Traveling waves in discrete models of biological populations with sessile stages

A common approach to describing invasions of non-native species into previously unoccupied habitat is to consider the speed of population expansion and the existence of traveling waves. Typical existence theorems for traveling waves require some compactness properties of the next-generation operator. Many realistic modeling assumptions, however, give rise to non-compact operators; for example the occurrence of sessile stages during the life cycle of an individual. Recent results have extended the existence theory of traveling waves to a large class of weakly compact operators, but conditions can be difficult to check and not easily accessible to theoretical ecologists. In this paper, we give a new proof for the existence of traveling waves in a large class of equations where the next generation operator is not compact, but rather the sum of an integral operator and a contraction. We illustrate our proof with a model for the dispersal of a plant species with a seed-bank and a model for dispersal of stream insects with larval stages.