Dynamics and Management of Sub-divided Populations

Abstract

Multi-site Leslie matrices for sub-divided populations are explored with respect to optimization of management goals and transient dynamics associated with implementing actions to achieve those goals. The management goals explored were minimizing the cost associated with controlling a pest species (Yellow Legged Herring gull, Larus cachinnans), and maximizing the yield from a commercially valuable species (Artco-Norwegian cod, Gadus morhua). Transient dynamics were evaluated for a representative r- and K-selected species, and time to convergence was compared between one-site versus multi-site models, and for different migration patterns, migration levels, and proportion of the population migrating.

In a density-independent model for the Yellow Legged Herring Gull, the most efficient control technique was to focus management actions on the better quality sites, because breeders at high quality sites had higher expected life-time reproductive values. The amount of harvest required to maintain equilibrium was a function of site quality and the balance between immigration and emigration-cost (and effort) increased as dispersal favored better quality sites. Given a choice between destroying eggs or culling adult breeders, culling required ten times less effort per-capita and would be the optimal strategy as long as per-capita culling cost is no more than ten times greater than the per-egg destruction cost.

A density-dependent model of Arcto-Norwegian cod revealed that the theoretical yield was maximized from harvesting age 6 individuals. If only the minimum age harvested could be controlled, then the constrained yield was maximized from harvesting ages five and older. Yields were compared between a reserve model with 25% of fishing area closed and a no-reserve model. Yields in the reserve model exceeded the non-reserve model when transfer rates out of the reserve were higher, when higher fecundity was realized in the reserve (which could result from improved habitat quality), and when fishing rates in the non-reserve model were 1.5 and 2.0 times the optimal level.

In both a density-independent and a density-dependent context, I showed that the optimal strategy could be determined from inspection of elements of the left eigenvector (i.e. reproductive value) divided by a vector of age specific harvest value (or cost of control action, in the case of a pest species). The maximum sustained yield was obtained when the age class with the smallest ratio was harvested; the minimum cost comes from removing individuals with the largest ratio. In one-site models, the optimal strategy involved the harvest of no more than two age classes, where the second (younger) age class had the second smallest ratio (for maximization) or largest ratio (for minimization). However, in multi-site models, the presence of migration permitted the replenishment of age classes beyond the one fully exploited in a harvested site, and thus the optimal strategy could involve the harvest of more than two age classes.

Optimal solutions to the above models corresponded to equilibrium conditions. However, the amount of time between the implementation of a management action and the attainment of equilibrium can be great. Analysis of transient dynamics revealed that the time to convergence is affected by many factors. I defined convergence as the time when a measured population growth rate (and the growth rate in all subsequent years) was within 1% of the asymptotic growth rate (corresponding to equilibrium for a given action). Comparing single-site versus multi-site models, the multi-site models converged more slowly. Multi-site models that incorporated low levels of migration and migration in only the first age class (as opposed to migration in all age classes) converged slowest. Models for the longer-lived K-selected species generally converged more slowly than the short-lived r-selected species, although for some migration patterns (particularly when emigration out of the site receiving a management action exceeded immigration into that site) models for both species converged quickly.