Internal dynamics of marine ecosystem models

In der vorliegenden Arbeit werden NPZD Modelle numerisch und theoretisch auf die ihnen innewohnende Dynamik hin untersucht. Im numerischen Teil wird die Parameteroptimierung eines nulldimensionalen Modells, beruhend auf der Formulierung von Oschlies and Garçon (1999), bezüglich zweier verschiedener Datensätze vorgestellt. Dabei wird insbesondere aufgezeigt, inwieweit das untersuchte Modell in der Lage ist, für das marine Ökosystem typische Daten wiederzugeben. Im zweiten, dem theoretischen Part dieser Arbeit, werden verwandte Modellformulierungen auf grundlegende mathematische Eigenschaften und ihre jeweils innewohnende Dynamik hin untersucht. Dies geschieht mittels Methoden der qualitativen Theorie gewöhnlicher Differentialgleichungen. Die Existenz und Eindeutigkeit von Lösungen der zu den Formulierungen gehörenden Anfangswertprobleme wird gezeigt, sowie stationäre Lösungen und deren Stabilitätsverhalten in Abhängigkeit von den gegebenen Modellparametern bestimmt.In this work, the dynamics of NPZD models are numerically and analytically investigated. The numerical part of this study comprises the parameter optimization of a zero dimensional NPZD model, based on the formulation presented in Oschlies and Garçon (1999), with respect to two different data sets. The extent to which the model formulation is capable of reproducing data typical for the marine ecosystem is identfied. In the subsequent, theoretical part of this study, basic mathematical properties of related model formulations are shown and their inherent dynamics are evaluated. The existence and uniqueness of solutions of the associated initial value problems is shown, and equilibrium solutions and their quality are determined as functions of the included parameters

Optimal Control Strategies in Ecosystem-Based Fishery Models

This dissertation considers the use of food chain models coupled with optimal control theory as a new approach for the problem of implementing ecosystem-based fishery management (EBFM) strategies. We consider the Black Sea anchovy on the southern part of the Black Sea as a case study of the implementation of EBFM. Because of the availability of temporal data, we build our first food chain model using ordinary differential equations to describe the anchovy dynamics, and then build our second food chain model using partial differential equations to include spatial features of the anchovy dynamics. In the study, we use the harvest rate of the anchovy fishery as our control that corresponds to number of fishing fleets.In the first model, the Black Sea anchovy stock was coupled with a prey and a predator species, using a system of nonlinear differential equations. The objective for the problem is to find the ecosystem-based optimal harvesting strategy that maximizes the discounted net value of the anchovy population with seasonal harvesting. In our numerical simulations, we obtained more profitable harvesting strategies for the southern part of the Black Sea, and also obtained much better structure for the related food web in terms of population biomasses via the optimal control strategy. Furthermore, we discussed the benefits of using food chain models in fishery management, and derived a schedule for ecosystem-based fishery management of the Black Sea anchovy.In the second model, a spatial food chain model on a bounded domain coupled with optimal control theory examined ecosystem-based harvesting strategies. Our system of nonlinear partial differential equations (PDEs) has logistic growth, movement by diffusion and advection, and Neumann boundary conditions. Numerical simulations were completed to illustrate several scenarios

Robust Parameter Estimation for Rational Ordinary Differential Equations

We present a new approach for estimating parameters in rational ODE models from given (measured) time series data. In typical existing approaches, an initial guess for the parameter values is made from a given search interval. Then, in a loop, the corresponding outputs are computed by solving the ODE numerically, followed by computing the error from the given time series data. If the error is small, the loop terminates and the parameter values are returned. Otherwise, heuristics/theories are used to possibly improve the guess and continue the loop. These approaches tend to be non-robust in the sense that their accuracy depend on the search interval and the true parameter values; furthermore, they cannot handle the case where the parameters are locally identifiable. In this paper, we propose a new approach, which does not suffer from the above non-robustness. In particular, it does not require making good initial guesses for the parameter values or specifying search intervals. Instead, it uses differential algebra, interpolation of the data using rational functions, and multivariate polynomial system solving. We also compare the performance of the resulting software with several other estimation software packages.Comment: Updates regarding robustnes

How models can support ecosystem-based management of coral reefs

Despite the importance of coral reef ecosystems to the social and economic welfare of coastal communities, the condition of these marine ecosystems have generally degraded over the past decades. With an increased knowledge of coral reef ecosystem processes and a rise in computer power, dynamic models are useful tools in assessing the synergistic effects of local and global stressors on ecosystem functions. We review representative approaches for dynamically modeling coral reef ecosystems and categorize them as minimal, intermediate and complex models. The categorization was based on the leading principle for model development and their level of realism and process detail. This review aims to improve the knowledge of concurrent approaches in coral reef ecosystem modeling and highlights the importance of choosing an appropriate approach based on the type of question(s) to be answered. We contend that minimal and intermediate models are generally valuable tools to assess the response of key states to main stressors and, hence, contribute to understanding ecological surprises. As has been shown in freshwater resources management, insight into these conceptual relations profoundly influences how natural resource managers perceive their systems and how they manage ecosystem recovery. We argue that adaptive resource management requires integrated thinking and decision support, which demands a diversity of modeling approaches. Integration can be achieved through complimentary use of models or through integrated models that systemically combine all relevant aspects in one model. Such whole-of-system models can be useful tools for quantitatively evaluating scenarios. These models allow an assessment of the interactive effects of multiple stressors on various, potentially conflicting, management objectives. All models simplify reality and, as such, have their weaknesses. While minimal models lack multidimensionality, system models are likely difficult to interpret as they require many efforts to decipher the numerous interactions and feedback loops. Given the breadth of questions to be tackled when dealing with coral reefs, the best practice approach uses multiple model types and thus benefits from the strength of different models types

