Given a set of multivariate, quadratic, non-homogeneous equations, is there a way to decide how many non-negative roots it have?

Some explanations:

- All the coefficients are real numbers.
- The number of variables is the same as the number of equations, and all the equations are independent.
- Some of the equations might actually be linear equations.
- By "non-negative" solution, I mean a solution in which all the variables take non-negative values.
- By some "physical" reasoning, we know it must have at least one non-negative solution.

Further explanations:

- I know generally a set of n quadratic equations with n variables has at most $2^n$ distinct roots.
- The background of this problem is: the set of quadratic equations is the right-hand side of the chemical rate equation. By equating is to zero, the steady-state case is being considered. As we only take one-body or two-body reactions into account, the degree of the equations are at most two. As the abundance of the molecules cannot be negative, we only care about the non-negative solutions.
- The number of variables can be up to 1000, so simple numerical test is not practical.

I am not a math student, and I am not sure whether this kind of question is allowed here.