Göran Högnäs: On some stochastic competition models

In his recent review, Schreiber (2012) describes the state of the art for stochastic competition models of the general form

X_{t+1}^i = f_i(X_t,\xi_t)X_t^i ,   i=1, ..., k, t=0, 1, 2, ...

where the state space S is a subset of $\Re_+^k$ and the union of the coordinate axes in $\Re^k_+ $ forms the extinction set  S_0; the $\xi$'s represent a randomly evolving environment.

I will look at some particular cases and variants of this model. Specifically, we will limit ourselves to two populations, i.e., k=2, and the functions f_i are chosen to be of the Ricker type

\exp(r_t^i - K_t^i(X_t^i + \alpha_j  X_t^j) ),   i,j=1,2, i \neq j.

Here the $r_t^i$'s model the average intrinsic per capita growth rate at time t. The growth is attenuated by the negative term  where the factor $K_t^i$ describes the intra-specific competition at time $t$ and  $a_j$ (assumed constant over time) the relative importance of the inter-specific competition.

Another variant of the model is concerned with demographic stochasticity. Here our aim is to study the evolution of two finite populations as size-dependent branching processes, which on average follow the above Ricker-type model (with non-random $r^i, K^i $). We want to describe the long-term behavior and compare it with that of the corresponding deterministic model.  The size-dependent branching processes will necessarily have finite life-times. Can anything be said about those?    

Joint work with Henrik Fagerholm, Mats Gyllenberg and Brita Jung.

Main reference: 
Sebastian J.\ Schreiber: Persistence for stochastic difference equations: A minireview, Journal of Difference Equations and Applications (2012),
now available via the author's web site www-eve.ucdavis.edu/sschreiber/pubs.shtml