Keywords
optimal control problem, harvest rate, population dynamics, equilibrium analysis
Start Date
1-7-2012 12:00 AM
Abstract
In this paper, we investigated an optimal harvesting problem in a simpleage structure population. We assume that by the end of each reproduction seasonthe population consists of two age groups: juveniles and adults. The increase in thepopulation number is regulated by density-dependent limitation of a younger classsurvival. We consider the strategy with a stationary character of exploitation notleading to the population extinction. The optimization problem is to determine theoptimal catch quotas and equilibrium population size to provide a sustainable yieldand maximum sales return taking price fluctuations into account. It is shown that asingle age class harvesting is the optimal one, and a choice of the age class isdetermined by the values of population parameters and prices ratio. It isdemonstrated that there is a domain of population parameters, characterized by theloss of optimal equilibrium stability at its transition into this domain, and theemergence of 2-cycles, in spite of harvesting with optimal constant catch quota. It isshown that the harvesting impact on the elder age class results in periodicdynamics only in that range of the parameters, where similar dynamics is observedfor the unexploited population. It is found that optimal harvesting with optimalconstant catch quota from the younger class changes the type of dynamicinstability, typical for the population, and also causes regular fluctuations in thenumbers at certain values of parameters. It leads to the necessity of transfer fromharvesting based on constant catch quotas to threshold harvesting. It is shown thatthe threshold strategy always stabilizes the systems dynamics.
Instability of the Exploited Population with a Simple Age Structure
In this paper, we investigated an optimal harvesting problem in a simpleage structure population. We assume that by the end of each reproduction seasonthe population consists of two age groups: juveniles and adults. The increase in thepopulation number is regulated by density-dependent limitation of a younger classsurvival. We consider the strategy with a stationary character of exploitation notleading to the population extinction. The optimization problem is to determine theoptimal catch quotas and equilibrium population size to provide a sustainable yieldand maximum sales return taking price fluctuations into account. It is shown that asingle age class harvesting is the optimal one, and a choice of the age class isdetermined by the values of population parameters and prices ratio. It isdemonstrated that there is a domain of population parameters, characterized by theloss of optimal equilibrium stability at its transition into this domain, and theemergence of 2-cycles, in spite of harvesting with optimal constant catch quota. It isshown that the harvesting impact on the elder age class results in periodicdynamics only in that range of the parameters, where similar dynamics is observedfor the unexploited population. It is found that optimal harvesting with optimalconstant catch quota from the younger class changes the type of dynamicinstability, typical for the population, and also causes regular fluctuations in thenumbers at certain values of parameters. It leads to the necessity of transfer fromharvesting based on constant catch quotas to threshold harvesting. It is shown thatthe threshold strategy always stabilizes the systems dynamics.
