  Multi-band emission from radio to ultra-high energy gamma-rays in the Crab
Nebula has been detected. To explain the observed results, non-thermal photon
production \textbf{in} the Crab Nebula is carefully studied in a spatially
dependent lepto-hadronic model. In our model, the dynamical evolution of the
PWN is simulated in a spherically symmetric system. Both electrons and protons
are accelerated at the termination shock. The relevant particle propagation
equations as well as the photon evolving equation are simultaneously solved.
For the Crab Nebula, our results reveal that the observed multi-band photon
spectra can be well reproduced with reasonable model parameters. In particular,
the photons with energy $\gtrsim 200$ TeV are mainly contributed by the
hadronic component via proton-proton interaction. The contribution of the
hadronic component depends on both proton spectral index $\alpha_{\rm p}$ and
number density $n_{\rm H}$ of medium within the PWN. Besides, high energy
neutrino fluxes are predicted with variable proton spectral indices. The
predicted fluxes are not only far below the sensitivities of current neutrino
observatories, but also beneath the atmospheric neutrino background with energy
less than $\sim 40$ TeV. Moreover, the calculated radial profiles of surface
brightness and spectral index are presented.
