Generalized models as a universal approach to the analysis of nonlinear dynamical systems

We present a universal approach to the investigation of the dynamics in generalized models. In these models the processes that are taken into account are not restricted to specific functional forms. Therefore a single generalized models can describe a class of systems which share a similar structure. Despite this generality, the proposed approach allows us to study the dynamical properties of generalized models efficiently in the framework of local bifurcation theory. The approach is based on a normalization procedure that is used to identify natural parameters of the system. The Jacobian in a steady state is then derived as a function of these parameters. The analytical computation of local bifurcations using computer algebra reveals conditions for the local asymptotic stability of steady states and provides certain insights on the global dynamics of the system. The proposed approach yields a close connection between modelling and nonlinear dynamics. We illustrate the investigation of generalized models by considering examples from three different disciplines of science: a socio-economic model of dynastic cycles in china, a model for a coupled laser system and a general ecological food web.Comment: 15 pages, 2 figures, (Fig. 2 in color

Mathematical models for chemotaxis and their applications in self-organisation phenomena

Chemotaxis is a fundamental guidance mechanism of cells and organisms, responsible for attracting microbes to food, embryonic cells into developing tissues, immune cells to infection sites, animals towards potential mates, and mathematicians into biology. The Patlak-Keller-Segel (PKS) system forms part of the bedrock of mathematical biology, a go-to-choice for modellers and analysts alike. For the former it is simple yet recapitulates numerous phenomena; the latter are attracted to these rich dynamics. Here I review the adoption of PKS systems when explaining self-organisation processes. I consider their foundation, returning to the initial efforts of Patlak and Keller and Segel, and briefly describe their patterning properties. Applications of PKS systems are considered in their diverse areas, including microbiology, development, immunology, cancer, ecology and crime. In each case a historical perspective is provided on the evidence for chemotactic behaviour, followed by a review of modelling efforts; a compendium of the models is included as an Appendix. Finally, a half-serious/half-tongue-in-cheek model is developed to explain how cliques form in academia. Assumptions in which scholars alter their research line according to available problems leads to clustering of academics and the formation of "hot" research topics.Comment: 35 pages, 8 figures, Submitted to Journal of Theoretical Biolog

The Social and Ecological Dimensions of Vertebrate Management: Reintroductions and Invasions

Conflicts between wildlife and humans continue to be a persistent problem across a wide spectrum of landscapes. In the body of work below, I focus on two classes of vertebrates in particular, invasive pest species and reintroduced species. Invasive species and reintroduced species are both species with which humans can conflict, which has profound consequences for the persistence of species across the landscape and long term human livelihoods. Populations of both invasive and native species typically exist at low densities at first, then establish, grow, and spread across the landscape. Both invasions and reintroductions can be strongly influenced by the human landscape and tolerance for the presence of particular species and their associated impacts to nature and their livelihoods. For invasive and pest species, opposition to eradication programs has the ability to stop or stall management, which has implications for the successful establishment and spread of an invasive species. Conversely, public support or opposition for reintroduction programs can dictate whether they happen at all. Understanding the human landscape of tolerance is important in understanding the success or failure of conservation programs more generally.The body of work below focuses on both of these classes of species and examines different problems associated with each, using techniques from both natural and social sciences. The first three chapters of this dissertation focus on the ecological and social dynamics of vertebrate pest species. I begin by exploring the utility of barn owls to reduce and control populations of vertebrate pests in agricultural landscapes. Next, I examined case of the wild pig, first comparing the population demographic characteristics of wild pigs and second, understanding what kinds of message frames can increase support for invasive wild pig management. Finally, I used social science techniques to understand attitudes toward grizzly bear reintroduction in California

Fighting cheaters: how and how much to invest.

Human societies are formed by different socio-economical classes which are characterized by their contribution to, and their share of, the common wealth available. Cheaters, defined as those individuals that do not contribute to the common wealth but benefit from it, have always existed, and are likely to be present in all societies in the foreseeable future. Their existence brings about serious problems since they act as sinks for the community wealth and deplete resources which are always limited and often scarce. To fight cheaters, a society can invest additional resources to pursue one or several aims. For instance, an improvement in social solidarity (e.g. by fostering education) may be sought. Alternatively, deterrence (e.g. by increasing police budget) may be enhanced. Then the following questions naturally arise: (i) how much to spend and (ii) how to allocate the expenditure between both strategies above. This paper addresses this general issue in a simplified setting, which however we believe of some interest. More precisely, we consider a society constituted by two productive classes and an unproductive one, the cheaters, and proposes a dynamical system that describes their evolution in time. We find it convenient to formulate our model as a three-dimensional ordinary differential equation (ODE) system whose variables are the cheater population, the total wealth and one of the productive social classes. The stationary values of the cheater population and the total wealth are studied in terms of the two parameters: φ (how much to invest) and s (how to distribute such expenditure). We show that it is not possible to simultaneously minimize the cheater population and maximize the total wealth with respect to φ and s. We then discuss the possibility of defining a compromise function to find suitable values of φ and s that optimize the response to cheating. In our opinion, this qualitative approach may be of some help to plan and implement social strategies against cheating

