Coupling the socio-economic and ecological dynamics of cyanobacteria: Single lake and network dynamics

Abstract

In recent decades freshwater lakes have seen an increase in human presence. A common byproduct of this human presence is eutrophication, which readily results in harmful cyanobacteria (CB) blooms. In this work we propose a model that couples the socio-economic and ecological dynamics related to cyanobacteria systems. The socio-economic dynamics considers the choices a human population makes regarding whether or not to mitigate their pollution levels. These choices are based on various costs related to social ostracism, social norms, environmental concern and financial burden. The coupled model exhibits bistable dynamics, with one stable state corresponding to high mitigation efforts and low CB abundance, and the other to low mitigation efforts and high CB abundance. Furthermore, we consider social interactions among a network of lakes and present dynamic outcomes pertaining to various associated costs and social situations. In each case we show the potential for regime shifts between levels of cooperation and CB abundance. Social ostracism and pressure are shown to be driving factors in causing such regime shifts.

Introduction

Cyanobacterial harmful algal blooms (CHABs) are an ever present global concern in aquatic environments. The presence of CHABs often leads to several adverse outcomes both ecologically and economically. For example, CHABs can decrease ecosystem productivity by creating anoxic conditions and producing toxins as metabolic byproducts (Orr and Jones, 1998; Kaebernick and Neilan, 2001). Economically, CHABs add costs to water treatment, lower recreational and tourism value, and add risks when using freshwater for agricultural purposes. Although CHABs occur for a variety of reasons they are most commonly a result of eutrophication. Eutrophic conditions occur when an excess amount of nutrients required for organismal growth is in an aquatic ecosystem. Furthermore, eutrophication often occurs as a result of anthropogenic nutrient pollution from agriculture, industrial and urban run-off (Paerl, 2014). In this sense there is a noteworthy connection between anthropogenic nutrient pollution and economic costs due to CHABs.

The study of systems where human and environmental dynamics are intertwined is beginning to receive more attention in the literature. For example, the importance of linking human and social dynamics to climate models to understand climate trajectories has been addressed (Beckage et al., 2020; Bury et al., 2019). Other researchers have used social processes to better understand disease outbreaks (Pedro et al., 2020; Fair et al., 2021). Ecologically, social dynamics have been coupled to forestry, fishery and other common-pool resource models to gain insight towards the balance between sustainable resource use and profit seekers (Satake et al., 2007; Farahbakhsh et al., 2021; Lee and Iwasa, 2011; Wang et al., 2016). Socio-ecological mechanisms to support persistent of native species of grasses that are under stress from anthropogenic nitrogen sources and invasive species have also been studied (Thampi et al., 2019). Finally, coupled socio-economic and ecosystem models for lake eutrophication have been considered by Iwasa et al. (Iwasa et al., 2007; Iwasa et al., 2010), but do not consider phytoplankton dynamics. In essence, human activities often result in changes in the ecological system, however changes in the ecological system will, in-turn, have an impact on the human behaviours thus creating a feedback loop. These types of systems are thought of as an integration between an ecological system and socio-economic system. Mathematical modelling of such systems typically involves the coupling of an ecological model that has terms dependent on human decisions to a human socio-economic model with outputs dependent on the state of the ecology (Satake et al., 2007; Iwasa et al., 2007).

Socio-economic models can be derived by considering social norms and pressures, monetary costs and psychology associated with the ecological system (Fransson and Gärling, 1999). As is the case in many current environmental issues, social ostracism can occur when an individual does not behave in a way that is environmentally favourable (Poon et al., 2015). Social ostracism happens when a group or individual excludes or slanders another group or individual based on an action, opinion or response. Psychologically, being ostracised is harmful as humans have a basic want of being accepted (Williams, 2007). As a response to ostracism humans often change behaviour to further avoid ostracism (Williams and Nida, 2011). In the context of environmental issues, such as lake pollution, groups who assume non-environmentally favourable roles are often ostracised more than those that do (Iwasa et al., 2007; Poon et al., 2015; Sun and Hilker, 2020) adding costs to the defection role. This means that modelling of socio-economic systems should include factors that account for social pressures. In addition, social norms often influence a person to assume a strategy regardless of its environmental impacts (Kinzig et al., 2013). Social norms are described as the set of rules and behaviours a society deems appropriate and are often established based upon the behaviour of the majority, regardless of any implications. Intrinsically, there exists pressure to adhere to these social norms although it is indirect. Socio-economic dynamics may be dependent on the frequency of each strategy, and not on the costs alone. Furthermore, the social costs due to ostracism and adherence to norms can be non-local and come from distanced social connections. Costs associated with pro-environmental roles often exceed the non-environmentally favourable role. These costs are often monetary and involve the investment in infrastructure to filter or treat urban water run-off. Additionally, lakes with low water quality and persistent CHABs face additional costs associated with decreased land value, recreation, tourism based on the presence of toxins, and the visually and olfactorily unpleasant nature of CHABs (Nicholls and Crompton, 2018; Wolf and Klaiber, 2017).

