Disclosed is a multilayer structure comprising a layer of humidity-dependent composition having a ratio of wet cup permeance to dry cup permeance greater than 10 and a layer of crystallizable polymer sufficiently permeable to oxygen, carbon dioxide and water vapor, exhibiting an increase in permeability to at least one of oxygen and carbon dioxide by a factor of at least 2.5 over a temperature range of 10° C. between 0° C. and 40° C. Also disclosed are packaging materials comprising the multilayer structure and methods for storing respiring materials using the packaging materials.