We study the relations between gas-phase metallicity ($Z$), local stellar mass surface density ($Sigma_*$), and the local star formation surface density ($Sigma_{rm SFR}$) in a sample of 1120 star-forming galaxies from the MaNGA survey. At fixed $Sigma_{*}$ the local metallicity increases as decreasing of $Sigma_{rm SFR}$ or vice versa for metallicity calibrators of N2 and O3N2. Alternatively, at fixed $Sigma_{rm SFR}$ metallicity increases as increasing of $Sigma_{*}$, but at high mass region, the trend is flatter. However, the dependence of metallicity on $Sigma_{rm SFR}$ is nearly disappeared for N2O2 and N2S2 calibrators. We investigate the local metallicity against $Sigma_{rm SFR}$ with different metallicity calibrators, and find negative/positive correlations depending on the choice of the calibrator. We demonstrate that the O32 ratio (or ionization parameter) is probably dependent on star formation rate at fixed local stellar mass surface density. Additional, the shape of $Sigma_*$ -- $Z$ -- $Sigma_{rm SFR}$ (FMR) depends on metallicity calibrator and stellar mass range. Since the large discrepancy between the empirical fitting-based (N2, O3N2) to electronic temperature metallicity and the photoionization model-dependent (N2O2, N2S2) metallicity calibrations, we conclude that the selection of metallicity calibration affects the existence of FMR on $Sigma_{rm SFR}$.
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