  Given a bounded domain $\Omega$ in $\mathbb{R}^N$, $N\geq 1$ we study the
asymptotic behavior as $\varepsilon \to 0$ of the eigencurves of $$
  -\Delta_p u_\varepsilon=\alpha_\varepsilon
m(\tfrac{x}{\varepsilon})(u_\varepsilon^+ )^{p-1} - \beta_\varepsilon
n(\tfrac{x}{\varepsilon})(u_\varepsilon^- )^{p-1} \quad \textrm{ in } \Omega $$
with Dirichlet boundary conditions, where $m$ and $n$ are bounded periodic
weights. In this work we obtain accurate bounds of the convergence rates of
these curves to some limit curves as $\varepsilon \to 0$.
