How does the sun manage to maintain an environment in its core that is conducive to nuclear fusion?

The answer is the core stays the same because of the gravitational equilibrium. It's in hydrostatic equilibrium. Gravity attempts to crush its mass down, forcing huge pressure at the core, whereas the explosive force of fusion attempts to blast it apart. When the two forces more or less match, you get a continuous rate of fusion, offsetting the huge gravitational pull.