Spatio-temporal statistical models are ubiquitous in the environmental sciences. Often the underlying spatio-temporal process can be written as a hierarchical state-space process that utilizes prior understanding of the physical process to formulate models for the state transition. In situations with complicated dynamics, such as wave propagation, or population growth and dispersal models, parameterizing the transition function associated with the high-dimensional state process may prove difficult or even impossible. One approach to overcoming this difficulty is through the use of polynomial stochastic integro-difference equation (IDE) models. Here, in the context of discrete time and continuous space, complicated dynamics can be accommodated through redistribution kernels that are allowed to vary with space. To facilitate computation and dimension reduction, we consider a basis-function-expansion representation of this model. Even in this framework there are too many parameters to estimate efficiently from a classical perspective. Therefore, we utilize the hierarchical Bayesian framework in order to implement a stochastic-search variable-selection algorithm that mitigates these estimation issues. Finally, the method is illustrated on the problem of long-lead prediction of equatorial sea surface temperature.