Inverse Modelling, Sensitivity and Monte Carlo Analysis in R Using Package FME

Mathematical simulation models are commonly applied to analyze experimental or environmental data and eventually to acquire predictive capabilities. Typically these models depend on poorly defined, unmeasurable parameters that need to be given a value. Fitting a model to data, so-called inverse modelling, is often the sole way of finding reasonable values for these parameters. There are many challenges involved in inverse model applications, e.g., the existence of non-identifiable parameters, the estimation of parameter uncertainties and the quantification of the implications of these uncertainties on model predictions. The R package FME is a modeling package designed to confront a mathematical model with data. It includes algorithms for sensitivity and Monte Carlo analysis, parameter identifiability, model fitting and provides a Markov-chain based method to estimate parameter confidence intervals. Although its main focus is on mathematical systems that consist of differential equations, FME can deal with other types of models. In this paper, FME is applied to a model describing the dynamics of the HIV virus.

Integrating ecology and economics in the mathematical modelling of marine ecosystems

This thesis focusses on the integration of economics and ecology in the mathematical modelling of marine ecosystems. The project addressed two aspects of the problem, 1) the relationship between fish price and landings, and 2) the formulation and solving a mean field game model for fisheries. Regarding price flexibility, we performed statistical analysis on ex-vessel fish prices, landings, and other market variables for the whole UK market, and estimated negative own-price flexibilities for numerous individual fish species, and for some broadly defined guilds of species. These provided relationships between the quantity landed and the price received, which generates a feedback between the marine ecosystem and human activity. Regarding mean field games, we developed a model which considers an economic game with a large number of players exploiting a common resource, whose payoffs depend on the collective the actions of all other players. The application of mean field game approaches to a common resource situation was novel and absent in the literature. Solving the mean field game model numerically, we were able to dynamically simulate the feedback between the marine ecosystem and fishing activity. This allowed us to investigate how the dynamics of the coupled economic-ecological system depended on ecological and economic factors, including the price flexibility identified earlier. We found interesting results relating to the impact of price flexibility and its interaction with stock growth rate, showing that higher price flexibilities resulted in increased fishing pressure for stocks with lower growth rates, but decreased fishing pressure for stocks with very high growth rates. Finally, we modelled the implementation of regulations in the mean field game, and demonstrated how these regulations affect the distribution of fishing activity in a mean field game model for a North Sea cod fishery.This thesis focusses on the integration of economics and ecology in the mathematical modelling of marine ecosystems. The project addressed two aspects of the problem, 1) the relationship between fish price and landings, and 2) the formulation and solving a mean field game model for fisheries. Regarding price flexibility, we performed statistical analysis on ex-vessel fish prices, landings, and other market variables for the whole UK market, and estimated negative own-price flexibilities for numerous individual fish species, and for some broadly defined guilds of species. These provided relationships between the quantity landed and the price received, which generates a feedback between the marine ecosystem and human activity. Regarding mean field games, we developed a model which considers an economic game with a large number of players exploiting a common resource, whose payoffs depend on the collective the actions of all other players. The application of mean field game approaches to a common resource situation was novel and absent in the literature. Solving the mean field game model numerically, we were able to dynamically simulate the feedback between the marine ecosystem and fishing activity. This allowed us to investigate how the dynamics of the coupled economic-ecological system depended on ecological and economic factors, including the price flexibility identified earlier. We found interesting results relating to the impact of price flexibility and its interaction with stock growth rate, showing that higher price flexibilities resulted in increased fishing pressure for stocks with lower growth rates, but decreased fishing pressure for stocks with very high growth rates. Finally, we modelled the implementation of regulations in the mean field game, and demonstrated how these regulations affect the distribution of fishing activity in a mean field game model for a North Sea cod fishery

Network resilience

Many systems on our planet are known to shift abruptly and irreversibly from one state to another when they are forced across a "tipping point," such as mass extinctions in ecological networks, cascading failures in infrastructure systems, and social convention changes in human and animal networks. Such a regime shift demonstrates a system's resilience that characterizes the ability of a system to adjust its activity to retain its basic functionality in the face of internal disturbances or external environmental changes. In the past 50 years, attention was almost exclusively given to low dimensional systems and calibration of their resilience functions and indicators of early warning signals without considerations for the interactions between the components. Only in recent years, taking advantages of the network theory and lavish real data sets, network scientists have directed their interest to the real-world complex networked multidimensional systems and their resilience function and early warning indicators. This report is devoted to a comprehensive review of resilience function and regime shift of complex systems in different domains, such as ecology, biology, social systems and infrastructure. We cover the related research about empirical observations, experimental studies, mathematical modeling, and theoretical analysis. We also discuss some ambiguous definitions, such as robustness, resilience, and stability.Comment: Review chapter