Optimal harvesting of an age-structured, two-sex herbivore–plant system

Abstract

This study presents an optimal harvesting model for the semi-domesticated reindeer (Rangifer t. tarandus) and its main winter energy source, ground lichens (Cladonia spp.). Females are divided into 17 age classes and males into 13. Reproduction is specified by a modified harmonic mean mating function and an age-specific reproduction success. Lichen availability determines individuals’ overwinter weight decrease, natural mortality, the number of calves per females and calves’ birth weight. The reindeer herding cooperative can choose the number of animals harvested from the 30 age/sex classes and is assumed to maximize the preset value of net income. The structured optimization model is solved as a fully dynamic system and for initial states that may not be close to the optimal steady state. This enables to study optimal recovery from overgrazed pastures and the optimality of the constant escapement policy. We show that given zero interest rate the optimal steady-state lichen density is less than 50% of the maximum sustainable yield level but nearly twice as high as empirically observed. Density dependence at optimal equilibrium is realized in calf weight and in the average number of calves per female. Optimal slaughtering is concentrated on six-month-old calves. Adult females are slaughtered at the age of 9.5 years and males at the age of 5.5 years. A moderate or high interest rate increases the steady state reindeer population but decreases pasture conditions. Dynamic solutions deviate from constant escapement, implying that the optimal recovery from initially overgrazed pastures is slower than suggested in existing studies and actual policy. The shadow value for males is three times higher than for females.