[en] Trapped electron mode (TEM) turbulence exhibits a rich variety of collisional and zonal flow physics. This work explores the parametric variation of zonal flows and underlying mechanisms through a series of linear and nonlinear gyrokinetic simulations, using both particle-in-cell (the GEM code) and continuum (the GS2 code) methods. The two very different and independent codes are compared in studies of TEM turbulence, providing verification. Zonal flows are important in TEM turbulence near the critical density gradient. Our previous work revealed a new nonlinear upshift in the TEM critical density gradient, associated with zonal flow dominated states, which increases strongly with collisionality. In contrast, zonal flows have little effect on the TEM saturation level in cases with strong electron temperature gradients and T_e = 3T_i. We have addressed this apparent contradiction in a series of linear and nonlinear simulations using GS2 and GEM. Zonal flows in TEM turbulence are sensitive to the electron temperature gradient, T_e/T_i, and other parameters. The link between zonal flows and the strength of the electron temperature gradient can be understood using both TEM linear stability properties and secondary instability theory. We have developed a comprehensive TEM/ETG stability diagram to illustrate that strong electron temperature gradients tend to drive fine scale structure, weakening zonal flow drive. When zonal flows are not the dominant saturation mechanism, a simple mode coupling model shows that stable zonal density fluctuations are driven to large amplitudes, at twice the growth rate of the dominant 'primary' mode. Simple estimates of the saturation level agree with GEM simulations. (author)