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 harvesting cases are constant harvesting and periodic harvesting.

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. In the case of periodic harvesting, however, both situations yield intractable equations.

The constant harvesting system that is considered has two equilibria. Initial populations tend either toward the greater equilibria or toward extinction. The lower equilibria marks the minimum surviving population, while the greater equilibria is the carrying capacity of the population.

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

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. For this experiment, we are using the values k = 0.5 and N = 5. The function f is either a constant value or a sinusoidal function (periodic harvest) based on time t.

Using phase lines, numeric solutions, and slope fields, we can attempt to discover solutions, numeric or analytic, to the differential equation. In most cases, analytic solutions are difficult or impossible to find.

Here, P represents the population as a function of time, N represents the carrying capacity, k represents the growth rate, and a represents the amount of harvesting. For this experiment, k and N will constant, held at values of .5 and 5 respectively.

Because this equation is autonomous, it is possible to solve directly for equilibria and bifurcation values. The domain that concerns us is all p and t greater than or equal to zero.

Equilibria:

For this to produce real roots, a must be greater than or equal to 5/8.

Bifurcations:
  

These values result from setting p’(t) and p’’(t) equal to zero.

Separation of Variables:

The integration of the left hand side varies depending on the value of parameter a. In general, all forms are exponential or closely related.

Slope Fields and Numerical Solutions:

The equilibria are at p=0 and p=5. These correspond to no initial population (therefore no reproduction) and the carrying capacity N. All solutions will approach the carrying capacity. This makes sense because a=0 simply corresponds to a case without harvesting.

There are two equilibria wedged between p=0 and p=5. As the amount of harvesting increases, these equilibria grow closer together, approaching p=2.5. The upper equilibrium value marks the adjusted carrying capacity. The lower equilibrium marks the lowest survivable population. All values above the lowest survivable population approach the adjusted carrying capacity, while any population below the lowest survivable population will become extinct.

A=5/8:

This value of harvesting creates a very unstable situation for the population. All values above a population of 2.5 approach 2.5. Any population below 2.5 will become extinct. A population in equilibrium at 2.5 cannot withstand any small population decrease without becoming extinct.

This represents over-harvesting. Any population will become extinct, regardless of how large it was initially.

The parameter a represents the contribution of the sine wave to the growth function. In other words, it is the coefficient that determines the total rate of periodic harvesting. Parameter b represents the wavelength of the sinusoidal function that dictates periodic harvesting.

From part 1, we know that the equilibrium points are at 0 and N if the sin function is not present.

Looking at various solutions, we can see that there are two sinusoidal equilibrium points. For small values of a, the function is similar to a standard logistic growth function, with the population collecting around the equilibrium solution N (the carrying capacity):

As a approaches the bifurcation point, the effects of the sin function are apparent in the slope field. Additionally, the sin function shifts the equilibrium state below N:

After the bifurcation point, both equilibrium states disappear and the population drops off rapidly for all initial values:

In part 1, we concluded that a = 5/8 is a bifurcation point. With this function, we can only guess that bifurcation will occur at some value of a near 5/8. Finding an explicit value is not possible due to the nonautonomous nature of the function.