Intra-specific competition and insect larval development: A model with time-dependent delay

We derive a stage-structured model for an insect population in which a larva matures on reaching a certain size, and in which there is intra-specific competition among larvae that hinders their development, thereby prolonging the larval phase. The model, a system of delay differential equations for the total numbers of adults and larvae, assumes two forms. One of these is a system with a variable state-dependent time delay determined by a threshold condition, the other has constant and distributed delays, a size-like independent variable replacing time t, and no threshold condition. We prove theorems on boundedness and on the linear stability of equilibria.