David A. Sánchez

Some additional economic aspects of ground water resources and replacement flows in semi-arid agricultural areas

Abstract This article considers some aspects of the integrated model of the demand function for irrigation water with a hydrologic model of a one-cell-aquifer. A comparison between an optimal control and free market is carried out. The issue of the optimal time for introducing imported water to the aquifer is also considered. The results are empirically applied to the Pecos Basin in New Mexico.

PERIODIC ENVIRONMENTS, HARVESTING, AND A RICCATI EQUATION

Publisher Summary This chapter focuses on periodic environments, harvesting, and a Riccati equation. It discusses the autonomous logistic equation of population growth in which where r ( t ) and K ( t ) are T -periodic functions. This can be regarded as a model for the growth of a population where the intrinsic growth rate r ( t ) and the carrying capacity K ( t ) show periodic fluctuations. This chapter highlights the existence of a T -periodic solution x *( t ) that satisfies inf K ( t ) ≤ x *( t ) ≤ sup K ( t ), and it also discusses a few of its asymptotic properties. The principal results can be obtained by a simple direction field argument and some known properties of the Riccati differential equation. The chapter presents an extension of the results to the case where the population is being proportionally harvested in a periodic manner.

Linear age-dependent population growth with seasonal harvesting

A population growth is modelled by the Von Foerster PDE with accompanying Lotka-Volterra integral equation describing the birth rate; the age specific death and fertility rates are assumed to depend only on age and not time. A harvesting policy where a fraction of the population of age greater than a given age is harvested for a fraction of a given season. This introduces a time dependence, but this difficulty is circumvented by devising approximate timeindependent models whose birthrates bracket the true birthrate-the standard renewal equation theory applies to the approximate models so quantitative results can be obtained.

Iteration and nonlinear equations of age-dependent population growth with a birth window

Abstract Several iteration schemes are developed for finding the age distribution satisfying a nonlinear McKendrick-Von Foerster equation of population growth. The scheme is based on the concept of matriarchal and temporal generation expansions developed by S.I. Rubinow, and it depends on the assumption that the birth and death rates depend on only a portion of the present population, not the entire population.

Periodic and constant solutions of matrix Riccati differential equations: n = 2

Several formulas are developed which can be used to determine constant solutions and the possible periods of periodic solutions (if any) of autonomous homogeneous matrix Riccati differential equations. These formulas are used to analyse some 2 × 2 cases, as well as to discuss the existence of periodic solutions under weak periodic forcing.