’s beliefs (Ranalli and Malcom, 2023).Inequalities such as societal class can amplify these echo chambers and in-crease polarization between groups (Menard et al., 2021).Vested interests occur when an individual has a personal stake in a situ-ation (Crano and Prislin, 1995). An individual might form a vested interestfor various reasons such as, e.g., financial gain (Jiang and Ling, 2013; Steindland Jonas, 2015), non-tangible benefits (Siegel et al., 2019), or avoiding fu-ture consequences like the addition of a comprehensive exam to an academicprogram (Crano and Prislin, 1995) or increased extreme event risk (De Do-minicis et al., 2021). These vested interests can influence behaviour (Cranoand Prislin, 1995; Steindl and Jonas, 2015), increase misinformation throughthe misrepresentation of scientific results (Gierth and Bromme, 2020), andshape policy (Jiang and Ling, 2013). When one or more individuals whohold the same vested interest gather, they can form a committed minority.It has been found that committed minorities can have a profound impacton shifting public opinion and upsetting social norms (Bolderdijk and Jans,2021; Moscovici, 1980).1.2 Climate SystemsWhen we refer to the “climate” in this thesis, we are referring to theaverage weather over a long timescale, typically 30 years (WMO, 2024).Arrhenius (1896) was one of the first climate modelers and found that in-creasing CO2 in the atmosphere will lead to increased global average surfacetemperatures. In particular, he calculated that a doubling of the atmo-spheric CO2 concentration will result in 4◦C warming (Arrhenius, 1896).Since early climate models relied on numerical methods before the adventof mechanical computers, initial progress was slow. For example, in 1922,it took Lewis Fry Richardson six weeks to compute an eight hour weatherforecast (Richardson and Lynch, 2007). Computer simulation of the weatherbegan in 1950 when Jule Charney and a team of researchers used a 700kmresolution grid to produce 24 hour weather forecasts of the Earth (Charneyet al., 1950). It took 24 hours to obtain these forecasts. Building on pre-vious research, the first global circulation model was proposed in 1956 andproduced long term forecasts of seasonal weather patterns (Phillips, 1956).31.3. SOCIAL-CLIMATE INTERACTIONSIn 1988, the United Nations and World Meteorological Organization cre-ated the Intergovernmental Panel on Climate Change (IPCC) with the goalof providing the general public with a comprehensive summary of scientificknowledge on the current state of the world under climate change (Hick-man, 2018). The IPCC released their First Assessment Report in 1990which highlighted the importance of climate change mitigation and led to thecreation of the United Nations Framework Convention on Climate Change(UNFCCC) (IPCC, 1990). The UNFCCC was later involved in proposingthe Paris Agreement to set global goals for climate change mitigation (UN-FCCC, 2015). The IPCC has since released regular assessment reports andgreatly advanced climate modeling (IPCC, 2021).1.3 Social-Climate InteractionsThere are many ways in which people and the climate interact. For ex-ample, anthropogenic greenhouse gas emissions have a large impact on theclimate (IPCC, 2021) and extreme weather events impact climate changerisk perception (Bergquist et al., 2019) and human behaviour (Demski et al.,2016). To achieve climate change mitigation, it is necessary to persuade poli-cymakers and the general population to act (Fisher and Nasrin, 2020; Kempet al., 2022). In this section, we discuss possible challenges and methodsto increase concern about climate change and increase climate mitigationbehaviours.In terms of challenges in achieving climate action, it has been found thatinequality between social classes can increase polarization due to asymmet-ric climate impacts, e.g., a poor person may be less able to mitigate climateimpacts to their life (Menard et al., 2021). When polarization occurs, pop-ulations are less willing to work together to mitigate climate change and theprojected global maximum temperature anomalies increase (Menard et al.,2021). Another study finds that collective action can be difficult to achieve(Andersson et al., 2021; White et al., 2019) due to the low likelihood thatan individual will voluntarily change their behaviours without policy inter-vention (von Borgstede et al., 2013). Further, even when collective action issuccessful, it can be difficult to maintain climate mitigation behaviours whenthere are not immediate climatic improvements (Christensen and Rapeli,2020; White et al., 2019). In particular, it can take up to a decade for thetemperature increase due to greenhouse gas emissions to fully take effect(Ricke and Caldeira, 2014).There are many ways to participate in and encourage climate action such41.4. MODELLINGas, e.g., decreasing car use, lobbying policymakers, shifting social norms,or holding climate strikes (Beckage et al., 2018; von Borgstede et al., 2013;Fisher and Nasrin, 2020; White et al., 2019). It has been found that lobbyingpolitical officials can lead to increased climate policy; however, businessgroups, such as fossil fuel associations, can have greater financial resourcesto influence politicians to reduce or block climate policies (Fisher and Nasrin,2020). Further studies suggest that, to achieve widespread climate action,we must shift attitudes (von Borgstede et al., 2013) or social norms (Beckageet al., 2018; White et al., 2019). This work corresponds with psychologicalstudies on the strong influence people have on one another (Bohns et al.,2013; Flynn and Lake, 2008; Melnyk et al., 2021; Morris et al., 2015). Inparticular, elite cues (i.e., influence from high power individuals) can have asubstantial impact on changing concern over climate change (Brulle et al.,2012).On the other hand, experiencing an extreme event can influence an indi-vidual’s recognition of climate change, but was not found to have an effect onthe average risk perception (Brulle et al., 2012). It is possible that this resultis due to a lack of rapid attribution studies at the time (Oakes et al., 2021;Pidcock and McSweeney, 2022). One study suggests that better understand-ing the worst case climate change scenarios can lead to action (Kemp et al.,2022). Specifically, they note that knowledge of extreme cases is requiredto make informed decisions and manage risk. Kemp et al. (2022) find thateven if we meet all the goals outlined in the Paris agreement, their modelpredicts 2.1◦C warming by 2100. Another study finds that in order to limitglobal warming to 2◦C, peak greenhouse gas emissions must occur no laterthan 2015 (von Borgstede et al., 2013). We have missed this requirement(IPCC, 2021). Thus, the new goal becomes limiting global warming above2◦C as much as possible.1.4 ModellingEach of the above factors contribute to forming our social-climate systemon Earth. We note that this list is far from exhaustive. Understanding themechanisms that make up such a complex system can be difficult (Smaldino,2017). This difficulty is where modeling comes in. Models can be used tofulfill a variety of needs, e.g., predict future outcomes, explain phenomena,direct data collection, or investigate possible solutions to crises (Epstein,2008). To understand complex systems, modelers make assumptions to buildmodels that are approximations of reality yet can yield insights into system51.5. THESIS OUTLINEbehaviour (Epstein, 2008; Haefner, 2005; Smaldino, 2017; Wimsatt, 2007).The most accurate models are complex, since they can better represent re-ality. However, researchers must be careful in their modeling choices sinceover-complexity can lead to over-fitting (Lever et al., 2016) and models thatuse so much computational power and time that it takes weeks to makea forecast for the next several hours may not be useful as predictive tools(Richardson and Lynch, 2007). On the other hand, simple models might notbe the most accurate representation of reality, but they can serve other pur-poses. For example, a simple model can be a starting point to build a bettermodel, provide insights into behaviours of more complex models, determinebounds for plausible ranges of the true behaviour, or suggest extensions orrefinements of an established model (Smaldino, 2017; Wimsatt, 2007).1.5 Thesis OutlineIn this thesis, we use toy models to investigate the impact of a com-mitted minority on a social-climate system. In Chapter 2, we replicate theresults of the social agent-based model in Centola et al. (2018) and studythe equilibria using an ordinary differential equation framework. To fill thegap in formal analysis and understand the driving behaviours of this socialmodel, we then introduce a new tool, opinion response functions. Theseopinion response functions allow us to determine the equilibria analyticallyand provides rigour to this social model. In Chapter 3, we leverage previ-ous research from IPCC (2021) to determine the CO2 emissions that we useas input for an existing climate model. We modify this model to includeclimatic variability by transforming it into stochastic differential equations.In Chapter 4, we connect our social and climate models in a social-climatefeedback loop. We note that while it is known that humans and the climateinteract in a social-climate feedback loop (Bergquist et al., 2019; Demskiet al., 2016), few models include the impact of the changing climate on hu-man behaviour, let alone include human behaviours at all (Beckage et al.,2020; Long and Tyson, 2014). As such, the main aim of this thesis is topropose a social-climate feedback loop framework and expand on previoussocial-climate models. In particular, we use the speaking rate of climate ac-tion in the social model to determine the CO2 emissions fed into the climatemodel. Then, extreme events occur in the model, at a rate dependent onthe current global average temperature. These events change the prevalenceof the climate action opinion in the social model. We note that since we are61.5. THESIS OUTLINEusing toy models, our results are qualitative and we focus on the generalbehaviours of these models and the insights that they can provide.7Chapter 2Social Models2.1 BackgroundPeople are strongly influenced by the opinions and behaviours of thosearound them and often underestimate the strength of their influence onothers (Bohns et al., 2013; Flynn and Lake, 2008). In particular, it has beenfound that, in specific scenarios, committed minorities making up only 10%of the total population have the ability to upset a social convention (Xieet al., 2011). Opinion dynamics and the study of tipping points of socialconventions are active areas of research in mathematical modelling (Centolaet al., 2018; Galam and Jacobs, 2007; Granovetter, 1978). Applications ofthese models can be found in, e.g., linguistics (Baronchelli et al., 2006),vaccination decisions (Papst et al., 2022), and climate action (Constantinoet al., 2022).Individuals in these social models are often assumed to be willing tochange their opinions based on decision making processes, e.g., a cost-benefitanalysis (Granovetter, 1978), following the local majority (Galam and Ja-cobs, 2007), or considering an external influence such as mass media (Ander-sson et al., 2021). Conversely, an individual who is unwilling to change theiropinion is called “committed” or “inflexible”. Galam and Jacobs (2007)state that they are the first to study the impact of adding committed indi-viduals to an opinion dynamics model. For a finite population size, it hassince been found that when a committed minority is large enough to reacha critical mass, it can trigger a tipping event (Centola et al., 2018; Galamand Jacobs, 2007; Xie et al., 2011), i.e. an abrupt change in the majorityopinion.Opinion dynamics systems are often studied using an agent-based model(ABM) approach. ABMs consider agents interacting under a set of rules toinvestigate the behaviour of the overall system. Examples of such modelsare bounded confidence models (Cao et al., 2015; Deffuant et al., 2000; Gra-novetter, 1978), the Voter model (Holley and Liggett, 1975; Jedrzejewski andSznajd-Weron, 2018; Majmudar et al., 2019; Sood and Redner, 2005), andthe naming game (Baronchelli et al., 2006; Centola et al., 2018; Xie et al.,82.2. METHODS2011). Each of these models can be used to study how social conventionsare formed through coordination in pairs or small groups. These modelscan have pairwise interactions in which both individuals update their opin-ions (Cao et al., 2015; Deffuant et al., 2000) or they can define a speaking-listening interaction where only the listeners update their opinions (Centolaet al., 2018). Other models define interactions as occurring in groups ofthree where all of the agents take on the most frequent opinion in the group(Galam and Jacobs, 2007). Results from each of these model types showthat consensus of opinion can be achieved, but generally provide little formalanalysis of the model dynamics.Ordinary differential equations (ODEs) can be used to study the mean-field behaviour of ABMs (Baumgaertner et al., 2018; Foxall, 2018; Sood andRedner, 2005; Tessone et al., 2013). This approach can allow for quantifi-cation and parameterization of ABM behaviours such as average consensustime, steady states, and bifurcation structure. Some authors suggest thatODE modelling is especially useful when there is limited empirical data sinceODE models require fewer assumptions on parameter values as comparedto ABMs (van den Bergh et al., 2019).In this chapter, we aim to fill the gap in formal analysis by using ODEmodels and a new tool, namely, opinion response functions. Opinion re-sponse functions consider a speaking-listening interaction as two separateprocesses (i.e., a speaking process and a listening process). This decouplingof processes (Diekmann et al., 2003) allows us to more easily analyze theequilibria of the system.2.2 Methods2.2.1 General FrameworkThe general framework we adopt for each of our social models is basedon the agent-based opinion dynamics model from Centola et al. (2018). Wenote that the ABM we use is the same as the one they used. The keyterms and definitions are listed in Table 2.1. Here, we present a detaileddescription of the