Abstract
We provide an overview of the results devoted to the analysis of the dynamics and economics of shallow lakes, spanning the period from 1999 until now. A shallow lake serves as a typical representative of an ecological system subject to (possibly irreversible) regime shifts. The dynamics of a shallow lake are described by a non-linear model with multiple steady states and multiple domains of attraction and is thus suitable to model the evolution of an ecosystem featuring both resilience within a domain of stability and an abrupt regime shift outside of it. Beyond this, the shallow lake model can also be viewed as a metaphor for many other ecological problems. Due to the broad applicability of this model, there is substantial interest in the management of shallow lakes and both their optimal regulation and competitive usage.
Highlights
Ecological systems frequently shift between different domains of stability, displaying discontinuous changes of their steady states over time, see, e.g., the classical paper by Scheffer et al [1] as well as more recent [2,3]
While the profit function (8) is quite standard in economic analysis, there are complementary approaches aiming at valuing the ecological services of the ecosystem
We mention the paper by Mooij et al [71] where they present an overview of extremely complex dynamic models aiming at providing a detailed description of the whole ecosystem of a lake in all its complexity. While such models are supposed to account for all processes that influence the dynamics of a shallow lake and the surrounding ecosystem, they are typically difficult to interpret and are hardly amenable to a qualitative analysis. This survey confirms that the qualitative analysis of the dynamics and the management of shallow lake-based ecological systems remains a vibrant research topic, even 20 years after it was initiated
Summary
Ecological systems frequently shift between different domains of stability, displaying discontinuous changes of their steady states over time, see, e.g., the classical paper by Scheffer et al [1] as well as more recent [2,3]. The dynamics of the transition(s) between the oligotrophic and eutrophic types of the lake can be modelled by a scalar non-linear differential equation which has multiple steady-states with separated domains of attraction in a certain range of the phosphorus load These models are deterministic and are convenient to analyse the different optimal paths under various parameter conditions. Using the lake management model of Carpenter et al, Brock and Starrett provided a fairly complete qualitative characterization of the steady states of the optimally controlled system, while Mäler et al extended this work to a dynamic game of common property and to possible tax policies aiming to internalize the resulting externalities Together, these papers provided a firm starting point and unleashed an impulse for an intensive study of the optimal management of shallow lakes. We do not consider those cases here, though, as that would clearly exceed the limits and dilute the focus of this survey
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