Standard transport theory is used to investigate the acceleration and modulation of anomalous cosmic rays during solar minimum periods. With the developed anomalous cosmic ray transport model, pickup ions are transformed into anomalous cosmic rays by diffusive shock acceleration at the heliospheric termination shock. The emphasis is on explaining the 1987 cosmic ray spectra at large radial distances in the equatorial plane. Energy density calculations of cosmic ray spectra at the termination shock suggest that the termination shock might be strongly mediated by anomalous cosmic rays. Therefore, in the spirit of nonlinear shock acceleration theory, the termination shock is modeled as a hyperbolic tangent. Calculations show that realistic anomalous cosmic ray intensity radial gradients can only be achieved, within the framework of our assumptions, for a termination shock structure with a compression ratio of 3.2<s<4.0 and a scale length of ~1.2 AU. In addition, simulations of anomalous and galactic cosmic ray spectra revealed as follows: (1) For radial distances of less than at least 24 AU from the Sun it should be generally difficult, although not impossible, to identify anomalous protons, for example, in the observed 1987 proton spectra. (2) Anomalous protons should be more easily detectable, however, in the proton spectra seen at 42 AU by Pioneer 10 during 1987 and should be clearly visible at larger radial distances. (3) The best way to identify anomalous protons in observed proton spectra is to look for a clear flattening in the spectral slopes in the energy interval of ~20--100 MeV. Recent 1994 observations by Pioneer 10 at ~60 AU and by Voyager 1 and Voyager 2 support the model prediction of very flat proton spectra in the above mentioned energy interval at large radial distances. Within the framework of our assumptions this might be an indication that the average termination shock position during 1994 was in the vicinity of 80 AU as assumed in the model. ¿ American Geophysical Union 1996