model.Individuals in the model can hold one of two opinions, A or B. The indi-viduals participate pairwise in one-sided interactions where one individual israndomly assigned as the speaker, making the other individual the listener.Each individual has a memory bank that is a string of memories of lengthM where each “memory” corresponds to one of the two available opinions.92.2. METHODSMemory banks are updated as follows. After each interaction, the lis-tener adds the speaker’s opinion to their memory bank in a first-in-first-out(FIFO) process. An individual’s memory bank is thus a finite history of theopinions they have heard most recently. In other words, each individual’smemory bank consists of the opinions spoken in that individual’s last Mlistening interactions.The opinion an individual holds and, when applicable, speaks, is thememory that appears most frequently in their memory bank. When M iseven, the frequencies of the A and B opinions can be equal. We assumethat these undecided individuals speak A and B equally often. The bulk ofthe population does not inherently favour one opinion over the other, andindividuals within this group are willing to update their memory banks and,thus, their opinions. We call this group the uncommitted population.Table 2.1: Key terms used in our social model.Term Variable DefinitionMemory An entry held by an individual,can be of type A or BMemory Bank An individual’s collection of memoriesMemory Bank Length M The number of memories in anindividual’s memory bankOpinion O The most frequent memory in anindividual’s memory bankSpeaking rate ri The speaking rate of opinion i inthe populationCommitted Minority CM A small group of individuals whorefuse to update their memorybanks and, thus, their opinionsSocial Convention The opinion held by the majorityof the populationConsensus The entire population holdingthe same opinionThe remainder of the population is the committed minority and we de-note the proportion of individuals in this group by CM ∈ [0, 1]. Theseindividuals hold only A memories in their memory banks, and do not up-date their memory banks or opinions. We aim to investigate the conditionsunder which this committed minority can overturn a pre-existing social con-vention, i.e., the opinion held by the majority of the population. As such,102.2. METHODSwe initialize the uncommitted individuals to only hold B in their memorybanks which means they also all hold opinion B. We say the social conven-tion is overturned when opinion B is no longer held by a majority of theuncommitted population. The minimal critical mass, C∗M, is the smallestproportion of the committed minority required to cause this tipping event.2.2.2 Agent-Based ModelHere we provide the details needed to rigorously define and simulate theABM as given in (Centola et al., 2018). We track the state of each agent us-ing a continuous-time Markov chain (CTMC) given by (X1,t, X2,t, . . . , XP,t)where Xi,t is the state of agent i at time t. The state space for each agentis S = {(x1, x2, . . . , xM ) : xi ∈ {A,B} ∀i} and the state space for the entirepopulation is SP where P is the total number of agents (i.e., the populationsize). Let the number of uncommitted agents who hold opinion A be PAand the number who hold opinion B be PB. When M is even, it is possiblefor some agents to have memory banks that contain an equal number of Aand B memories. These agents are undecided, and their number is denotedPU . Undecided agents speak A or B at each time step each with probabil-ity 1/2. An agent with state (x1, x2, . . . , xM ) updates to (A, x1, . . . , xM−1)at rate CM + (PA + 0.5PU )/P and updates to (B, x1, . . . , xM−1) at rate(PB + 0.5PU )/P .Simulations of the ABM are run as follows. We use the initial conditionsas described in Section 2.2.1. That is, the committed minority proportionof the population is CM and has memory banks full of opinion A, while theproportion of the rest of the population is 1 − CM and has memory banksfull of opinion B. In each time step, we perform P/2 interactions so thateach agent has, on average, one interaction (speaking or listening) per timestep. Each interaction consists of the following steps. First, we randomlychoose two agents and set one to be the speaker and the other to be thelistener. Second, we determine whether or not the listener is a committedagent and update the listener’s memory bank accordingly. Listeners in thecommitted minority do not update their memory bank, while listeners in theuncommitted population drop their oldest memory and insert the speaker’sopinion as the newest entry in their memory bank. Third, and finally, wecheck whether the new memory changes the listener’s opinion and updatetheir opinion accordingly. These three steps are repeated until all P/2 inter-actions are complete. This process is repeated until the simulation reachesa pre-determined end time.112.2. METHODS2.2.3 Ordinary Differential Equation ModelsWhen an ABM population is large and well-mixed, the mean-field be-haviour of the model can be understood using compartmental ODE modelswhere the rate of movement between compartments is based on the law ofmass action (Kurtz, 1976). Below, we develop the specific ODE modelscorresponding to our ABM model for M = 1, M = 2 (two versions), andM = 3. In this section, we normalize the total population to 1, i.e., the un-committed population has size 1−CM. We note that throughout this sectionwe use the notation X with subscripts to represent the proportion of thepopulation holding each memory bank arrangement. The number of lettersin the subscripts is the memory bank length, M , and the subscripts denotethe memory bank of the individuals in that compartment. For example,XABA is the proportion of individuals in compartment ABA.2.2.3.A M = 1 ModelIn the case where individuals hold one memory, there are two possiblestates for the uncommitted population. We call the corresponding compart-ments A and B, where each member of the population holds, respectfully,opinion A or opinion B. When M = 1, an individual’s memory bank andopinion are the same. Individuals only change their opinion when they hearthe opposite opinion. That is, the transition rate from A to B is XA + CMand the transition rate from B to A is XB. The movement between groupsis shown in Figure 2.1. The system of ODEs is:dXAdt= XB(XA + CM)−XAXB, (2.1a)dXBdt= XAXB −XB(XA + CM). (2.1b)Figure 2.1: Compartmental diagram for the uncommitted compartments oftheM = 1 ODE model. The compartments that speak opinion A are A andCM (not shown) and the compartment that speaks opinion B is B.122.2. METHODS2.2.3.B M = 2 ModelIn the M = 2 case of the model, there are four compartments (AA, AB,BA, andBB) and thus there are four equations. We note that compartmentsAB and BA are undecided and speak each opinion equally often. Thespeaking rate of opinion A (i.e., the “hears A” transition rate) is rA =XAA +12XAB +12XBA + CM and the speaking rate of opinion B (i.e., the“hears B” transition rate) is rB = XBB +12XAB +12XBA. Hence, the fullsystem of equations for the M = 2 model is:dXAAdt= rAXAB − rBXAA, (2.2a)dXABdt= rAXBB + rAXBA − (rA + rB)XAB, (2.2b)dXBAdt= rBXAA + rBXAB − (rA + rB)XBA, (2.2c)dXBBdt= rBXBA − rAXBB. (2.2d)We provide a compartmental diagram in Figure 2.2 that depicts the tran-sitions between compartments. We describe other possible choices for thespeaking behaviour of undecided individuals in Section 2.4. The case wherethese undecided groups are combined is discussed in Section 2.2.3.C.Figure 2.2: Compartmental diagram for the uncommitted compartments ofthe M = 2 ODE model. The compartments that speak only opinion A areAA and CM (not shown). The compartment that speaks only opinion B isBB. Compartments AB and BA speak both opinions equally often.132.2. METHODS2.2.3.C M = 2 with Combined Undecided Compartments ModelIn this variation, we assume that the AB and BA populations have thesame behaviour and combine types AB and BA into an “undecided” type,U . Moreover, we assume that U switches to whatever it hears next, i.e.,to AA if it hears A, and to BB if it hears B. This process differs fromthat used in the original M = 2 model above since, upon hearing A, ABtransitions to AA but BA transitions to AB. This assumption allows us tosimplify the previous M = 2 model to the compartmental diagram given inFigure 2.3 and the system of equations is:dXAAdt= rAXU − rBXAA, (2.3a)dXUdt= rBXAA + rAXBB − (rA + rB)XU , (2.3b)dXBBdt= rBXU − rAXBB, (2.3c)where the speaking rate of opinion A is rA = XAA +12XU + CM and thespeaking rate of opinion B is rB = XBB+12XU . The only difference betweenthis simplified M = 2 model and the original M = 2 model is the loss of theflow between the AB and BA compartments. The transition rates amongthe remaining compartments do not change as a result of the simplificationsince XU = XAB +XBA.Figure 2.3: Compartmental diagram for the uncommitted compartments ofthe M = 2 with combined undecided compartments ODE model. Compart-ments AA and CM speak only opinion A, compartment BB speaks onlyopinion B, and compartment U speaks opinions A and B equally often.2.2.3.D M = 3 ModelThe last ODE model we explicitly write is the model where individualscan hold 3 memories. A compartmental diagram to show the flow betweencompartments is given in Figure 2.4. The speaking rate of opinion A isrA = XAAA+XAAB+XABA+XBAA+CM and the speaking rate of opinion142.2. METHODSB is rB = XBBB + XBBA + XBAB + XABB. Hence, the full system ofequations for the M = 3 model is:dXAAAdt= rAXAAB − rBXAAA, (2.4a)dXAABdt= rAXABB + rAXABA − (rA + rB)XAAB, (2.4b)dXABAdt= rAXBAB + rAXBAA − (rA + rB)XABA, (2.4c)dXBAAdt= rBXAAA + rBXAAB − (rA + rB)XBAA, (2.4d)dXABBdt= rAXBBA + rAXBBB − (rA + rB)XABB, (2.4e)dXBABdt= rBXABA + rBXABB − (rA + rB)XBAB, (2.4f)dXBBAdt= rBXBAA + rBXBAB − (rA + rB)XBBA, (2.4g)dXBBBdt= rBXBBA − rAXBBB. (2.4h)Figure 2.4: Compartmental diagram for the uncommitted compartments ofthe M = 3 ODE model. Compartments AAA,AAB,ABA,BAA, and CMspeak opinion A and compartments BBB,BBA,BAB, and ABB speakopinion B.The ODE model becomes unwieldy for large M since 2M equations arerequired to describe all the possible permutations of memories that an indi-vidual can hold in their memory bank. We use opinion response functionsto study the deterministic behaviour for M = 1, 2, 3 as well as for M > 3.152.2. METHODS2.2.4 Opinion Response FunctionsIn this section we characterize the equilibria of the ODE model for anyM , using a tool we created called the opinion response function (ORF). Webegin with the case CM = 0, i.e., there is no committed minority. To definethe ORF for any of the ODE models, we view the opinion dynamics systemas a decoupled speaking-listening process, i.e., each interaction is interpretedas a speaking event followed by a separate listening event. When the systemis well-mixed and at equilibrium, the speaking rate and the listening rateof each opinion are equal and roughly constant throughout the populationand across time. In other words, at equilibrium, the average per-capita rateat which opinion A or B is spoken is, respectively, the same as the rates atwhich A or B is heard. We denote these listening rates by ri where i ∈ O,the opinion set; here, O = {A,B}. To define the speaking rate, consider theset S of possible memory bank states for each individual (which is differentfrom O whenM > 1) and let Xt denote the opinion state of an individual attime t. Then, if the listening rate is held constant, (Xt) is a Markov chainwith transition rates determined by ri. If this Markov chain is irreducible,it has a stationary distribution that we denote by πr. Let qi(x) denote theprobability that an individual in state x speaks opinion i. Then πr and(qi(x))i∈O,x∈S determine a map Φ = (Φi)i∈O defined byΦi(r) :=∑x∈Sπr(x)qi(x). (2.5)This function Φi is the ORF. We can read it as Φi(r) = E[qi(X(r))] whereX(r) is a random variable with distribution πr. That is, since ri is thelistening rate of opinion i, then Φi(r) defines the expected speaking rate ofopinion i. We note that Φi is vector-valued and is defined over Y → Y whereY is the subset of [0, 1]O whose entries sum to 1. Further, πr is a functionof M and thus Φi(r) varies as a function of memory bank length. Since wehave decoupled the speaking-listening process into two separate processes,the fixed-point equationri = Φi(r) (2.6)characterizes the equilibria of our models. That is, (ux)x∈S is an equilibriumof the ODE formulation of the model if and only if u = πr and ri :=∑x∈X uxqi(x) satisfies ri = Φi(r). In particular, when irreducible, there isat most one equilibrium for each choice of ri. By studying the properties ofΦi, we can determine the steady state behaviour of the ODE models for anyvalue of M .162.2. METHODSWe assume each individual hears opinions at a constant rate of 1, sothat∑i ri = 1. Here, (ri) is specified by either of rA or rB, i.e., there iseffectively one parameter r ∈ [0, 1] since the other parameter can be definedas 1 − r. Without loss of generality, we choose r = rA and can define Φ(r)from [0, 1]→ [0, 1] to be the average speaking rate of A, when the listeningrate of A is r. When A and B are interchangeable in distribution, we expectΦ(1 − r) = 1 − Φ(r) to hold and, in general, such functions are frequentlysigmoidal, i.e., Φ is increasing, Φ(0) = 0, Φ(1) = 1, and Φ is convex on[0, p1/2] and concave on [p1/2, 1], where pa is defined by Φ(pa) = a.The memory bank states are S = {(x1, x2, . . . , xM ) : xi ∈ {A,B}} andwe define N(x) =∑Mi=1 1(xi = A) where 1 is the indicator function. Thatis, N(x) is the number of A memories that an individual x holds in theirmemory bank. The probability of a given individual x speaking opinion AisqA(x) =1 if N(x) > M/2,1/2 if N(x) =M/2,0 otherwise.