The Logistic Equation with Harvesting: Qualitative Methods


How many fish in the sea ?

Ecologists are often asked to try to estimate the size of a population and the extent to which it can be used as a resource without being depleted. They may have some idea about the way that the population behaves if left on its own, and are asked to predict the effect of harvesting on the population size. On this page we will look at a simple "model" in which harvesting is taken into account. Our goal will be to predict acceptable levels of harvesting under a variety of assumptions. To do so, we will use some of the ideas that we have already developed for differential equations, but this time, taking a fresh look at the analysis of these equations.

We have examined the behaviour of the differential equation for density-dependent growth, called the Logistic Equation:

\[ \frac{dN}{dt} = r N \frac{(K - N)}{K} \]


This equation provides a good basic description of a population that tends to stabilize at some constant level ( $ K, $ the carrying capacity). We have analyzed its behaviour on the last page. Our analysis included the following steps: rescaling the equation by defining the new variable

\[ y = \frac{N}{K}, \]

and then, using separation of variables to understand the behaviour of the solutions to
\[ \frac{dy}{dt} = r y (1-y). \]

The process was rather long and arduous, involving integration by partial fractions and a lot of algebraic manipulation to arrive at the form of the solution, and to understand the behaviour of $ y(t) $ and, thus, of the whole population, $ N(t) $ . Even when we had the functional form corresponding to the solution $ y(t) $ , it was a nontrivial exercise to sketch and understand this solution.

In many cases, if we want to get some appreciation of the Qualitative Behaviour of a solution, rather than its exact numerical value at every instant, then we need not go through this elaborate process. Indeed, on this page, we will see some geometric techniques that provide wonderful shortcuts. These are simple but powerful methods, avoiding the need for complicated calculation, and helping us to visualize behaviour simply from "careful inspection" of the differential equation itself.

We will apply these methods to a second look at the Logistic Equation, and to a generalization of this equation with harvesting,


\[ \frac{dN}{dt} = r N \frac{(K - N)}{K} - H \]

where H is a positive constant.


For your consideration:



Our ultimate goal on this page will be to understand the predictions of this new equation. Before we do this, let us develop some geometric tools for describing the behaviour of a differential equation.


Qualitative methods applied to the Logistic Equation
Consider the case of no harvesting, as a start, i.e. set $ h=0 $ . Then
\[ \frac{dy}{dt} = r y (1-y). \]

Remember that the constant $ r $ , which is a reproductive parameter is positive, and that $ y $ , a variable that measures the population in units of the carrying capacity, is also positive.

From the form of the equation itself, we can see at least two "special solutions", which correspond to $ dy/dt=0 $ . These are the values of $ y $ that satisfy $ r y (1-y)=0 $ , namely $ y =0, y=1 $ . These values will not change ! (Because $ dy/dt=0 $ ). They represent population levels that are constant in time. If we start the population at either of these values, it will stay at that level forever, at least according to our "model". Such values are called steady states (also sometimes known as fixed points or equilibria).

We also know that $ dy/dt>0 $ whenever $ r y (1-y)>0 $ . Since both $ r $ and $ y $ are positive, this whole term is positive if and only if $ (1-y)>0 $ , in other words, $ y<1 $ . This tells us that any value of $ y $ that starts out in the range $ 0<y<1 $ will increase.

We can make similar conclusions about range of $ y $ values that result in decreasing $ y $ . The table below summarizes the properties of the function $ y(t) $ implied directly by the differential equation.
$ \frac{dy}{dt}=0 $ whenever

$ r y (1-y)=0 $

i.e. $ y=0, 1 $

at these values there is no change
$ \frac{dy}{dt}>0 $ whenever

$ r y (1-y)>0 $

i.e. $ 0<y<1 $

at these values y is increasing
$ \frac{dy}{dt}<0 $ whenever

$ r y (1-y)<0 $

i.e. $ y >1 $

at these values y is decreasing


An even better way of summarizing this information is shown in the following diagram. Here, we plot the expression on the RHS (Right Hand Side) of the logistic equation versus the variable $ y $ . As you see from this graph, the expression that represents the rate of change, $ dy/dt=0 $ is positive for some regions (where the parabola is above the horizontal axis), and negative for other regions where the parabola is below the axis. Since we know which of these regions lead to increasing and decreasing $ y $ , we indicate this on the same diagram with our arrows.
Note that $ y $ always increases to the right, and decreases to the left on the horizontal axis. The arrows along the horizontal axis (the $ y $ axis) now tell us everything that happens for various starting values of y. Note that the horizontal axis is not time! However the direction of changes that take place over time is indicated by the direction of the arrows in the diagram above.
We can use this diagram to "describe the behaviour of y" completely. Here we have redrawn the y axis, vertically, showing only the steady states and the direction of the flow.
We can add a time axis to our picture to show a qualitative picture of what the trajectories - i.e. the functions that represent the full behaviour of y versus t - might look like. We have shown here a whole family of curves that represent $ y(t) $ for various starting values $ y_0 $ . Ofcourse, in general, our sketch might not be numerically accurate, but the point is that we are getting a lot of this information "for free": without having to do long and careful calculations. If we want the exact values of y corresponding to each instant of time, we would use our more detailed solution process techniques - or else harness the computer to help us calculate numerical values.

The blue curves on this diagram are the graphs of the solutions $ y(t) $ as functions of time. These are often called the "trajectories". You should notice that the behaviour shown on this graph is also shown (in a compressed version) on the y axis that we drew just up above.


For your consideration:


The effect of harvesting
With our geometric ideas at hand, we can now turn to the effect of including a harvesting term in the logistic equation. Let us examine the equation
\[ \frac{dy}{dt} = r y (1-y) - h. \]

where $ h $ is a positive constant. Treating this with the same qualitative methods would lead us to plot the RHS of this equation (which represents the rate of change) versus the state variable, $ y $ .

We are already familiar with the plot of the expression $ r y (1-y) $ versus $ y $ , and the effect of the constant $ h $ is simply to lower the graph of this parabola. A typical example of what happens when $ h $ is some small constant is shown below.

r=2, h=0.25

You will notice that the parabola is lowered when harvesting is included. This means that the flow will be different and the values of the steady states will also change. Where are the new steady states? How do they compare with the previous ones? You should use this picture to draw your own version of what the new flow would look like before going on to the paragraphs below.

We can study the effect of gradually increasing the harvesting on the flow and the qualitative behaviour. In the demonstration you will see again that as the harvesting is turned on, the positions of the steady states changes! One decreases and one increases. Now there is a steady state at low population densities below which the population decreases ! If there are not enough fish for the population to be self-renewing, the whole population will crash, and go extinct. Move the red rectangle to the position at which the parabola just dips below the axis. Examine the behaviour of this system, and then think about the questions below.




For your consideration:



The example above has demonstrated an interesting transition in the behaviour of the solutions to the Logistic Equation with Harvesting: namely, that beyond some critical value of the parameter $ h, $ the population crashes no matter what its starting level!. A transition of this kind separating one behavioural regime from another is called a bifurcation We will be exploring many other examples and kinds of bifurcations in the future.