  We explore numerically the morphological patterns of thermo-diffusive
instabilities in combustion fronts with a continuum fuel source, within a range
of Lewis numbers and ignition temperatures, focusing on the cellular regime.
For this purpose, we generalize the model of Brailovsky et al. to include
distinct process kinetics and reactant heterogeneity. The generalized model is
derived analytically and validated with other established models in the limit
of infinite Lewis number for zero-order and first-order kinetics. Cellular and
dendritic instabilities are found at low Lewis numbers thanks to a dynamic
adaptive mesh refinement technique that reduces finite size effects, which can
affect or even preclude the emergence of these patterns. This technique also
allows achieving very large computational domains, enabling the study of
system-size effects. Our numerical linear stability analysis is consistent with
the analytical results of Brailovsky et al. The distinct types of dynamics
found in the vicinity of the critical Lewis number, ranging from steady-state
cells to continued tip-splitting and cell-merging, are well described within
the framework of thermo-diffusive instabilities and are consistent with
previous numerical studies. These types of dynamics are classified as
"quasi-linear" and characterized by low amplitude cells and may follow the mode
selection mechanism and growth prescribed by the linear theory. Below this
range of Lewis number, highly non-linear effects become prominent and large
amplitude, complex cellular and {\it{seaweed}} dendritic morphologies emerge.