(2.7)The transition rate from state (x1, . . . , xM ) to state (i, x1, . . . , xM−1) isri. Additionally, the product measure πr withπr(x) = rN(x)(1− r)M−N(x) (2.8)is stationary. If X(r) has distribution π, then (1(Xi(r) = A) : i = 1, . . . ,M)are independent Bernoulli(r) trials so N(X(r)) is Binomial(M, r) andΦ(r) = P (N(X(r)) > M/2) +12P (N(X(r)) =M/2). (2.9)Altogether, Φ(r) describes the speaking rate of opinion A as a functionof r, the listening rate of opinion A. This function computes, for a givenhearing rate r, the probability that individuals in the population have amemory bank containing greater than or equal to M/2 memories of typeA. If the individuals have greater than M/2 memories of type A, thenthey speak opinion A in every speaking interaction. If the individuals haveexactly M/2 memories of type A, then they speak opinion A in half of theirspeaking interactions. The fixed points of Φ(r) (i.e., the roots of Φ(r) = r)are the same as the equilibria of the corresponding ODE model with thesame memory bank length, namely M , and no committed minority.If there is a committed minority of size CM that speaks only A, thenthe uncommitted population is size 1− CM. Each individual in the uncom-mitted population speaks A at a rate dependent on the number of type A172.2. METHODSmemories in their memory bank. Thus, the opinion response function witha committed minority isΨCM(r) = CM + (1− CM)Φ(r) = CM + (1− CM)∑x∈Xπr(x)qA(x) (2.10)since the speaking rate of A is CM plus the contribution from the rest of thepopulation, which is (1 − CM)Φ(r). The inclusion of CM has the effect of“pulling up” Φ towards 1 as CM is increased (Figure 2.5). As long as Φ isstrictly sigmoidal there is a value CM∗ > 0 such that for CM < CM∗, ΨCMhas three equilibria, for CM = CM∗ it has two equilibria, and for CM > CM∗it only has the equilibrium r = 1. That is, CM∗ is the location of a saddlenode bifurcation. We provide a proof for this claim in Section 2.3.3.In Section 2.3.3, we show that Φ(r) is a strictly sigmoidal function for allM ≥ 3. Since these opinion response functions characterize equilibria, bysolving for the roots of r = ΨCM(r), we can: obtain bifurcation diagrams forany M , compute the value of CM∗, and investigate the limiting behaviourof the system as M →∞.182.2. METHODS(a) CM = 0 (b) CM = 0.31(c) CM = 0.4Figure 2.5: Sample opinion response functions, ΨCM(r), for M = 25 andthe given CM. We note that ΨCM(r) = Φ(r) when CM = 0 and the tippingpoint for this model occurs at CM = 0.31. An identity line is provided toshow the fixed points of ΨCM(r).192.3. RESULTS2.3 Results2.3.1 Agent-Based ModelWe simulate the ABM to estimate the tipping point for various M usinga binary search algorithm and an error bound of 10/P (i.e., the tippingpoint is found within an accuracy of 10 agents). In order for our ABMresults to match those obtained using the ORF approach, we find that thenumber of agents and time steps must be sufficiently large. When thereare too few, we obtain tipping points that occur at a committed minoritysize smaller than CM∗. On the other hand, when CM > CM∗, too fewtime steps results in the model not having enough time to overturn thesocial convention. The larger CM is, the faster the social convention can beoverturned and the tipping only occurs for committed minority sizes largerthan CM∗. We find that 10,000 agents and 1000 time steps results in tippingpoints that have an error on the order of 0.001 (10/10,000) when comparedto the opinion response functions tipping points. The relationship betweenmemory bank length and committed minority size required to overturn thesocial convention is an increasing, concave function (Figure 2.6).Figure 2.6: A plot of CM∗, the minimal committed minority size required tooverturn a social convention, versus M , the memory bank length, from theABM.202.3. RESULTS2.3.2 Ordinary Differential Equation Models2.3.2.A M = 1We can simplify the M = 1 system of equations by setting the totalpopulation to one. After applying the substitution XB = 1−XA −CM andsimplifying, we obtain:dXAdt= CM(1−XA − CM). (2.11)This ODE is always non-negative (XA is non-decreasing) and is zero whenCM = 0 or 1−XA − CM = XB = 0. When there is no committed minority,the initial state is always stable. We note that the flow rates from A to Band from B to A in Equations 2.1 cancel each other, and there is only anet change if CM > 0. Additionally, when CM = 0, Equation 2.11 is alsozero. For a non-zero committed minority, the system always tends towardsconsensus on opinion A by the entire population. The general solution is:XA = 1− CM −De−CMt, (2.12a)XB = De−CMt, (2.12b)where D is a constant of integration. The B proportion of the population isexponentially decreasing and goes to zero as t→∞. Overall, we have foundthat when CM ̸= 0 and for any initial conditions, the committed minoritysucceeds in overturning the social convention. This result means the M = 1ODE model only has a trivial tipping point at CM = 0. Time series for thismodel (Figure 2.7a) show convergence to consensus on A when CM > 0.The only difference between different values of CM ∈ (0, 1] is the rate ofconvergence to consensus. In particular, larger values of CM result in fasterconvergence.2.3.2.B M = 2Using the initial condition (XAA, XAB, XBA, XBB) = (0, 0, 0, 1 − CM)and various values of CM ∈ [0, 1], we run time series of the M = 2 model.Sample time series are given in Figure 2.7b. Similar to the M = 1 ODEmodel, we find that when CM = 0, there is no movement between com-partments and the proportions of the uncommitted population remain thesame as in the initial conditions. When CM ∈ (0, 1], XBB displays exponen-tial decay. The undecided population proportions XAB and XBA initiallyincrease with XAB having a slightly larger proportion throughout the sim-ulation. Then, these undecided populations transition to compartment AA212.3. RESULTSand XAA increases until AA makes up the entirety of the uncommitted pop-ulation. As CM is increased, the rate of convergence increases and the thedegree of centering decreases, i.e., the undecided populations reach a smallermaximum.(a) M = 1 model for CM = 0 (left) and CM = 0.05 (right).(b) M = 2 model for CM = 0 (left) and CM = 0.05 (right). Note that XAA, XAB , andXBA all overlap when CM = 0.Figure 2.7: Sample time series for the ODE models with a trivial tippingpoint. Note that the legends are the same within each row.2.3.2.C M = 2 with Combined Undecided CompartmentsWe use the initial condition (XAA, XU , XBB) = (0, 0, 1 − CM) to runtime series for the M = 2 model with combined undecided compartments.Sample time series are given in Figure 2.8a. The results of this model differfrom the previous two ODE models since this model has a non-trivial tip-ping point. When CM is small, the social convention is not overturned andthe BB population remains dominant with a small increase in XU and an222.3. RESULTSeven smaller increase in XAA. However, when CM is large enough, there isa sharper increase in XU which increases the speaking rate of A and pro-vides enough force for the AA population to eventually overturn the socialconvention. Similar to the previous M = 2 case, increasing CM beyond thetipping point increases the rate of convergence to consensus on A and thecentering behaviour decreases.2.3.2.D M = 3Once again, we use the initial condition where the uncommitted pop-ulation consists of only BBB and all the other compartments are empty.Sample time series are given in Figure 2.8b. Similar to theM = 2 combinedundecided compartments model, there is a non-trivial tipping point. WhenCM is less than the tipping point the solution trajectories go to a coexistenceequilibrium with most of the uncommitted population still holding opinionB. The committed minority is not large enough to overturn the social con-vention and only a small proportion of the population holds opinion A. Onthe other hand, when CM is larger than the tipping point, the compartmentswith two B memories (e.g., ABB) increase which causes an increase in thecompartments with two A (e.g., ABA) memories which causes the AAApopulation to increase until all of the uncommitted population goes to theAAA compartment and there is a consensus on opinion A throughout thepopulation.Note that the overturning of the social convention appears to be slightlyfaster in the M = 3 model than in the M = 2 model with combined un-decided compartments. This difference in convergence rate occurs becausethe value of CM used for the M = 2 model with combined undecided com-partments is closer to the tipping point. In particular, values of CM justabove the tipping point have less force pushing the overturning of the socialconvention than values of CM much greater than the tipping point.2.3.3 Opinion Response FunctionsOn our way to proving Φ is a sigmoidal function, we start by provingΦ(1− r) = 1− Φ(r).Proof. We haveΦ(1− r) = P (N(X(1− r)) > M/2) + 12P (N(X(1− r)) =M/2)= P (N(X(r)) < M/2) +12P (N(X(r)) =M/2)232.3. RESULTS(a) M = 2 with combined undecided compartments model for CM = 0.07 (left) andCM = 0.08 (right).(b) M = 3 model for CM = 0.11 (left) and CM = 0.12 (right).Figure 2.8: Sample time series for the ODE models with a non-trivial tippingpoint. Note that the legends are the same within each row.242.3. RESULTS= 1−(P (N(X(r)) > M/2) +12P (N(X(r)) =M/2))= 1− Φ(r).Since N(X(r)) = 0 when r = 0, we have Φ(0) = 0. This result combinedwith Φ(1− r) = 1−Φ(r) gives us Φ(1) = 1. To prove Φ is convex on [0, pa]requires more work. We start with proving the M ≤ 3 cases are convex. Wealso include the M = 2 with combined undecided individuals case since itdoes not follow the typical Φ(r) representation.M = 1:Φ(r) = P (N(X(r)) = 1) = rThis function is increasing and trivially sigmoidal.M = 2:Φ(r) = P (N(X(r)) = 2) + 12P (N(X(r)) = 1) = rSame as M = 1.M = 2 with combined undecided individuals:Using Equation 2.7, we have qA(A) = 1, qA(U) =12 , and qA(B) = 0. Inaddition, the transitions A → U and U → B have rate rB and B → U ,U → A have rate rA. With r = rA and rB = 1−r, the stationary distributionsatisfies detailed balance, i.e., at equilibrium, the movement of individualsfrom one compartment to a second compartment is exactly balanced withindividuals moving in the reverse direction. So, (1− r)πr(A) = rπr(U) and(1− r)πr(U) = rπr(B). Using α = r/(1− r), we obtain πr(A) = απr(U) =α2πr(B) which we rewrite in terms of α to obtain the stationary distribution:(πr(A), πr(U), πr(B)) =11 + α+ α2(α2, α, 1).Then the speaking rate of opinion A isΦ(r) = πr(A) +12πr(U) =α2 + α/21 + α+ α2=r2 + r2(r2 − r + 1) .We haveΦ′(r) =−2r2 + 2r + 12(r2 − r + 1)2andΦ′′(r) =(−2r + 1)(−2r2 + r + 2)(r2 − r + 1)3 .252.3. RESULTSΦ′ is positive for r ∈ [0, 1] and is an increasing function. Further, Φ′′ ispositive for r ∈ [0, 1/2), so Φ is convex over [0, 1/2) and sigmoidal.M = 3:Φ(r) = P (N(X(r)) ≥ 2) = r3 + 3r2(1− r) = 3r2 − 2r3We then have Φ′(r) = 6r − 6r2 = 6r(1 − r) which is positive on (0, 1) andincreasing on [0, 1/2). Hence, together with symmetry, Φ is strictly convexon [0, 1/2) and a sigmoidal function.Now that we have shown Φ(r) is convex for certain cases, we show thatit is convex in general. Since Φ(1 − r) = 1 − Φ(r), showing Φ is convex on[0, 1/2] is equivalent to showing Φ is concave on [1/2, 1]. Throughout thefollowing proof, we also show that Φ is an increasing function. Altogether,we find that Φ is a sigmoidal function for all M ∈ N.Proof. Let’s start by considering oddM . In this case, Φ(r) = P (N(X(r)) >M/2). We’ll proceed by computing Φ′(r) but we’d like to do it without aton of algebra. To do so let’s represent X(r) by first fixing i.i.d. uniformrandom variables (ui)mi=1 and declaring that Xi(r) = A if and only if ui ≤ r.This way, if Xi(r) = A and s > r then Xi(s) = A. Further, if Xi(r) = Band Xi(r + h) = A then r < ui ≤ r + h. If N(X(r + h)) > M/2 andN(X(r)) ≤ M/2, up to an error of size O(h2), the most likely event forsmall h is that exactly one entry changed between r and r + h, i.e., thereis a set I ⊂ {1, . . . ,M} of size (M − 1)/2 and an element j /∈ I such thatui ≤ r for all i ∈ I, r < uj ≤ r + h and ui > r + h for all i /∈ I ∪ {j}. Theprobability of this event is(M(M − 1)/2)M + 12hr(M−1)/2(1− (r + h))(M−1)/2,since there are(M(M−1)/2)choices for I, (M + 1)/2 choices for j, and prob-ability hr(M−1)/2(1 − (r + h))(M−1)/2 that the entries are in the requiredintervals. Dividing by h and letting h→ 0, we obtainΦ′(r) =(M(M − 1)/2)M + 12r(M−1)/2(1− r)(M−1)/2,which is always positive for r ∈ (0, 1) (Φ is increasing) and is increasing on[0, 1/2), so Φ is sigmoidal for odd M .For even M , first note that Φ can be written as Φ(r) = 12P (N(X(r)) >M/2) + 12P (N(X(r)) ≥ M/2). Applying the same reasoning as before on262.3. RESULTSeach term separately, we findΦ′(r) =(MM/2− 1)(M/2+1)rM/2−1(1−r)M/2+(MM/2)M2rM/2(1−r)M/2−1.By symmetry, we haveΦ′(r) =(MM/2)M2(rM/2−1(1− r)M/2 + rM/2(1− r)M/2−1).Factoring out r and 1− r respectively, since r + 1− r = 1 we find thatΦ′(r) =(MM/2)M2rM/2−1(1− r)M/2−1which is, again, positive on (0, 1) and increasing on [0, 1/2). Altogether, Φis a sigmoidal function for all M .When CM = 0, we have ΨCM(r) = Φ(r) and the equilibria are character-ized as the roots of Φ(r) = r for r ∈ [0, 1]. Further, when Φ is non-triviallysigmoidal (M ≥ 3 and M = 2 with combined undecided individuals), thereare three roots, r = {0, 1/2, 1}. When CM = 1, we have ΨCM(r) = 1 and theonly root of ΨCM(r) = r is r = 1. So there exists at least one CM∗ ∈ (0, 1)where there is a bifurcation. We now prove that such a unique CM∗ existsand the bifurcation is a saddle node bifurcation when Φ(r) is non-triviallysigmoidal.Proof. We want to study the relation ΨCM(r) = r. Since both sides of thisrelation are polynomials, we know they are continuous and we defineF (r, CM) = ΨCM(r)− r = CM + (1− CM)Φ(r)− r.We know that in a saddle node bifurcation, the upper steady state decreaseswith a slope approaching −∞ and the lower steady state increases with aslope approaching +∞. Since our bifurcation occurs in r as a function ofCM, we want to study drdCM . Since F is a polynomial, we can take thederivative of both sides to obtaindF = 0 = dCM + (1− CM)Φ′(r)dr − Φ(r)dCM − drand rearranging givesdrdCM =1− Φ(r)1− (1− CM)Φ′(r) .272.3. RESULTSWe know that drdCM approaches ±∞ as 1 − (1 − CM)Φ′(r) approaches 0 oras Φ′(r) approaches 11−CM . Hence, we obtainCM∗ = 1− 1Φ′(r∗).This result means that two of the steady states collide at CM∗ and we havea saddle node bifurcation.Altogether, we have shown that Φ is non-trivially sigmoidal for M ≥ 3and we have a saddle bifurcation for M ≥ 3 and varying CM in ΨCM .There is a one-to-one correspondence between the equilibria of the ODEmodels and the fixed points of the opinion response functions. Hence, wecan use the roots (fixed points) of ΨCM(r) = r to find the