| Summary: | This thesis consists of the manuscripts of three research papers studying stochastic ODEs (ordinary differential equations) and PDEs (partial differential equations) that arise in biological models of interacting multi-type populations.
In the first paper I prove uniqueness of the martingale problem for a degenerate SDE (stochastic differential equation) modelling a catalytic branching network. This work is an extension of a paper by Dawson and Perkins to arbitrary networks. The proof is based upon the semigroup perturbation method of Stroock and Varadhan. In the proof estimates on the corresponding semigroup are given in terms of weighted Hölder norms, which are equivalent to a semigroup norm in this generalized setting. An explicit representation of the semigroup is found and estimates using cluster decomposition techniques are derived.
In the second paper I investigate the long-term behaviour of a special class of the SDEs considered above, involving catalytic branching and mutation between types. I analyse the behaviour of the overall sum of masses and the relative distribution of types in the limit using stochastic analysis. For the latter existence, uniqueness and convergence to a stationary distribution are proved by the reasoning of Dawson, Greven, den Hollander, Sun and Swart. One-dimensional diffusion theory allows for a complete analysis of the two-dimensional case.
In the third paper I show that one can construct a sequence of rescaled perturbations of voter processes in d=1 whose approximate densities are tight. This is an extension of the results of Mueller and Tribe for the voter model. We combine critical long-range and fixed kernel interactions in the perturbations. In the long-range case, the approximate densities converge to a continuous density solving a class of SPDEs (stochastic PDEs). For integrable initial conditions, weak uniqueness of the limiting SPDE is shown by a Girsanov theorem. A special case includes a class of stochastic spatial competing species models in mathematical ecology. Tightness is established via a Kolmogorov tightness criterion. Here, estimates on the moments of small increments for the approximate densities are derived via an approximate martingale problem and Green's function representation.
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