Lea F. Murphy

Optimal harvesting of an age-structured population

Here we investigate the optimal harvesting of an age-structured population. We use the McKendrick model of population dynamics, and optimize a discounted yield on an infinite time horizon. The harvesting function is allowed to depend arbitrarily on age and time and its magnitude is unconstrained. We obtain, in addition to existence, the qualitative result that an optimal harvesting policy consists of harvesting at no more than three distinct ages.

On the optimal harvesting of persistent age-structured populations

SummaryThis paper discusses optimal harvesting policies for age-structured populations harvested with effort independent of age.

On the optimal harvesting of age-structured populations: Some simple models☆

Abstract In this paper we develop optimal harvesting policies for age-structured populations using a model for which the basic equations reduce to a pair of ordinary diffential equations for the total population and the per-capita birth-rate. Our assumptions insure the existence of a critical size Pc(t) which maximizes the instantaneous growth-rate at time t. We study the infinite-horizon problem, using the overtaking criterion of optimality, and show that: for a large population with ample per-capita birth-rate the optimal policy is to instantly reduce the stock to the critical value Pc(0), and then to harvest along the path Pc(t); for a sufficiently small population it is optimal to refrain from harvesting until the population reaches Pc(t), and then to harvest along Pc(t) for all subsequent time; for a large population with a small per-capita birth-rate, it is generally best to initially remove a given amount of stock, then to refrain from harvesting until the population reaches Pc(t), and finally to harvest alongPc(t).

A nonlinear growth mechanism in size structured population dynamics

Abstract The dynamics of a population depend upon the sizes and/or maturity of the individual members of the population. In many species (the most popular examples are fish and trees) the growth and maturation rate of an individual is controlled by the ability of that individual to obtain necessary resources. For such a species, individual growth rates depend upon the overall condition of the population. The purpose of this paper is to develop a continuous model with size structure, in which individual growth rates are not known a priori. An individuals size depends not only upon its age, but upon the history of the general population during its past life. The resulting model is a nonlinear variation of the classical McKendrick-Von Foerster model. The mathematical descriptions of growth processes which originate in the biological literature can be used as constitutive equations in our model. In some cases, such substitutions allow the model to be reduced to a system of ordinary differential equations.

Maximum sustainable yield of a nonlinear population model with continuous age structure

Here we investigate the maximum sustainable yield problem for an age-structured population whose dynamics are density dependent. We use the nonlinear version of the McKendrick model of population dynamics that was introduced by Gurtin and MacCamy and do not constrain the magnitude of the harvesting term. We show that this problem has an optimal solution and that the optimum is attainable by a bimodal harvesting policy. This result is consistent with the results obtained by Grey for the nonlinear Leslie model.

Density dependent cellular growth in an age structured colony

Abstract The McKendrick-Von Foerster model is often used to model cell colonies with both constant and arbitrarily varying generation times. Our purpose here is to extend the model by linking the fluctuation of the generation time to the history of the colony. Such a linkage is natural, since, for example, when a colony grows large enough to severely tax its environment, the generation time lengthens accordingly. The resulting model consists of a pair of hyperbolic balance laws with a boundary condition of the form u (0, t ) = 2(1 − m ′( t )) u ( m ( t ), t ), where m depends functionally on the solution u . We show the model to be well posed and demonstrate its ability to duplicate observed biological phenomena in a simple case.

A mathematical analysis of small mammal populations

Populations of Microtus montanus, the montane vole, have been extensively studied. It is known that their reproductive activity is closely linked to the availability of the chemicals in growing plants. We use a mathematical model here to study how the length of the vegetative season and the natural reproduction rhythm of voles are involved in the long term dynamics of the population numbers. In particular, we use data obtained from Timpie Springs, Utah, and from Jackson Hole, Wyoming, to formulate a model. The novelty of this model is its use of littering curves that highlight the temporally discrete nature of vole reproduction. The model shows how the timing of the vegetative season can influence vole population sizes.