critical value CM∗for various M (Figure 2.9) and plot bifurcation diagrams (Figure 2.10).We note that the tipping points found using opinion response functionsare analytically computed and closely match the estimates found using theABM. Additionally, using these figures, we can infer the behaviour for allmemory bank lengths. In particular, CM∗ increases with M towards somelimit that must be ≤ 1/2 as M →∞.Figure 2.9: A plot of CM∗, the minimal committed minority size required tooverturn a social convention, versus M , the memory bank length, from theopinion response functions.We also note that the values of CM∗ are grouped in pairs of M values(e.g., M ∈ {1, 2} has CM∗ = 0). This pattern suggests that increasing thememory bank length from oddM = 2L−1 to evenM = 2L does not change282.3. RESULTSthe roots of Ψ(r) = r for L ∈ N. In particular, P (N(X(r)) > Modd/2) whereModd = 2L − 1 is equal to P (N(X(r)) > Meven/2) + (1/2)P (N(X(r)) =Meven/2) where Meven = 2L. We now prove this equality.Proof. Assume that M is even. Then M − 1 is odd. We want to prove thatΦ(M, r) = Φ(M − 1, r). Let N(X(r)) have distribution Binomial(M, r),then by definitionΦ(M, r) = P (N(X(r)) > M/2) +12P (N(X(r)) =M/2).Let Y1, ..., YM be independent and identically distributed Bernoulli(r) sothat Z1 := Y1 + ... + YM has distribution Binomial(M, r), i.e., Φ(M, r) =P (Z1 > M/2)+12P (Z1 =M/2). If we randomly remove one of the Yi wherei ∈ {1, ...,M} from Z1, then we obtain a random variable Z2 with distri-bution Binomial(M − 1, r). More formally, let I be uniformly distributedon {1, ...,M} independent of (Yi)Mi=1, then Z2 :=∑i ̸=I Yi has distributionBinomial(M −1, r), i.e., Φ(M −1, r) = P (Z2 > (M −1)/2). There are threecases to consider:1. If Z1 > M/2, then since M is even Z1 − 1 > (M − 1)/2. Moreover,Z2 ≥ Z1 − 1, so Z2 > (M − 1)/2.2. Similar to 1., if Z1 < M/2, then Z1 − 1 < (M − 1)/2, and Z2 <(M − 1)/2.3. If Z1 = M/2, then P (YI = 1) = 1/2. So, with equal probability weremove either a 0 or a 1 from Z1 to obtain Z2. Since Z1 > (M−1)/2 >Z1 − 1, we obtain P (Z2 > (M − 1)/2) = 1/2.Altogether, P (Z2 > (M − 1)/2) = P (Z1 > M/2) + 12P (Z1 =M/2).In order to determine limM→∞ CM∗, we must study the limiting be-haviour of the model as M → ∞. We know that limM→∞ P (N(X(r)) =M/2) = 0, so Φ(r) = P (N(X(r)) > M/2) for both the odd and even cases.Additionally, as M becomes large, we can approximate this binomial dis-tribution with a normal distribution that has a width of 1/√M . WhenM → ∞, the width of our distribution goes to 0, meaning the cumulativedensity function is 0 up until r = 1/2 and 1 after r = 1/2. By definition,the cumulative density function is the same as Φ(r) and we havelimM→∞Φ(r) =0 if r < 1/2,[0, 1] if r = 1/2,1 if r > 1/2.(2.13)292.4. DISCUSSIONSince we have defined limM→∞Φ(r), we can also compute thatlimM→∞ΨCM(r) =CM if r < 1/2,[CM, 1] if r = 1/2,1 if r > 1/2.(2.14)Hence, we can define the behaviour of Ψ(r) asM →∞. When CM < 1/2,we have three roots of ΨCM(r) = CM + (1− CM)Φ(r) = r. These roots are{CM, 1/2, 1}. When CM = 1/2, we have roots {1/2, 1}. When CM > 1/2,we only have the consensus on opinion A root (r = 1). Hence, the tippingpoint CM∗ approaches 1/2 as M → ∞. The behaviour of the equilibriafollow a trend where, as M increases, CM∗ increases and the shape of thebifurcation diagrams becomes more triangular (Figure 2.10).Figure 2.10: Bifurcation diagrams for various values of M . Note that alldiagrams have the consensus on A opinion equilibrium (A only) for all CMand M = 2∗ is the M = 2 with combined undecided individuals case.2.4 DiscussionStudying equilibria and, if applicable, bifurcations of mathematical mod-els can provide a framework for the expected deterministic model behaviour.Diekmann et al. (2003) introduce a method to decouple model mechanismsand compute the equilibria of a model. The authors apply their frameworkto a predator-prey model where they decouple the populations from theirrespective environments. In particular, the prey are part of the predator’s302.4. DISCUSSIONenvironment and vice versa. Making this distinction between population andenvironment allows the authors to more easily study equilibria and bifurca-tion structure. In this work, we leverage this idea to study the deterministicbehaviour of our opinion dynamics model when the ABM or ODE forms areunwieldy.Our models provide similar qualitative results to previous papers thatstudy the naming game, even though our individuals interact following dif-ferent rules (Baronchelli et al., 2006; Xie et al., 2011). In particular, ourM = 2 with combined undecided individuals case has a tipping point simi-lar to that of the complete graph model in Xie et al. (2011). Additionally,our results compare well with those from Centola et al. (2018) who use thesame interaction rules as our models. While our result appears differentfrom their non-trivial tipping point when M = 2, this apparent discrepancyis an artifact of their choice of a larger error bound. Saddle node bifurcationsappear in various opinion dynamics models and our M ≥ 3 models have thesame qualitative behaviour as the work in Galam and Jacobs (2007) whouse a third distinct set of rules for agent interaction.It is instructive to compare the behaviour of our model with the Votermodel, which is a foundational model for the mathematical study of opiniondynamics (Holley and Liggett, 1975). The Voter model considers a two-state (i.e., two possible opinions) model where agents switch to the opinionopposite the social convention with probability p if they have a neighbourholding the opposing opinion (Holley and Liggett, 1975). The authors findthat consensus on the social convention is an absorbing state if p < 0.31 ona square lattice. The M ∈ {1, 2} cases of our social models have the samemodel set-up and behaviour where p = 0.5 when CM = 0. We also find thatthe consensus on the social convention state is not absorbing. Mobilia et al.(2007) studies the Voter model with committed minorities and finds thatcommitted minorities of equal size and opposite opinion prevent consensus.On the other hand, different sized committed minorities result in an overallpopulation shift towards agreement with the larger of the committed mi-norities (Biswas and Sen, 2009; Mobilia et al., 2007). This result matchesthe overturning social convention behaviour in ourM ∈ {1, 2} social modelswhen CM > 0.In our M ∈ {2, 3} ODE models, we note that the moderate opiniongroups (i.e., those not holding all A or all B in their memory banks) initiallyincrease when the tipping point is crossed and then decrease as the consensuson A behaviour takes over. This initial behaviour is called “centering” andhas been found in other opinion dynamics models (Baumgaertner et al.,2018, 2016; Iacopini et al., 2022). In Baumgaertner et al. (2018, 2016), the312.4. DISCUSSIONauthors assume that an interaction between two agents holding the sameopinion will result in amplification of that opinion. When the probabilityof this amplification increases, they find that the system reaches consensusfaster and there is a smaller degree of centering. This behaviour is the sameas our ODE models when we increase CM. While studies of the naming gamewith more than two individuals in each interaction also find this centeringbehaviour, but in some cases the intermediate populations remain and donot decay (Iacopini et al., 2022).In Section 2.2.3 we mention that the ODE M = 2 models (with andwithout combined undecided compartments) are defined similarly and theonly difference between the two models is the exclusion/inclusion of the flowbetween the AB and BA compartments. However, the resulting dynamicsof the models are very different. In particular, the M = 2 model only hasa trivial tipping point, while the M = 2 with combine undecided compart-ments has a non-trivial tipping point. Using opinion response functions inSection 2.3.3, we find that when the undecided compartments are combinedΦ(r) = (r2 + r)/2(r2 − r + 1) which is a strictly sigmoidal function, unlikewhen these groups are not combined and Φ(r) = r. This difference in func-tional form results in the different tipping point behaviour. A reason forthis difference in functional form is the loss of movement between the ABand BA populations (i.e., the interactions where AB switches to BA andvice versa). In theM = 2 model, these switches between AB and BA canceleach other out and Φ(r) = r for all r. However, when the undecided com-partments are combined, Φ(r) < r when r < 0.5. This decreased speakingrate slows down the shift towards the consensus on A state and the resultis a non-trivial tipping point.Tipping points can also occur in social models as functions of variablesother than committed minority size. For example, Andersson et al. (2021)study a bounded confidence model with an external field that representsfactors such as government policy, or mass media. The agents in this modelhold an opinion to cooperate with or defect from the other agents. Theauthors find that increasing the external field causes a tipping event wherebelow the tipping point, agents are more likely to defect and above thetipping point, agents are more likely to cooperate. A similar cooperationversus defection model is studied in Nowak et al. (2004). In this model,the population size can determine whether a committed agent is capable ofupsetting the defection social convention to achieve consensus on coopera-tion. In particular, the authors note that it is difficult to reach the minimumcritical mass in large populations and spite plays a role in small populations(i.e., there are not enough others agents to cooperate with and offset the322.4. DISCUSSIONdisadvantage from helping another individual).Our current social models represent a simplification of the real world, yetthey allow us to study the underlying mechanisms of more complex opiniondynamics systems (Epstein, 2008; Smaldino, 2017). We assume that allindividuals in our models behave the same and have the same characteristicsand willingness to update their memory banks and possibly change theiropinions. Bounded confidence models often have a set threshold that isthe same for every individual and determines whether or not this individualinteracts with another specific person (Deffuant et al., 2000; Weisbuch et al.,2002). On the other hand, some models have studied the dynamics whenthis threshold varies between individuals in the population (Granovetter,1978). It is also known that individuals hold different numbers of memoriesbased on factors such as age (Small, 2001) or sometimes sex (Asperholmet al., 2020).In this thesis, we use “listening” and “hearing” interchangeably since thelisteners in our models always listen to and take in the opinions that theyhear. Previous models define a threshold where individuals do not listen toeach other if their opinion difference is greater than the threshold (Deffuantet al., 2000; Weisbuch et al., 2002). In these models, high thresholds can re-sult in consensus and low thresholds can result in multiple polarized groups.Opinion thresholds can be used to study confirmation bias (Allahverdyanand Galstyan, 2014) or the backfire effect (Chen et al., 2021). Confirmationbias (i.e., the tendency to acquire information that aligns with an individ-ual’s current beliefs) can result in overconfidence (Rabin and Schrag, 1999)or viewing a neutral statement as confirmation of their beliefs (Lord et al.,1979). While the backfire effect (i.e., individuals that become more en-trenched in their opinion if they hear the opposing opinion) can producepolarization or consensus depending on the level of opinion entrenchment(Chen et al., 2021). An extension on the current work is formulating a rulewhere individuals or do not listen when their memory banks are too differentfrom each other.Our choice to have undecided individuals remain undecided and “flip acoin” when they speak is commonly used in modelling (Balenzuela et al.,2015; Centola et al., 2018; Xie et al., 2011). There are a few additional wayswe could have chosen to determine the behaviour of undecided individuals.Following recency effects, an individual might weigh recent memories as moreimportant (Cromwell, 1950). On the other hand, an individual might weigholder memories as more important (i.e., primacy effect) if reinforcementlearning or the anchoring effect plays a larger role (Knower, 1936; Lund,1925). Lastly, an individual may choose to not speak when undecided or a332.4. DISCUSSIONmodeler may choose to not allow undecided individuals exist (Baumgaertneret al., 2016). Since we found different model behavior for the M = 2 modelswhen we combined the undecided compartments, we hypothesize that thechoice of undecided speaking behaviour can have a strong effect on modeldynamics and warrants further study.Our model assumes that individuals hold opinions A and B with thesame intensity and that uncommitted individuals are always willing to up-date their memory banks. Other models have studied the system dynamicswhere, instead of holding memories, agents hold their opinions with differentstrengths (Baumgaertner et al., 2016). The authors find that when agentsare allowed to increase their opinion strength and become stubborn, the re-sult is polarization of opinions, i.e., a group of each opinion forms and thereis no consensus on either opinion. In our social models, the proportion ofeach memory that is held in an individual’s memory bank acts as a proxy foropinion strength. Our results differ from Baumgaertner et al. (2016) sinceour committed minority prevents polarization and there exist intermediateopinions (when the social convention is not overturned) or there is consensuson opinion A (when the social convention is overturned).In our models, we assume the system is well-mixed and that every in-dividual is able to interact with any other individual. Other models useincomplete graphs to model opinion dynamics systems when it is not guar-anteed that every individual can interact with every other individual (Mar-iano et al., 2020). These incomplete graphs can represent echo chambersand the authors find model dynamics result in polarization. More complexnetwork models can use asymmetric graphs (Su et al., 2021; Talebi et al.,2024). In such cases, there exist