Evolution of the density response of juvenile hormone concentration in the determination of wing type

Question: What is the evolutionarily stable density-dependent juvenile hormone concentration that determines wing type in wing-polymorphic insects? Key assumptions: An asexual species with larval and adult stages living in many patches. Larval growth rate depends on within-patch resources that are depleted by feeding larvae. Dispersal-type adults migrate out of the natal patch just before the reproductive stage, whereas reproductive-type adults do not disperse but are more fertile than dispersal types. The proportions of the two wing types are determined by juvenile hormone concentration, which follows a normal distribution with the mean decreasing linearly with density (larval biomass per unit resource). The carrying capacity of resources fluctuates between high and low values. Method: We studied the evolution of three quantities controlling juvenile hormone concentration: (1) baseline value of the mean, γ₀; (2) density dependence of the mean, γ₁; and (3) variance, σ². Results: If γ₀ and γ₁ evolved with σ² fixed, juvenile hormone production was high at low density and decreased with increasing density. If σ² evolved with γ₀ and γ₁ fixed, the variance evolved to be smaller for larger environmental fluctuations. If all three quantities (γ₀, γ₁, and σ²) evolved simultaneously, the variance attained a minimum value when environmental fluctuation was large. In all three cases, evolution increased the proportion of dispersal-type individuals, which increased with density in an accelerating manner.