In many cases socio-economic models often have a game-theoretic component in which players choose one of several strategies based on the associated utility differences to the other strategies (Farahbakhsh et al., 2021; Iwasa et al., 2007; Iwasa et al., 2010; Sun and Hilker, 2020; Suzuki and Iwasa, 2009). Each strategy then has an associated disturbance of the ecological system, i.e. high vs. low pollution or deforestation rates. Individuals assume strategies at rates that are dependent on the perceived costs of each strategy, or fitness in game theory literature, and can be modelled in many different forms. For example, the logit best-response dynamics assumes there is a probability an individual assumes a strategy based on associated costs alone, where as the replicator dynamics assumes that the individual first learns of an alternate strategy and chooses it with a probability proportional to the cost differences allowing for strong conformity (Bury et al., 2019; Farahbakhsh et al., 2021; Iwasa et al., 2007; Sun and Hilker, 2021). By explicitly considering distinct strategies and their associated costs ecosystem managers can use these models to gain insight towards policy implementation to obtain a favourable outcome.

Many phytoplankton models have been used for the study of algal dynamics and take various forms including discrete time models, ordinary and partial differential equations (ODEs and PDES, respectively). In this study we extend a stoichiometric model that has been well established in the literature (Heggerud et al., 2020; Wang et al., 2007; Berger et al., 2006). Ecological stoichiometry is defined as the study of the balance of energy and resources in ecological systems (Sterner and Elser, 2002). This is a powerful tool as it allows the study of large scale phenomena, like cyanobacteria (CB) abundance, by considering small scale components like internal energy and nutrients. The use of ecological stoichiometry has become increasingly common because of its ability to mechanistically capture the effects of resource limitations on ecological systems. For example, ecological stoichiometry has been used to study predator prey systems (Mitra and Flynn, 2005; Branco et al., 2018), producer-grazer systems (Wang et al., 2008; Loladze et al., 2000), phytoplankton dynamics (Wang et al., 2007; Klausmeier et al., 2004), toxicology (Peace et al., 2021) and plant-disease dynamics (Lacroix et al., 2017) with great success. Ecological stoichiometry has been used to discuss the timescale separation between nutrient uptake and both algal growth and available nutrient depletion in Heggerud et al. (Heggerud et al., 2020). Separation of timescales allowed for the in-depth study of algal transient dynamics and driving mechanisms. This, along with many other studies, has established a solid modelling framework for phytoplankton dynamics (Wang et al., 2007; Berger et al., 2006; Huisman and Weissing, 1994). Additional complexity arises when coupling such ecological models to socio-economic models, both mathematically and in terms of timescales (Hastings, 2016; Hastings, 2010). Human behaviour may change slower than the ecological dynamics and furthermore, the response of the ecological systems to human management strategies may be delayed (Hastings, 2016; Carpenter, 2005).

Phosphorus is commonly considered to be a nutrient of interest in aquatic systems (Carpenter, 2005; Whitton, 2012). Furthermore, the Redfield ratio (C:N:P = 106:16:1) (Redfield, 1934) implies that CB demands phosphorus more than other elements, except perhaps nitrogen (Sterner and Elser, 2002; Whitton, 2012). However, since the demand for phosphorus is high the uptake rates and cell quotas for phosphorus will also be larger than other elements, expect perhaps nitrogen, and thus the corresponding phosphorus dynamics in the media occur on similar timescales to other ecological processes (Heggerud et al., 2020; Whitton, 2012). Other nutrients, such as iron, can limit phytoplankton growth in a significant way by limiting photosynthesis, such as the case of peat lakes in the Netherlands (Smolders and Roelofs, 1993) and regions of the Antarctic (Koch et al., 2019). The extended Redfield ratio implies the requirement of iron is much less than phosphorus and as a result cell quota values are small compared to those for phosphorus (Cunningham and John, 2017). This means that the iron dynamics in the media may occur on a different timescale than the remaining ecological dynamics (Wurtsbaugh and Horne, 1983). Thus, the timescale of the ecological dynamics depends on the study species and the nutrient being considered as uptake and growth rates can vary among species and nutrient (Whitton, 2012).

In this paper we couple the ecological dynamics of cyanobacteria with the socio-economic dynamics of humans at each lake. We consider a network of lakes which are connected via social interactions only, allowing for presence of social norms and ostracism to influence human decision making. The ecological dynamics are given by extending the well established stoichiometric CB model of (Heggerud et al., 2020; Wang et al., 2007). The socio-economic model is an extension of the models discussed in (Iwasa et al., 2007; Iwasa et al., 2010; Sun and Hilker, 2020; Suzuki and Iwasa, 2009) in which individuals in a population choose to either cooperate by lowering pollution rates, or defect, by continuing to pollute at higher rates. The individuals choose their strategy based on costs associated with social pressure, concern for CB, tourism and recreation value, and infrastructure investment (Iwasa et al., 2007). We fully derive the network model and offer several useful simplifications in Section 2. Our analysis begins in Section 3 where we consider the coupled dynamics at a single lake. The analysis of the single lake case is done by utilizing the separation in time scales in several different ways, including a phase line analysis for when phosphorus is the polluting nutrient in Section 3.1 and phase plane analysis when iron is the polluting nutrient in Section 3.2. In each case we observe bistable behaviour and gain insight towards the socio-economic parameter regions that lead to favourable outcomes. Lastly, in Section 4, we revisit the network model. We simplify the network model to allow the system to be studied in the restricted phase plane showing three possible equilibria corresponding the low, high, and mixed levels of cooperation regimes throughout the network. Finally, discuss several two-parameter bifurcation plots which highlight under which parameter regions each regime occurs.