pairs of individuals where one individualcan speak their opinion to another individual, but the other individual isless able to or cannot speak back due to, e.g., lack of influence (Talebi et al.,2024), disability (Liu et al., 2018), or social status (Hasani-Mavriqi et al.,2016; Su et al., 2021). It has been found that the overall behaviour of theseasymmetric systems becomes more similar to the initial behaviour of highlyinfluential or leader individuals (Hosseini et al., 2018; Talebi et al., 2024).There are various other extensions to our social models. A model couldconsider more than two possible opinions. It has been found that, whileconsensus on a single opinion is still possible, the time it takes to reachconsensus increases in a saturating manner as a function of the number ofpossible opinions (Starnini et al., 2012). We could choose to let individualsinteract in groups instead of pairs. One paper studies a model similar to ourM = 2 with combined undecided compartments model and finds that a smallcommitted minority (∼ 0.3% of the population) is capable of overturning342.4. DISCUSSIONa social convention when individuals interact in groups of arbitrary size(Iacopini et al., 2022). Assuming there are multiple committed minoritiescould reveal insights on human behaviour when there are vested interests onboth sides of an argument. A previous study compares the proportion of thepopulation holding opinion A at time t versus time t+ 1 with a committedminority on both sides (Galam and Jacobs, 2007). They produce a plotsimilar to our opinion response functions that shows the same pulling uptowards 1 behaviour caused by an increasing committed minority holdingopinion A (see Figure 2.5). However, the authors also include a committedminority holding opinion B. The result is a pulling down behaviour towardsΨC(r) = 0. If the committed minorities are of the same size, consensusof the uncommitted population on one opinion is not possible. When oneof the committed minorities is slightly larger than the other, the result isthe majority of the population holding the opinion of the larger committedminority. Since the authors do not produce a bifurcation diagram for thecase where one committed minority is larger than the other, it is unclear ifthis is a reversible tipping point.We find that tipping points in our social models occur as CM increasesand are a function of memory bank length. In particular, the longer thememory bank, the larger the committed minority must be to cause a tippingevent. When M ∈ {1, 2}, we obtain CM∗ = 0 which represents a lowerbound for the committed minority size required to cause the overturningof a social convention in the real world. We obtain CM∗ = 0 as an upperbound when M → ∞. Other studies suggest that the tipping point occursat CM = 0.25 (Centola et al., 2018) or CM = 0.1 (Xie et al., 2011) in thecase of the naming game when individuals are less likely to hold vestedinterests. It has been found that vested interests influence an individual’sbehaviour (Crano and Prislin, 1995; De Dominicis et al., 2021) and candrive human behaviour more than, e.g., adaption considerations in the faceof climate change (Gussmann and Hinkel, 2021). Altogether, depending onthe memory bank length and the investment in the opinion held, we canexpect tipping points to occur for some CM ∈ [0, 0.5].35Chapter 3Climate Model3.1 BackgroundAt the current warming of 1.45 degrees Celsius (WMO, 2024), we areexperiencing extreme heat waves, flooding, fires, and storms (IPCC, 2021).Studies suggest that the current rate of carbon release due to human be-haviours is unprecedented (Zeebe et al., 2016) and impacts may be felt untilat least the year 2500 (Lyon et al., 2021; Meinshausen et al., 2020). TheParis Agreement sets a goal of keeping temperature below 2◦C above pre-industrial times and aims to limit this increase to 1.5◦C (UNFCCC, 2015).Current mitigation strategies are insufficient to limit warming to 2◦C (Leachet al., 2018; Roelfsema et al., 2020) and even meeting these goals is unlikelyto prevent sea level rise of at least 1.5 metres by 2300 (Mengel et al., 2018)or save coral reefs (Frieler et al., 2012).A comparatively simple way of modelling climate systems is using ordi-nary differential equations (ODEs). Examples of such ODE models includeenergy balance models (Budyko, 1969; Geoffroy et al., 2013), soil carbon-temperature models (Luke and Cox, 2010), and global CO2 models (McHughet al., 2015). These models assume that the Earth is is a well-mixed sys-tem, i.e., they assume that the climate is the same across the Earth. Somemodels divide Earth into compartments (e.g., lower vs. upper atmosphere)with the assumption that these compartments may behave differently fromeach other (McHugh et al., 2015). While it is known that different partsof the Earth are impacted differently by climate change (Lyon et al., 2021;Rantanen et al., 2022; Wang et al., 2022), in which case partial differentialequation models may be more suitable (Gordon et al., 2000; Hasselmann,1976), ODE models allow us to study the average behaviour and gain anunderstanding of the mechanisms that may be driving more complex models.The Intergovernmental Panel on Climate Change (IPCC) was created toassess and further research on climate change through regular assessmentreports (IPCC, 2021). The IPCC uses a variety of models, from simple en-ergy balance models to highly complex Earth system models (Flato et al.,2014). To include the impact of human behaviours in their models, they363.2. MODELSstudy five “high priority” climate scenarios under the shared socio-economicpathways (SSPs) framework (Meinshausen et al., 2020). The five scenariosthey consider are SSP1-1.9, SSP1-2.6, SSP2-4.5, SSP3-7.0, and SSP5-8.5.The first number represents the SSP family where SSP1 is called sustain-able development, SSP2 is middle-of-the-road development, SSP3 is regionalrivalry, SSP4 is inequality, and SSP5 is fossil-fuel development. The rest ofthe numbers describe the radiative forcing level by 2100 given in W/m2.For example, SSP1-1.9 is in the sustainable development family and pre-dicts 1.9W/m2 radiative forcing in 2100. SSP1-1.9 represents the best-casescenario where we reach net-zero CO2 emissions by 2050, and SSP5-8.5 rep-resents the worst-case of these five scenarios.Many climate models describe stochasticity using a white noise processthat some researchers call a “mathematical idealization” (van der Bolt et al.,2018). There has since been a shift towards using pink or red noise, i.e., au-toregressive noise, to account for temporal and spatial autocorrelation inthe climate (Di Cecco and Gouhier, 2018). In particular, studies have foundthat temporal autocorrelation, or climate memory, is increasing over timewhich corresponds to a reddening of climate stochasticity for various climatemetrics (van der Bolt et al., 2018; Di Cecco and Gouhier, 2018). Increasedautocorrelation means the climate is more likely to persist in its currentstate. If this current state involves poor environmental conditions, the cli-mate system may cross a tipping point which may be irreversible (van derBolt et al., 2018). One study found that the duration of extreme weatherevents increases as a function of the autocorrelation in the system (van derBolt et al., 2018). On the other hand, a system with less temporal auto-correlation may cross a tipping point during poor environmental conditions,but then the conditions may improve fast enough for the tipping event tobe reversed, and the worst of the impacts may be avoided. This pink or rednoise can be included in climate models through the use of stochastic dif-ferential equations (SDEs) such as the Ornstein-Uhlenbeck process (Benthand Sˇaltyte˙-Benth, 2005; Moreles and Mart´ınez-Lo´pez, 2016) or autoregres-sive models (Lin et al., 2023; Mustin et al., 2013). The Ornstein-Uhlenbeckprocess and autogressive models are both mean-reverting stochastic models.3.2 ModelsHere, we introduce an existing energy-balance ODE climate model. Fol-lowing Tung (2007), we then simplify this model and extend it to include theeffect of greenhouse gas (GHG) emissions. Lastly, we introduce stochasticity373.2. MODELSby modifying the model into a SDE form.3.2.1 Ordinary Differential Equation ModelsA commonly used non-spatial energy-balance model for Earth describesenergy entering the system via absorbed solar radiation (ASR) and en-ergy leaving the system via outgoing longwave radiation (OLR). Multiplyingthese energy terms by the specific heat capacity of water, C, produces anequation that models the temperature of the Earth’s surface over time,CdTdt=S(1− α)4− τσT 4, (3.1)where S(1 − α)/4 is the ASR and −τσT 4 is the OLR (Tung, 2007). Here,τ is the transmissivity of the atmosphere and σ is the Stefan-Boltzmannconstant. The rest of the parameters are described in Table 3.1.This model, however, does not account for the impact of GHG emissionson parameters such as the transmissivity of the atmosphere. In particular, asGHGs increase, the atmospheric transmissivity decreases (Rose and Scott-Brown, 2024). To include the impact of GHG emissions in our model, wefollow Tung (2007) and linearize the OLR term to obtain:CdTdt=S(1− α)4− (η − βT ) + a log CO2CO2,P I. (3.2)The ASR term remains the same as in Equation 3.1. The term η − βTrepresents energy lost in outgoing radiation where both η and β are climatefeedback parameters. Different sources provide different values for η andβ, and these values change across the globe (Graves et al., 1993; Meieret al., 2020; Tung, 2007). We use values from Graves et al. (1993) basedon empirical data. The term a log (CO2/CO2,P I) describes the temperatureforcing from increased CO2 emissions as compared to pre-industrial levels.In particular, CO2 is the current atmospheric concentration of CO2 in partsper million (ppm). However, the data we use are CO2 emissions projections(Table 3.2, IPCC (2021)) which are given in gigatonnes per year once adecade. First, we linearize between the given values to obtain annual CO2emissions in gigatonnes per year for each year (IPCC, 2021). Then, weconvert the data to units of ppm. Lastly, we compute the current CO2concentration by summing the 2020 CO2 concentration with the emissionsup to the current time step.383.2.MODELSTable 3.1: Variables and parameters used in the climate models. Note that while C is given in unit of seconds(J=W*seconds), we make a conversion to years to match the IPCC CO2 emission projections.Parameter Definition Value Units SourceT temperature variable ◦CTPI pre-industrial temperature 13.7◦C (Lenssen et al., 2019)T2020 2020 temperature 14.9◦C (IPCC, 2021)t time variable yearsC specific heat capacity of ocean water 3.985× 106 J/m2◦C Cox and Smith (1959)to a depth of 50 mS solar radiation 1372 W/m2 Tung (2007)α albedo 0.33 unitless Tung (2007)η climate feedback parameter 202 W/m2 Tung (2007)β climate feedback parameter 1.9 W/m2◦C Tung (2007)a CO2 forcing coefficient 5.35 W/m2 Myhre et al. (1998)CO2 projected CO2 atmospheric concentration variable Gt/year IPCC (2021)CO2,P I pre-industrial CO2 atmospheric 280 ppm Gulev et al. (2021)concentrationCO2,2020 2020 CO2 atmospheric concentration 410 ppm Gulev et al. (2021)393.3. RESULTSTable 3.2: Projected CO2 emissions for each IPCC scenario given in 10 yearincrements. Data from IPCC (2021).Year SSP1-1.9 SSP1-2.6 SSP2-4.5 SSP3-7.0 SSP5-8.52020 39.69 39.80 40.65 44.81 43.712030 22.85 34.73 43.48 52.85 55.302040 10.48 26.51 44.25 58.50 68.782050 2.05 17.96 43.46 62.90 83.302060 -1.53 10.53 40.20 66.57 100.342070 -4.48 4.48 35.24 70.04 116.812080 -7.31 -3.29 26.84 73.41 129.652090 -10.57 -8.39 16.32 77.80 130.582100 -13.89 -8.62 9.68 82.73 126.293.2.2 Stochastic Differential Equation ModelTo include climatic variation in our model, we modify Model 3.2 byadding a stochastic term as follows:dT =(S(1− α)4− (η − βT ) + a log CO2CO2,P I)dtC+ dW, (3.3a)dW = −θWdt+ σdB. (3.3b)Equation (3.3b) defines an Ornstein-Uhlenbeck process W . Here, θ = 1/5corresponds to a mean-reverting time of 5 years and σ = 1 (Singh, 2023).Since 0 < θ < ∞, our SDE formulation is an energy balance model withadditive pink noise (Mustin et al., 2013). In this model, dB =√dtZ is aGaussian random variable where Z ∼ N(0, 1).3.3 Results3.3.1 Ordinary Differential Equation ModelThe initial conditions we use for Model 3.2 are obtained from tempera-ture and CO2 concentration data from 2020 (Gulev et al., 2021; IPCC, 2021).We use projections of future CO2 emissions (Table 3.2) from the IPCC tocalculate projections for CO2 concentration levels for the five IPCC climatescenarios (IPCC, 2021). Since these projections were only run until 2100,our simulations are also restricted to end in year 2100. We solve the system403.3. RESULTSusing Euler’s method (solution trajectories given in Figure 3.1). The firstten years of the temperature change projections are about the same acrossthe five climate scenarios. As time passes, differences in emissions levelshave an effect. Scenarios SSP2-4.5, SSP3-7.0, and SSP5-8.5 show increasingtemperatures until 2100 with the worst case scenario (SSP5-8.5) reachinga maximum temperature change of 5.36◦C by 2100. In the best case sce-nario (SSP1-1.9), the maximum temperature change is 2.46◦C and occursin 2059. After that date, the global surface temperature starts decreasinguntil it reaches 2.25◦C by 2100. Scenario SSP1-2.6 has similar behaviourto SSP1-1.9 though the maximum temperature is greater (2.86◦C), occurslater (year 2079), and the final temperature change is also greater (2.79◦C).Figure 3.1: Projections of global surface temperature change from the ODEmodel for the five IPCC climate scenarios.3.3.2 Stochastic Differential Equation ModelTo simulate trajectories of Model 3.3, we use the same initial conditionsas in the ODE formulation. At each time time step, we sample dB ∼ N(0, 1)and solve the system using the Euler-Maruyama scheme. We run 10,000simulations to obtain the average behaviour and 95% confidence intervals foreach of the five IPCC climate scenarios (Figure 3.2). The average behaviourof the SDE model for each climate scenario is the same as the ODE solution.The 95% confidence intervals start out with a zero width since the initialconditions for every simulation are the same. The width