The transverse momentum spectra of identified particles produced in high energy proton-proton ($p+p$) collisions are empirically described by a new method with the framework of participant quark model or the multisource model at the quark level, in which the source itself is exactly the participant quark. Each participant (constituent) quark contributes to the transverse momentum spectrum, which is described by the TP-like function, a revised Tsallis--Pareto-type function... The transverse momentum spectrum of the hadron is the convolution of two or more TP-like functions. For a lepton, the transverse momentum spectrum is the convolution of two TP-like functions due to two participant quarks, e.g. projectile and target quarks, taking part in the collisions. A discussed theoretical approach seems to describe the $p+p$ collisions data at center-of-mass energy $\sqrt{s}=200$ GeV, 2.76 TeV, and 13 TeV very well. (read more)