The constants b and d are control parameters of the system and will control the gross structure of the solution. Before we go much further, however, it is worth looking at the equations and noting that we could make this simpler. By looking at Equation (1) we can see that the only important thing that affects the population growth is the difference between the birth and death rate which is (b-d)N. Therefore we we could write the model in a simpler form
where r=(b-d). Now we only have one adjustable parameter, the net growth rate r. In modeling, it is always useful to reduce the number of true parameters to their smallest number or else you will waste your time solving apparently different problems that are actually the same.
Now that we have simplified the model, let's ask the crucial question
At this point, we could jump in to STELLA, slap together a model, and solve for the population as a function of time and parameters (see Section 2.1.3). To answer our question, however, we would need to explore a wide range of values of r and and it would be pretty tedious. Here we will show you that with a bit of algebra and a bit of sketching you can analyze the qualitative behavior of the solution without touching the keyboard. To do this we need a graphical representation of exactly what Equation (2) means.
For single variable systems (also known as one-dimensional systems), a useful representation is given by Direction Sets. The important message of Equation 2 is that if we know the population at any time then we know that how it will change locally in time. For example, if we made a graph of population vs. time (e.g. Figure 1) then we could go to any point on the graph (e.g. N=40 and t=10) and ask how the population will change. According to Equation (2), for this value of population, the population should increase at a rate of rN=8 bunnies per year. That is, if we took a small step forward in time, we expect that population would increase. We can plot this change in time and population as an arrow starting at the point N=40 and t=10. The arrow would point to the right and up indicating increasing time and increasing population (this is much easier to do than to say so I will show it to you). If we started at a higher population (e.g. N=80, the growth rate would be higher and the arrow would increase more steeply for the same amount of time. This way, we could go to every point in the graph and put a small arrow saying how things would change if we happened to be at that point. The set of arrows is called a direction set and Figure 1 shows some direction sets for the growth problem for r=0.2 and r=-0.2.
Figure: 1 Direction sets for the simple growth model in equation 2
Inspection of the direction sets gives an immediate feeling for how this problem will evolve. We can think of the field of arrows as a flow field like currents in a river. In the case of flow in a river, if we dropped a leaf into the flow it would trace out a trajectory in space. In exactly the same way, if we started to solve Equation (2) at any initial population and time, it would also track out a trajectory in the graph, such that at any point, the direction set arrows would be tangent to the trajectory. In general, finding the exact trajectory for a given initial condition can be difficult or impossible analytically, but computer programs like STELLA do exactly that. Thus a combination of direction sets for gaining intuition and Stella for solving specific instances make for a very powerful combination.