  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.
