The paper deals with a model for water freezing in a deformable elastoplastic container. The mathematical problem consists of a system of one parabolic equation for temperature, one integrodifferential equation with a hysteresis operator for local volume increment, and one differential inclusion for the water content. The problem is shown to admit a unique global uniformly bounded weak solution.
The cocoons characteristic of the prepupal and pupal stages of many insects vary widely in size, durability, structure, shape and colour, as well as in other features such as orientation and attachment to the substrate. In some species they vary seasonally. Most cocoons provide little direct insulation, although they may reduce the rate at which temperature changes, but many provide the mechanical protection required for overwintering beneath insulating substrates such as soil and snow. The cocoons of some terrestrial species prevent inoculative freezing by isolating the integument from ice crystals on the cocoon surface or its surroundings. In some aquatic species, cocoons appear to limit damage by providing mechanical protection during the freezing of surrounding water. Some cocoons help in the acquisition of solar heat: dark structures are especially effective because dark pigments absorb heat, and surrounding layers trap this heat. Insects are immobilized when it is cold and so cannot move in response to environmental threats, and protective cocoons made for winter tend to be more robust than their summer counterparts. Such cocoons protect against abrasion of the waterproof layer of the cuticle. In some species, robust cocoons or complex structures impede natural enemies. Cocoon silk has anti-bacterial and anti-fungal actions. Other cocoons are more or less waterproof. These and other features withstand simultaneous constraints in addition to cold. Therefore, cocoons enhance survival during cold conditions in many species. However, this conclusion is based on fragmentary evidence, and there has been relatively little explicit examination of the roles of cocoons during winter. Therefore, specific work is required to assess resistance to or enhancement of inoculative freezing, resistance to penetration by natural enemies and water, the roles of particular cocoon silks and silk constituents, and the quantitative contributions of cocoons to winter survival in nature.