Evolution of the N₂H⁺a), b) and HC₃N c), d) abundances, and the HC₃N to N₂H⁺ abundance ratio e), f) as a function of time for free-particle densities of 1 × 10⁵ cm^-3 (dotted), 3 × 10⁵ cm^-3 (solid), and 1 × 10⁶ cm^-3 (dashed line) and a kinetic temperature of 9 K. Each column shows predictions from one specific model (static, contraction). The predictions from the contraction models correspond to abundances derived from integrated (along all depth points) column densities (see Garrod et al. 2006a,b). The horizontal dot-dashed lines indicate the range of observed values for the N₂H⁺ (top row), HC₃N abundances (middle row), and the HC₃N to N₂H⁺ ratio (bottom row). The maximum H₂ column densities for the contraction models are achieved at 1 Myr and they are ~4 × 10²¹ cm^-2, ~ 1.3 × 10²² cm^-2, and ~ 4 × 10²² cm^-2, respectively.