  We calculate the number of dark matter particles that a neutron star
accumulates over its lifetime as it rotates around the center of a galaxy, when
the dark matter particle is a self-interacting boson but does not
self-annihilate. We take into account dark matter interactions with baryonic
matter and the time evolution of the dark matter sphere as it collapses within
the neutron star. We show that dark matter self-interactions play an important
role in the rapid accumulation of dark matter in the core of the neutron star.
We consider the possibility of determining an exclusion region of the parameter
space for dark matter mass and dark matter interaction cross sections based on
the observation of old neutron stars with strong dark matter self-interactions.
We show that for a dark matter density of $~10^3$ GeV/cm$^3$ and dark matter
mass $m_\chi$ less than approximately 10 GeV, there is a potential exclusion
region for dark matter interactions with nucleons that is three orders of
magnitude more stringent than without self-interactions. The potential
exclusion region for dark matter self-interaction cross sections is many orders
of magnitude stronger than the current Bullet Cluster limit. For example, for
high dark matter density regions, we find that for $m_\chi\sim 10$ GeV when the
dark matter interaction cross section with the nucleons ranges from
$\sigma_{\chi n}\sim 10^{-52}$ cm$^2$ to $10^{-57}$ cm$^2$, the dark matter
self-interaction cross section limit is $\sigma_{\chi\chi}< 10^{-33}$ cm$^2$,
which is about ten orders of magnitude stronger than the Bullet Cluster limit.