increases to about0.5◦C, after which it remains constant, which suggests a standard deviationof about 0.25◦C.413.3. RESULTSFigure 3.2: Projections of global surface temperature change from the SDEmodel for the five IPCC climate scenarios. The solid lines show the averagebehaviour across 10,000 simulations and the windows around each solid lineare the 95% confidence intervals.3.3.3 Comparison to IPCC ProjectionsTo investigate the accuracy of our model, we compare it to results fromthe IPCC (Figure 3.3). We note that our model only uses two equationsand CO2 projections. Our model is significantly simpler than other climatemodels, yet it produces very similar temperature predictions as the IPCCmodels. In particular, the qualitative shape for years 2030 onwards is thesame between both models and we obtain similar temperature predictionsfor 2100. The main difference in model behaviour is the first decade ofprojections. In this region, our model shows a sharper increase in globalsurface temperature than the IPCC model. It is possible that adjustingparameters, such the climate feedback parameters, η and β, would shift ourmodel behaviour to better align with the IPCC results, although simulationsusing other values of η and β did not appear to produce a better fit. Forour purposes, the match between the two models is acceptable.423.4. DISCUSSIONFigure 3.3: IPCC projections of global surface temperature change comparedto pre-industrial times. The windows around SSP1-2.6 and SSP3-7.0 showthe “very likely ranges”. Image from IPCC (2021).3.4 DiscussionIn the classic energy balance model (Equation 3.1), temperature in-creases until it equilibrates within about 10 years at 1.42◦C above pre-industrial temperatures, for a given set of parameters provided in Rose andScott-Brown (2024). The resulting temperature is less than the results wefind in our ODE model since the authors do not consider the impact ofchanging CO2 emissions and climate change. One can study the impact ofclimate change on the equilibrium temperature by changing parameters inthe model such as τ , transmissivity of the atmosphere, or α, the planetaryalbedo. However, this model does not directly consider increased greenhousegas emissions or changing human behaviour, and only describes Earth’s sys-tem when the climate forcing from emissions is constant on average.The soil carbon-temperature model (Luke and Cox, 2010) provides anavenue for an extension of our current work. This model takes the atmo-spheric temperature as an input and studies respiration of carbon from soil.When the atmospheric temperature increases too much, the soil suddenlycombusts and releases all of its carbon causing an extreme increase in soiltemperature (on the magnitude of 200◦C). This effect has been called the“compost bomb”. The model predicts that such an increase in soil tempera-ture will cause a further increase in the atmospheric temperature. Creatinga feedback loop between our current energy balance model and this soilcarbon-temperature model would allow us to investigate whether any of our433.4. DISCUSSIONclimate simulations result in the occurrence of a compost bomb.When it comes to stochastic contributions in our climate model, wechose a mean-reversion time of 5 years (θ = 1/5). More precisely, the au-tocorrelation of W , the noise we use in Model 3.3, can be described bya(∆t) = exp (−θ|∆t|) where ∆t is the difference in time between two timesteps (van der Bolt et al., 2018). Our choice of parameter value is compara-ble to the 3.88 year mean-reversion time used in a two-box energy balancemodel (Fredriksen and Rypdal, 2017). Other studies suggest that envi-ronmental noise is well-described using pink noise which consists of noisewith frequencies between white noise (a(∆t) = 0, θ = ∞) and red noise(a(∆t) = 1, θ = 0) (Halley, 1996; Mustin et al., 2013). The stochasticityin our climate model falls in the range of pink noise since θ = 1/5. Asa result, our climate model is autocorrelated, but not as autocorrelated asBrownian motion (red noise). In our climate model, dW is additive since itis added onto the base ODE model and does not depend on T , i.e., the noiseprocess W does not depend on the current temperature. Additive noisehas been used in climate models to describe weather disturbances (Hassel-mann, 1976) and temperature fluctuations (Nicolis and Nicolis, 1981). Anextension on the current work is studying the impact of multiplicative noisewhich has been found to produce different model behaviour from additivenoise (Perez et al., 2005) and considers how climate variability may be morelikely to show up as a changed rate (multiplicative noise) than as a directtemperature fluctuation (additive noise) (van der Bolt et al., 2018).We note that the trajectories from our model for the five SSP scenariosgive similar results for the first decade. The IPCC model also displays asimilarity in temperature projections for the first decade, even though theemissions projections between the five scenarios differ. A possible reasonfor this similarity between different scenarios is it takes about one decadefor the maximum warming caused by a CO2 emission to occur (Ricke andCaldeira, 2014). Since the five scenarios are assumed to be the same upuntil 2020 in our model and until 2015 in the IPCC model, we expect tostart seeing the different amounts of warming caused by different emissionsin each scenario by 2030 and 2025, respectively.By using an ODE and SDE model, we assume that the Earth is a well-mixed system. This approach is a simplification of the real system, however,and more accurate models include spatial effects (Lyon et al., 2021; Mein-shausen et al., 2020; Rantanen et al., 2022; Wang et al., 2022). These modelsall predict increased warming of polar regions, a phenomenon called Arcticamplification. Studies have found that over the last four decades, the Arc-tic has warmed at least four times faster than the rest of the Earth, with443.4. DISCUSSIONsome regions warming at a rate that is seven times faster than the globalaverage (Rantanen et al., 2022). Some authors suggest that this Arctic am-plification is a result of sea ice loss and decreased albedo (Dai et al., 2019;Screen and Simmonds, 2010). Heat stress is also not uniformly distributedaround the globe, with some regions being far more vulnerable than oth-ers. In particular, one model projects that Africa, the Amazon, the ArabianPeninsula, Southeast Asia, the Maritime Continent, and northern Australiawill experience heat stress for more than half of the year by 2500 underthe representative concentration pathway (RCP) 6.0 scenario (Lyon et al.,2021).Mathematical climate models, including the models in this chapter, com-monly provide projections up to the year 2100 and no further (IPCC, 2021;Leach et al., 2018). Recent studies suggest that this time span is no longersufficient to understand the impacts of climate change, and make projectionsuntil the year 2500 (Lyon et al., 2021; Meinshausen et al., 2020; Mengel et al.,2018). In fact, they find that the global average surface temperature is ex-pected to keep increasing past 2100 for all SSP scenarios except SSP1-1.9and SSP1-2.6 (Lyon et al., 2021). Impacts beyond 2100 include sea levelrise even when net-zero GHG emissions are reached by 2050 (Mengel et al.,2018), heat stress that could prove fatal (Lyon et al., 2021), and significantdecreases in crop yields (Tigchelaar et al., 2018). The CO2 projections upto year 2500 for each of the five SSP scenarios (Meinshausen et al., 2020)provide an avenue to expand our current work and make projections on alonger time scale.Many models assume that, within each SSP or RCP scenario, humansfollow a fixed behaviour (IPCC, 2021; Lyon et al., 2021; Meinshausen et al.,2020). That is, researchers make the modelling assumption that humansdo not update their behaviours dynamically in response to the changingclimate. However, it is known that extreme events impact human perceptionof climate change risk (Bergquist et al., 2019) and mitigation behaviour(Demski et al., 2016). We discuss extensions to our climate model andinclude a social-climate feedback loop in Chapter 4.45Chapter 4Social-Climate Model4.1 BackgroundDue to the dynamic nature of interactions between the Earth’s climateand public opinion, there has been a call from modellers to include a social-climate feedback loop in climate models (Beckage et al., 2020; Long andTyson, 2014). Modelling the climate in this way allows modellers to ac-count for changes in human behaviour due to changes in extreme eventsor social norms (Beckage et al., 2018). Other social-climate models studya variety of factors such as, e.g., cooperation to conserve a common re-source (Andersson et al., 2021; Milinski et al., 2008), unequal distributionof resources impacting ability to participate in climate mitigation strategies(Menard et al., 2021; Tavoni et al., 2011), or the impact of delayed damagesfrom emissions (Ghidoni et al., 2017).Extreme events are increasing in frequency and intensity as the Earth’sclimate continues to change (Buttke et al., 2023; Cai et al., 2014; IPCC,2021; Martel et al., 2021; Mokria et al., 2017). Work on predicting theseextreme events is becoming increasingly important in light of their impacts,e.g., health risks (Buttke et al., 2023), cyclones (Cai et al., 2014), heatwaves(Domeisen et al., 2022), and extreme rainfall (Westra et al., 2014). Modelerspredict extreme events using various methods, e.g., statistical models fitto historical data (Buttke et al., 2023; Mokria et al., 2017), models basedon known chemical properties such as the Clausius-Clapeyron relationship(Martel et al., 2021; Westra et al., 2014), partial differential equation models(Farazmand and Sapsis, 2019), and highly complex Earth system models(Cai et al., 2014; Domeisen et al., 2022).Additionally, extreme events are a known driving force of human be-haviour (Bergquist et al., 2019; Demski et al., 2016). These extreme eventsare included in social-climate models in a variety of ways. Extreme eventscan be the end behaviour of a system if there is not enough cooperation toprevent further climate change (Milinski et al., 2008; Tavoni et al., 2011),while other models indirectly include extreme events through functions ofclimate change cost (Bury et al., 2019; Menard et al., 2021; Moore et al.,464.2. MODEL2022) or directly in a dynamic feedback loop (Beckage et al., 2018; Ghidoniet al., 2017). It has also been found that the impact of extreme events onhuman behaviour decays over time (Ray et al., 2017; Walshe et al., 2020)and can even be forgotten within a couple generations (Fanta et al., 2019).The exact relationship between extreme events and human behaviour is un-known, so some modelers define several different reasonable functional forms(Beckage et al., 2018).A source of hope for climate activists is the influence a committed mi-nority can have on overturning social conventions (Otto et al., 2020). Infact, social change is often initiated by minorities (Bolderdijk and Jans,2021). Previous work suggests that these committed minorities can exertthe same amount of influence as a majority population and this minority in-fluence may have a larger proportional effect on the population (Moscovici,1980). A committed minority prevents group consensus (Chapter 2) andcan change or challenge previous social conventions (Bolderdijk and Jans,2021). Further, changing current social conventions has been found to bea feasible route towards achieving widespread climate action (Constantinoet al., 2022) and a previous social-climate model considers social norms tobe a key variable in determining our future climate (Beckage et al., 2018).4.2 ModelIn this chapter, we combine our models from Chapter 2 and Chapter3 to create a social-climate feedback loop. We propose this toy model asa framework to guide future social-climate models and choose parametervalues that provide insight into the possible model behaviours we mightexpect from a future model with parameters that are fit to data. That is,unless stated otherwise, the parameters in this chapter were chosen so thatthere is a social model tipping event in about half of the simulations. Wenote that the results in this chapter are qualitative and we focus on thegeneral model behaviour rather than quantitative values.To begin our study of the social-climate model, we assume that the socialmodel equilibriates much faster than the climate model. This assumptionallows us to study a simpler model and draw insights that could inform thebehaviour of more complex social-climate models. As a result, we assumethe social model is at equilibrium throughout this chapter and we use theopinion response functions to solve for the equilibrium speaking rates. Tostudy the conditions under which a committed minority can shift publicopinion towards climate action, we choose opinion A as the climate action474.2. MODELopinion. We call opinion B the climate inaction or business as usual opinion.We set memory bank lengthM = 25 which has a tipping point at committedminority proportion CM∗ = 0.3086. We choose an initial condition CM = 0.1for our simulations. The speaking rate of the climate action opinion, r(t),is calculated by solving ΨCM(r(t)) = r(t) with the current CM(t) value.On the climate model side, we use our SDE with linearized outgoinglongwave radiation formulation (Equations 3.3a and 3.3b). The parametersand initial conditions in this model are given in Table 3.1.Figure 4.1: An outline of our social-climate model feedback loop.The novel work in this chapter is connecting these models in a social-climate feedback loop (Figure 4.1). We start by building the social-to-climate model connection (i.e., the arrow labeled CO2 emissions in Figure4.1). At each time step, we use the equilibrium speaking rate of the climateaction opinion to determine the level of current CO2 emissions, E. We usea weighted average between the best-case emissions, Ebest, and the worst-case emissions, Eworst, from our SSP scenarios. In particular, SSP1-1.9represents the best-case emissions and SSP5-8.5 represents the worst-caseemissions. We obtain:E(t) = r(t)Ebest(t) + (1− r(t))Eworst(t). (4.1)We follow the same steps as in Section 3.2.1 to convert these CO2 emissionsinto