The problem considered in the following analysis deals with an isolated population of fish growing in an environment with a fixed carrying capacity. From this population, fish are harvested at some predetermined rate. The first case yields an autonomous equation that can be solved directly for equilibria. A phase line and accompanying phase portrait are the final tools needed to solidify a qualitative solution. The constant harvesting system that is considered has two equilibria.
Initial populations tend either toward the greater equilibria or toward extinction.
The periodic harvesting systems have two equilibrium states. These states are values that the population tends toward and oscillates around. The two systems have equilibrium values close to those of the constant harvesting system, yet the values vary with amplitude and are therefore not equal. As the amplitude increases, the equilibria values approach one another. Introduction A problem involving logistic population growth with harvesting can be described by the differential equation Here, k is the growth rate of the population being studied, N is the carrying capacity of the population, and the function f dictates how the function is harvested.