This thesis consists of four papers, three papers about modelling animal populations and one paper about an area integral estimate for solutions of partial differential equations on non-smooth domains. The papers are:
I. Å. Brännström, Single species population models from first principles.
II. Å. Brännström and D. J. T. Sumpter, Stochastic analogues of deterministic single species population models.
III. Å. Brännström and D. J. T. Sumpter, Coupled map lattice approximations for spatially explicit individual-based models of ecology.
IV. Å. Brännström, An area integral estimate for higher order parabolic equations.
In the first paper we derive deterministic discrete single species population models with first order feedback, such as the Hassell and Beverton-Holt model, from first principles. The derivations build on the site based method of Sumpter & Broomhead (2001) and Johansson & Sumpter (2003). A three parameter generalisation of the Beverton-Holtmodel is also derived, and one of the parameters is shown to correspond directly to the underlying distribution of individuals.
The second paper is about constructing stochastic population models that incorporate a given deterministic skeleton. Using the Ricker model as an example, we construct several stochastic analogues and fit them to data using the method of maximum likelihood. The results show that an accurate stochastic population model is most important when the dynamics are periodic or chaotic, and that the two most common ways of constructing stochastic analogues, using additive normally distributed noise or multiplicative lognormally distributed noise, give models that fit the data well. The latter is also motivated on theoretical grounds.
In the third paper we approximate a spatially explicit individual-based model with a stochastic coupledmap lattice. The approximation effectively disentangles the deterministic and stochastic components of the model. Based on this approximation we argue that the stable population dynamics seen for short dispersal ranges is a consequence of increased stochasticity from local interactions and dispersal.
Finally, the fourth paper contains a proof that for solutions of higher order real homogeneous constant coefficient parabolic operators on Lipschitz cylinders, the area integral dominates the maximal function in the L2-norm.
2004. , 18 p.
population model, stochastic population model, population dynamics, discrete time model, Beverton-Holt model, Skellam model, Hassell model, Ricker model, first principles, coupled map lattice, CML, area integral, square function