a current CO2 concentration. We input this concentration into Equation3.3 and run one time step (one day) of the climate model.Now, we build the climate-to-social model connection (i.e., the arrowlabeled extreme events in Figure 4.1). We use data (Table 4.1) from climateattribution studies (Pidcock and McSweeney, 2022) to choose a base rate ofextreme weather events if there was no climate change, λ0. In particular,we take the average of the number of extreme events each year that werefound to not be due to human influence. We obtain an average of 4 extremeweather events per year which we scale to units of days to match our time484.2. MODELstep length. We note that it is likely that there are more extreme eventsthat occur in the later years of the study period and are not included in thedataset due to the time it can take to publish work in an academic journal.In general, models predict that extreme events will increase in frequencyand intensity as mean global temperature increases (Buttke et al., 2023;Cai et al., 2014; IPCC, 2021; Martel et al., 2021; Mokria et al., 2017). TheClausius-Clapeyron relationship predicts that rainfall will increase by 7% foreach 1◦C increase in global average temperature above pre-industrial tem-peratures (Martel et al., 2021). An empirical study suggests that the rainfallwill increase by as much as 14% as global warming progresses (Westra et al.,2014). Following (Singh, 2023), we use a 10% in extreme precipitation eventsas a proxy for the increase in frequency of all extreme weather events. Hence,the rate of extreme events, λ(t), is:λ(t) = λ01.1T (t)−TPI , (4.2)where TPI and T (t) are the global average surface temperatures from pre-industrial times and at time t, respectively.Table 4.1: Count data from attribution studies on extreme weather events.Extreme events are categorized into three categories: human influence found,human influence not found, and inconclusive. Data from Pidcock and Mc-Sweeney (2022).Year Human influence No human influence Inconclusive2011 23 9 112012 28 14 192013 40 10 162014 45 6 172015 57 8 132016 45 7 122017 45 0 42018 39 1 102019 22 2 32020 24 1 12021 6 1 12022 6 1 1To determine the number of extreme events at each time step, we sam-ple from a Poisson distribution with rate λ(t). If our sampling procedurereturns zero extreme events, we return to the social model and repeat the494.3. RESULTSfeedback loop. If our sampling procedure returns an extreme event, then weincrease the speaking rate of the climate action opinion in the social modelby increasing CM(t). We note that increasing r(t) directly would be undoneas soon as we return to the social model since our first step in the socialmodel is computing the equilibrium r(t) based on the committed minoritysize. Instead, we indirectly increase r(t) by increasing CM(t). We let thisincrease decay over time to account for extreme events having less impactor being forgotten over time (Ray et al., 2017; Walshe et al., 2020). Hence,we obtain the function which we call the decaying cumulative sum:CM(t) = CM(0) +∑s∈EEµ exp (−δ(t− s)), (4.3)where µ = 0.014 is the size of the temporary increase to the committedminority, δ = 0.002 is the decay rate of this increase, and EE is the set oftime indices for each extreme event up until time t. We assume that theimpact of extreme events that occur before our simulations start in 2020 areaccounted for in the initial condition CM = 0.1. We input this shifted CM(t)back into the social model and repeat the procedure until year 2100.4.3 ResultsWe run 10,000 simulations to investigate the dynamics of our social-climate model. From these simulations, we focus on four representative sam-ple simulations: minimum final temperature, maximum final temperature,largest maximum temperature, and a case where the social model tippingevent occurs late in the simulation. We describe each of these sample simu-lations in more detail in the following paragraphs. We do not consider thecase of smallest minimum temperature since the minimum temperature inall of our simulations is the initial temperature. Thus, all of our simulationsobtain the smallest minimum temperature.The simulation with the minimum final temperature (Figure 4.2) repre-sents the best case scenario from our simulations. We find that a high enoughfrequency of extreme events over a short time causes enough of the uncom-mitted population to be temporarily shifted into the committed minority tocause a tipping event in the social model (i.e., the decaying cumulative sumcrosses the threshold C∗M). In this simulation, the high number of extremeevents and the tipping event occur in about 2030. Following this tippingevent, there is an immediate shift towards climate action and r(t) = 1. Thisshift in opinion means Equation 4.1 simplifies to E(t) = Ebest(t) and the504.3. RESULTSemissions for the rest of the simulation are exactly the emissions from SSP1-1.9. We note that the tipping point in our social model is irreversible andthere is no shift back towards any individuals speaking B, and thus no shiftback towards higher emissions. As a result, extreme events after the tippingevent no longer have an effect since the committed minority and uncommit-ted population have the same behaviour and everyone is already speakingA.Figure 4.2: Annual counts of extreme events, decaying cumulative sum (left),speaking rate of A, and global surface temperature (right) in the minimumfinal temperature scenario.When the social model tipping event occurs late in the simulation time(roughly 2090 in Figure 4.3), the temperature change from pre-industrialtimes remains large. We note that when this tipping point is crossed, there isa substantial change in emissions from approximately 102 to -10 gigatonnesCO2 per year within one time step. This is a result of of the form of Equation4.1 and the bifurcation structure of our social model. That is, within onetime step, there is a mass shift from r(t) ≈ 0.2 to r(t+1) = 1 which resultsin a large change in emissions. While this change in emissions is immediate,the change in atmospheric concentration of CO2 is not as immediate sincethe concentration of CO2 has been built up over time.In the maximum final temperature scenario (Figure 4.4), extreme eventsdo not happen frequently enough and the climate inaction social conventionis not overturned. We note that while, around 2060, there is a year with 12extreme events, there are very few extreme events in the following years andthese extreme events are not frequent enough to cause a tipping event. Theresult is emission levels close to the SSP5-8.5 scenario and a much greaterincrease in temperature than the minimum final temperature case. Theresults from the largest maximum temperature case (Figure 4.5) are similar.514.3. RESULTSFigure 4.3: Annual counts of extreme events, decaying cumulative sum (left),speaking rate of A, and global surface temperature (right) in the late socialmodel tipping event scenario.We note that there are regions where the temperature decreases (2030-2060in Figure 4.3, 2030-2050 in Figure 4.4, and 2095-2100 in Figure 4.5). Thesedecreases are unrelated to improved climate action since there is not anassociated increase in the speaking rate of A, r(t). Hence, the decrease intemperature is a result of stochasticity in our social-climate model.Figure 4.4: Annual counts of extreme events, decaying cumulative sum (left),speaking rate of A, and global surface temperature (right) in the maximumfinal temperature scenario.When we consider all of the simulations in which the climate inactionsocial convention is not overturned (left plot in Figure 4.6), we find thatthe social-climate model produces projections similar to the maximum finaltemperature and largest maximum temperature scenarios, and the mediantemperature prediction in 2100 is 4.9◦C above pre-industrial levels. These524.3. RESULTSFigure 4.5: Annual counts of extreme events, decaying cumulative sum (left),speaking rate of A, and global surface temperature (right) in the largestmaximum temperature scenario.temperature projections are slightly lower than the projections from ourSSP5-8.5 climate model (Figure 3.2) since the committed minority speaks Aand decreases the emissions fed into the climate model. When we considerall of the simulations in which the social convention is overturned (right plotin Figure 4.6), the final temperature depends on when the tipping point iscrossed, i.e., the earlier the tip, the lower the temperature in 2100. We findthat about 40% of our simulations cross the tipping point in the social modeland overturn the climate inaction social convention. However, in general, thesimulations that do cross the tipping point do so quite late. In these cases,there is not enough time to reverse the effects of increased CO2 emissionsbefore 2100 and the final temperatures remain similar to those when thereis not a tipping event. The average behaviour from our simulations (Figure4.7) is most similar to that of the SSP3-7.0 climate model.534.3. RESULTSFigure 4.6: Left: A violin plot showing the 2100 temperature projectionsfrom those simulations wherein the social convention is not overturned.Right: A heat map showing the 2100 temperature projections as a func-tion of the time the social convention is overturned. The colour indicatesthe number of simulations.Figure 4.7: Projections of global surface temperature change from our social-climate model. The solid lines show the average behaviour and the shadingindicates the 95% confidence intervals.544.4. DISCUSSION4.4 DiscussionThe field of rapid extreme event attribution has grown substantially inthe last decade (Caldwell et al., 2014; van Oldenborgh et al., 2021; Philipet al., 2022). Further, rapid extreme event attribution is decreasing thelag between an extreme event and it’s attribution to climate change (Oakeset al., 2021; Reed and Wehner, 2023) and may influence human behaviours(McClure et al., 2022; Ogunbode et al., 2019). However, this rapid attri-bution can be biased by modelling choices such as how an extreme eventis defined (Miralles and Davison, 2023; van Oldenborgh et al., 2021). Forexample, since extreme weather events tend to be outliers, the choice toinclude an extreme event or not can bias a model fit. Miralles and Davison(2023) suggest a method to account for timing and spatial biases that arosein previous attribution studies. We note that the number of extreme eventsin any single year for each of representative sample simulations in Section4.3 are lower than the number of extreme events per year in Table 4.1 (ex-cluding 2021 and 2022). Additionally, choosing λ0 = 4 as the base rate ofextreme events in the absence of climate change is also low. Since we areinterested in the qualitative behaviour of this model rather than quantita-tive predictions, we leave to future work the task of using data from rapidattribution studies to find a better fit for model parameters. We suggestusing a discrete process such as Poisson or Negative Binomial to fit the ex-treme event rate parameters, λ0 and λ(t). Analyzing data for overdispersioncan determine whether Poisson or Negative Binomial distributions shouldbe used to sample extreme event occurrence. In particular, if data is foundto be overdispersed, then a Negative Binomial distribution may be used, asit allows for a larger variance relative to the mean (Bhunya et al., 2013).In our social-climate model, we assume that extreme events affect every-one with the same intensity and people forget at the same rate. In Chapter3, we discuss how climate change impacts different parts of the Earth dif-ferently (i.e., extreme events affect different parts of the global population)and in Chapter 2, we discuss differences in the memory dynamics of individ-uals. The impact of an extreme event on human behaviour can vary basedon factors such as financial inequality (Menard et al., 2021; Tavoni et al.,2011). It has been found that inequality can prevent cooperation (Tavoniet al., 2011) and increase global surface temperatures (Menard et al., 2021).In particular, when the individuals in a model are divided into poor and richgroups, the within-group cooperation increases but there is increased polar-ization between groups as a result in the asymmetric impacts of extremeevents (Menard et al., 2021). Previous work suggests that redistributing554.4. DISCUSSIONwealth and resources eliminates these inequalities and promotes cooperation(Tavoni et al., 2011). In addition, climate action can result in a short-termfinancial loss that people may be unwilling to take on. One study sug-gests that a way to convince people to invest in climate action is convincingthem that not investing now is likely to cause a larger future financial loss(Milinski et al., 2008). A framework has been proposed to motivate climateaction in the short-term by providing financial benefits to governments thatpartake in mitigation behaviors (Cseh, 2018). Possible extensions to ourcurrent work include studying spatial effects by considering our agent-basedsocial model on a network, using different types of agents (e.g., poor versusrich), including economic effects, and using a spatial climate model so thatindividuals in areas such as the Arctic (Rantanen et al., 2022) are impactedmore by extreme events.We assume in our social-climate model that extreme events are the onlydrivers of climate action behaviour. Some scientists conjecture that majordisasters are required to motivate climate action (Ghidoni et al., 2017) andit has been found that experiencing an extreme event alters an individual’sbehaviour (Demski et al., 2016; Konisky et al., 2015). Additional studieshave found that the framing of an extreme event (e.g., whether an extremeevent is more or less probable due to human influence) impacts an indi-vidual’s risk perception (Spence and Pidgeon, 2010) and willingness to act(Hurlstone et al., 2014; Lidskog and Sjodin, 2015). However, it has beenfound that co-benefits of action can also change human behaviour (Bainet al., 2015; Marshall et al., 2023). In particular, Bain et al. (2015) findthat scientific and economic development and care for a greater commu-nity can motivate people to commit to climate action regardless of whetherthey believe climate change is a risk. The authors suggest that co-benefitsof action have a similar impact on human behaviour as the recognition ofthe importance of mitigating climate change. A similar result is found byMarshall et al. (2023), who found that people responded most positively toclimate action messaging related to helping future generations. A recent pa-per (Roggero et al., 2023), however, suggests that the relationship betweenco-benefits and climate change mitigation is more complex. They studythree cities (Moscow, Paris, and Montreal) that reduced local greenhousegas (GHG) emissions and, as a result, improved air quality which led toa co-benefit of improved health. However, the authors found that driverssuch as international recognition or adaptation to climate change impactsaffected policy changes more than health benefits from improved air quality.We leave the study of other driving factors of climate action to future work.In this thesis, we assume that the committed minority holds the climate564.4. DISCUSSIONaction opinion and that individuals will react to extreme weather events andperform climate mitigation actions. However, many factors (e.g., politicalideology, misinformation, oversimplification) can influence climate changedenial and create a committed minority on both sides of an issue (Bliucet al., 2015; McCright and Dunlap, 2011a,b). In particular, Bliuc et al.