ISBN:
9783319077284
Language:
English
Pages:
1 online resource (123 pages)
Edition:
1st ed.
Series Statement:
Springer Theses Ser.
Parallel Title:
Erscheint auch als
DDC:
304.60151922
Keywords:
Statistical physics ; Data processing..
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Stochastic analysis ; Congresses
;
Electronic books
Abstract:
The dynamics of population systems cannot be understood within the framework of ordinary differential equations, which assume that the number of interacting agents is infinite. With recent advances in ecology, biochemistry and genetics it is becoming increasingly clear that real systems are in fact subject to a great deal of noise. Relevant examples include social insects competing for resources, molecules undergoing chemical reactions in a cell and a pool of genomes subject to evolution.?When the population size is small, novel macroscopic phenomena can arise, which can be analyzed using the theory of stochastic processes. This thesis is centered on two unsolved problems in population dynamics: the symmetry breaking observed in foraging populations and the robustness of spatial patterns. We argue that these problems can be resolved with the help of two novel concepts: noise-induced bistable states and stochastic patterns.
Abstract:
326645_1_En_OFC -- 326645_1_En_BookFrontmatter_OnlinePDF -- Supervisor's Foreword -- Abstract -- Acknowledgments -- Contents -- 326645_1_En_1_Chapter_OnlinePDF -- 1 Introduction -- 1.1 The Failure of Reductionism -- 1.1.1 Langton's Ant -- 1.2 Emergent Phenomenon in Real Systems -- 1.2.1 Aggregation of Slime Mould Amoebae -- 1.2.2 The Coats of Mammals -- 1.2.3 The Belousov-Zhabotinsky Reaction -- 1.3 The Origin of Intrinsic Noise -- 1.3.1 A Pedagogical Example -- 1.3.2 The Law of Mass Action -- 1.3.3 The Stochastic Approach -- References -- 326645_1_En_2_Chapter_OnlinePDF -- 2 Methods -- 2.1 Stochastic Formulation of Chemical Systems -- 2.1.1 Stochastic Processes -- 2.1.2 Markovian Processes -- 2.1.3 Homogeneous Processes -- 2.1.4 The Master Equation -- 2.1.5 Choosing the Transition Rates -- 2.1.6 Formalism for a General Network -- 2.2 Approximation Schemes for the Master Equation -- 2.2.1 The Deterministic Limit -- 2.2.2 The Kramers-Moyal Expansion -- 2.2.3 The Langevin Picture -- 2.2.4 On the Noise Matrix mathcalB -- References -- 326645_1_En_3_Chapter_OnlinePDF -- 3 Noise-Induced Bistability -- 3.1 The Simplified Togashi--Kaneko Model -- 3.1.1 Analysis in the Deterministic Limit -- 3.2 The Togashi--Kaneko Numerical Experiment -- 3.3 Analytical Treatment of the Togashi--Kaneko Experiment -- 3.3.1 Obtaining an Approximate Equation -- 3.3.2 The Stationary Distribution -- 3.4 Time-Dependent Analysis for λ=λc/2 -- 3.4.1 A Map into the Diffusion Equation -- 3.4.2 The Time-Dependent Distribution P(z,t) -- 3.4.3 The Statistics of Switches -- 3.5 A General Time-Dependent Analysis -- 3.5.1 The Mean Switching Time for a General λ -- 3.5.2 The Mean Switching Time for ε=0 -- 3.5.3 An Experimental Test for Noise-Induced Bistable States -- 3.6 Beyond the Simplified Togashi--Kaneko Scheme -- 3.6.1 A Class of Chemical Schemes -- 3.6.2 A Lyapunov Function.
Note:
Description based on publisher supplied metadata and other sources