The interplay of migration and population dynamics in a patchy world

One of the important issues in spatial ecology is how explicit considerations of space alter the prediction of population models. In this thesis we scrutinized some classical theories associated with the issue via spatial population models derived mechanistically. Incorporation of assumptions concerning the behavioural details of individuals of species living in a patchy habitat naturally gives rise to cross-migration models in which the per-capita rate of migration of one species depends on the density of some other species. To look into the impact of such a cross-migration factor on population dynamics we first studied a specific two-patch predator-prey crossmigration model while focusing on the hypothesis that space reduces predator-prey oscillations. We then investigated some general multi-patch multi-species crossmigration models while concentrating on the well-known theory of Turing Instability. We obtained new insights into these theoretical issues

Why Money Trickles Up - Wealth & Income Distributions

This paper combines ideas from classical economics and modern finance with the general Lotka-Volterra models of Levy & Solomon to provide straightforward explanations of wealth and income distributions. Using a simple and realistic economic formulation, the distributions of both wealth and income are fully explained. Both the power tail and the log-normal like body are fully captured. It is of note that the full distribution, including the power law tail, is created via the use of absolutely identical agents. It is further demonstrated that a simple scheme of compulsory saving could eliminate poverty at little cost to the taxpayer.Comment: 45 pages of text, 36 figure

Artificial Societies of Intelligent Agents

In this thesis we present our work, where we developed artificial societies of intelligent agents, in order to understand and simulate adaptive behaviour and social processes. We obtain this in three parallel ways: First, we present a behaviours production system capable of reproducing a high number of properties of adaptive behaviour and of exhibiting emergent lower cognition. Second, we introduce a simple model for social action, obtaining emergent complex social processes from simple interactions of imitation and induction of behaviours in agents. And third, we present our approximation to a behaviours virtual laboratory, integrating our behaviours production system and our social action model in animats. In our behaviours virtual laboratory, the user can perform a wide variety of experiments, allowing him or her to test the properties of our behaviours production system and our social action model, and also to understand adaptive and social behaviour. It can be accessed and downloaded through the Internet. Before presenting our proposals, we make an introduction to artificial intelligence and behaviour-based systems, and also we give notions of complex systems and artificial societies. In the last chapter of the thesis, we present experiments carried out in our behaviours virtual laboratory showing the main properties of our behaviours production system, of our social action model, and of our behaviours virtual laboratory itself. Finally, we discuss about the understanding of adaptive behaviour as a path for understanding cognition and its evolution

Decision-Making in Agent-Based Modeling: A Current Review and Future Prospectus

All basic processes of ecological populations involve decisions; when and where to move, when and what to eat, and whether to fight or flee. Yet decisions and the underlying principles of decision-making have been difficult to integrate into the classical population-level models of ecology. Certainly, there is a long history of modeling individuals' searching behavior, diet selection, or conflict dynamics within social interactions. When all the individuals are given certain simple rules to govern their decision-making processes, the resultant population–level models have yielded important generalizations and theory. But it is also recognized that such models do not represent the way real individuals decide on actions. Factors that influence a decision include the organism's environment with its dynamic rewards and risks, the complex internal state of the organism, and its imperfect knowledge of the environment. In the case of animals, it may also involve complex social factors, and experience and learning, which vary among individuals. The way that all factors are weighed and processed to lead to decisions is a major area of behavioral theory.While classic population-level modeling is limited in its ability to integrate decision-making in its actual complexity, the development of individual- or agent-based models (IBM/ABMs) (we use ABM throughout to designate both “agent-based modeling” and an “agent-based model”) has opened the possibility of describing the way that decisions are made, and their effects, in minute detail. Over the years, these models have increased in size and complexity. Current ABMs can simulate thousands of individuals in realistic environments, and with highly detailed internal physiology, perception and ability to process the perceptions and make decisions based on those and their internal states. The implementation of decision-making in ABMs ranges from fairly simple to highly complex; the process of an individual deciding on an action can occur through the use of logical and simple (if-then) rules to more sophisticated neural networks and genetic algorithms. The purpose of this paper is to give an overview of the ways in which decisions are integrated into a variety of ABMs and to give a prospectus on the future of modeling of decisions in ABMs