(2015) note that 97% of climate science papers hold a consensus on climatechange being caused by human behaviour, yet a survey found that about50% of the global population agrees (Kohut et al., 2013). Studies suggestthat there is a public divide based on factors such as political ideology, race,and gender where part of the population are staunch deniers of climatechange while another part of the population recognizes the need for climateaction (Bliuc et al., 2015; McCright and Dunlap, 2011a). Misinformationmagnifies this political divide and can lead to climate inaction movements toprevent climate mitigation policies (Brulle, 2013; Treen et al., 2020). Whilework is being done to investigate how to prevent and reverse the spreadof misinformation, clear answers do not exist so far (van der Linden et al.,2017; Treen et al., 2020). It has also been found that laypeople tend tosimplify the causes for extreme weather events and can overlook climatechange (McClure et al., 2022). For example, they may attribute a forest fireto a discarded cigarette instead of considering multiple interacting causesincluding climate change. Extending our social model to include committedminorities holding each opinion would allow us to study the impact of climatechange denial. It is possible that this extension would result in the socialmodel tipping point becoming reversible.Since it takes a decade to experience the full effects of a GHG emission(Ricke and Caldeira, 2014), there is a substantial delay between GHG emis-sions and the associated extreme weather events. As a result, it may bedifficult to sustain climate action if there is not an immediate improvementin the climate. In particular, it has been found that people tend to preferpolicies that produce results in 0-2 years over policies that produce resultsin 20-30 years (Christensen and Rapeli, 2020). Additionally, when extremeevents occur with a delay, there is a larger increase in cumulative emissionsthan if extreme events occur immediately after a GHG emission (Ghidoniet al., 2017). It becomes even harder to maintain climate change mitigationefforts when extreme events fade from memory (Fanta et al., 2019; Ray et al.,2017; Walshe et al., 2020). One model finds that only extreme events thatoccurred in the previous month have a statistically significant effect on sup-port for climate adaptation policies (Ray et al., 2017). In contrast, anotherstudy found that it takes approximately two generations for a population tomove back into an area previously devastated by a catastrophic flood (Fanta574.4. DISCUSSIONet al., 2019). If delay dynamics were included in our social-climate modelor if the social tipping point were reversible (i.e., if individuals commitmentto climate action could wane), we might expect oscillatory dynamics wherethe tipping point is crossed and reversed multiple times (van der Bolt et al.,2018).While many studies have found a correlation between experiencing ex-treme events and climate action (Demski et al., 2016; Konisky et al., 2015;Ray et al., 2017), the exact relationship is not known. In one study (Beckageet al., 2018), the authors find that if the relationship between extreme eventsand perceived risk of climate change is sigmoidal, people respond quickly toslightly more or slightly fewer extreme events than usual and there is moredensity in the tails of the distribution of final temperature projections. If therelationship is cubic, significantly more or significantly fewer extreme eventsthan usual are required to push people into action (for or against climatechange mitigation) and the majority of the resulting 2100 temperatures fallnear the median. Other models also use a similar sigmoidal shape to modelcost of climate change as a function of the temperature increase comparedto pre-industrial times (Bury et al., 2019). This functional form is the sameform as the cubic relationship in (Beckage et al., 2018) since Bury et al.(2019) only consider the right hand side of the cubic function, making itappear sigmoidal. The best and worst cases from the social-climate modelin (Bury et al., 2019) are comparable to our minimum and maximum finaltemperature cases. Altogether, the final temperature distribution from oursocial-climate model (Figure 4.6) best match results from models using acubic functional form.Climate inaction social conventions can prevent wide-scale climate action(Beckage et al., 2018; Constantino et al., 2022). If a committed minority isparticularly influential, their opinion is spread through the population fasterand wider than the opinion of a less influential minority (Nyborg et al., 2016)and can upset an inaction convention (Beckage et al., 2018; Bolderdijk andJans, 2021; Constantino et al., 2022). In addition, while committed minori-ties can be a driving force for social change via a bottom-up mechanism,policy makers can contribute by offering top-down support (Bolderdijk andJans, 2021). If decision makers don’t represent all generations, then resolv-ing issues can be challenging. Studies suggest that decision makers haveless incentive to help future generations (Sherstyuk et al., 2016) and aremore focused on short-term goals (Cseh, 2018). An extension to the currentwork is considering how social interactions between leaders (i.e., highly in-fluential people or decision makers), committed minorities, and citizens (i.e.,everyone else) impact climate action and the future climate.58Chapter 5ConclusionsIt is well-established that human activity is a driving force for climatechange (IPCC, 2021) and that extreme weather events influence risk per-ception of climate change (Bergquist et al., 2019) and mitigation behaviour(Demski et al., 2016). However, few models allow for human behaviours andthe climate to dynamically interact, and there has been a call to action tomodel climate change as a social-climate feedback loop (Beckage et al., 2020;Long and Tyson, 2014). As an initial step to answer this call to action, thisthesis demonstrates that a social-climate feedback loop may be importantin accurately predicting future temperatures, in contrast to the standardapproach where, for each shared socioeconomic pathways scenario, humanbehaviour is considered fixed. In particular, we find that the overturning ofa social convention and the timing of this tipping event have a large impacton future temperatures.It has previously been found that the number of memories an individualholds determines the committed minority size needed to overturn a pre-established social convention (Centola et al., 2018). The authors use anagent-based model (ABM) to estimate the minimum committed minoritysize required for this tipping event as a function of memory bank length. InChapter 2, we replicate their numerical simulations of this ABM and con-firm the resulting model behaviour. Then, we reformulate the ABM as aset of ordinary differential equations (ODEs) and propose a new tool, opin-ion response functions, to study the social model dynamics. Using opinionresponse functions, we analytically compute the steady states of our so-cial model and determine the conditions under which a committed minoritycan influence the greater population and overturn a social convention, as afunction of memory bank length. We find that, if and only if the commit-ted minority size, CM, is greater than some threshold value CM∗ (that is atmost 0.5), a social convention will be overturned. These results compare wellwith other social models (Baronchelli et al., 2006; Galam and Jacobs, 2007;Holley and Liggett, 1975; Mobilia et al., 2007; Xie et al., 2011). We notethat factors such as vested interests (Crano and Prislin, 1995; De Dominiciset al., 2021), level of influence (Hosseini et al., 2018; Talebi et al., 2024),59and listening threshold (Andersson et al., 2021; Chen et al., 2021) changethe committed minority size required to overturn a social convention. Dataon the number of memories held and attachment to a social convention isrequired to better estimate the location of this tipping point.To our knowledge, Chapter 3 represents the first study of a linearized en-ergy balance model modified to include a dynamic CO2 concentration inputterm and an additive autoregressive stochastic term (Equations 3.3). Thelinearized energy balance ODE model (Equation 3.2) has previously beenused to deterministically study the climate’s steady states under changingemissions scenarios (Meier et al., 2020; Tung, 2007). Previous stochastic cli-mate models consider an energy balance model (Nicolis and Nicolis, 1981)or higher complexity models (IPCC, 2021). Our toy model is simplistic, yetour results compare well with the higher complexity models. Some avenuesfor future work include studying multiplicative noise which may be a moreaccurate representation of the climate system (van der Bolt et al., 2018) orspatial effects (Lyon et al., 2021; Meinshausen et al., 2020; Rantanen et al.,2022; Wang et al., 2022).In Chapter 4, we propose a novel social-climate model. To create thesocial-to-climate side of the feedback loop, we assume that the speaking rateof the climate action opinion in the social model determines the level of CO2emissions that are fed into the climate model. To complete the feedback loop,we use a Poisson process to sample the occurrence of extreme weather eventsat a rate that depends on the current temperature change from pre-industrialtimes. When an extreme event occurs, we increase the speaking rate of theclimate action opinion and allow this increase to decay over time. We notethat our social-climate model provides temperature projections up to theyear 2100 which is no longer far enough into the future and projections witha longer timescale are necessary (Lyon et al., 2021; Meinshausen et al., 2020;Mengel et al., 2018). Since this is a toy model, we focus on the qualitativemodel behaviour.Our social-climate model suggests that it is important to disrupt thecurrent climate inaction social convention and start climate change mitiga-tion strategies sooner rather than later. This result aligns with the majorityof climate change studies (Beckage et al., 2018; IPCC, 2021; Moore et al.,2022; Otto et al., 2020). A committed minority plays a vital role in shiftingpublic opinion towards climate action and causing a tipping event (Ottoet al., 2020). In particular, the later the social model tipping event occurs,if a tipping event even occurs, the greater the temperature in 2100 in oursocial-climate model. Sociological studies find that committed minoritiesoften initiate social change and can change social conventions (Bolderdijk60and Jans, 2021), while previous mathematical models find that challengingsocial conventions can be key to increasing climate action (Beckage et al.,2018; Constantino et al., 2022; Otto et al., 2020).We find that enough extreme events occurring over a short time arerequired to cause a tipping event in the social model. This result alignswith previous work on the requirements to shift a population into action(Bergquist et al., 2019; Demski et al., 2016; Ghidoni et al., 2017). Attribu-tion studies (Caldwell et al., 2014; van Oldenborgh et al., 2021; Philip et al.,2022; Pidcock and McSweeney, 2022) are the key to fitting parameters suchas the base and increased rates of extreme events, while studies on the effectof extreme events on the population (Bergquist et al., 2019; Demski et al.,2016; Menard et al., 2021; Tavoni et al., 2011) help to determine the impactsof extreme events on the social model such as increased willingness to act(Hurlstone et al., 2014; Lidskog and Sjodin, 2015) or how fast the extremeevent decays from memory (Fanta et al., 2019; Ray et al., 2017; Walsheet al., 2020). Recent work suggests that co-benefits of action such as carefor community or future generations may also motivate and increase climateaction (Bain et al., 2015), although this relationship is likely more complexand vested interests may be a confounding factor (Roggero et al., 2023). Weleave the study of other driving factors of human behaviour to future work.The growing frequency and intensity of extreme weather events such asheat waves, flooding, fires, and storms make it clear that immediate climateaction is necessary (IPCC, 2021). While there are countless paths for futurework, our social-climate model serves as an initial framework to expandon current models studying climate change using a dynamic social-climatefeedback loop. Studying toy models, such as those in this thesis, providesnecessary insights into the behaviour of more complex systems (Epstein,2008; Haefner, 2005; Smaldino, 2017; Wimsatt, 2007). 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Thesis/Dissertation
eng
Mathematics
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Foxall, Eric
Tyson, Rebecca
The role of committed minorities in climate change action : qualitative insights from a social-climate model
Text