The Influence of Demographic Stochasticity on Population Dynamics: A Mathematical Study of Noise-Induced Bistable States and Stochastic Patterns
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.
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A.J. McKane amoebae analysis analytical treatment autocatalytic behaviour Belousov-Zhabotinsky reaction Biancalani bottom panel boundary conditions Brusselator calculation cell Chap chemical clock chemical reactions chemical species chemical system concentration corresponding defined denoted derivative deterministic equations deterministic limit deterministic system diffusion eigenvalues expression ﬁxed point fluctuations Fokker-Planck equation Fourier transform Gaussian Gillespie algorithm homogeneous initial condition interaction intrinsic noise Kampen expansion Langevin equation Laplacian Lyapunov function mass action master equation mathematical matrix mean switching microscopic nodes noise-induced bistability non-local non-zero number of molecules obtained occurs oscillations panel of Fig parameter pattern formation Phys physics population power spectrum previous chapter probability density function reaction-diffusion regular lattice rescaled spatial spectra stable fixed point stochastic differential equations stochastic patterns stochastic process stochastic Turing patterns stochastic waves thesis time-dependent Togashi–Kaneko transition rates travelling waves upper panel values variable volume wave instability z-equation