d. For the rest of the exercise, we assume , , and . What are the steady-state levels for this configuration of parameters?

I more or less computed them in the graph, but I assumed a specific value for , let’s do it again. We have . As for and :

So , and .

e. Show that the model dynamics can be summarized by a first-order difference equation in (of the type , with some function that you need to find; you can also look for an equation of the type with some function to find, if it is easier for you to do so).

This question is a very involved way of asking: what are the time dynamics of ? You know from your lecture notes that . However, we are in discrete time here, as the question asks for a difference equation (not differential), therefore in this case is substituted by . We just have to work out the expression above and plug values for the parameters.

f. Study the convergence of population to its steady state starting from an initial value of population close to for the following values of : , , .

This question basically asks you to study the dynamics of population for different values of . It is more or less about plugging numbers. Let's start from and see what the dynamics look like. Since is a bit uncomfortable I substitute it with . Let’s start easy and substitute numbers time by time.

You see the pattern. By thinking a little bit you should realise that we can express in the following way:

For , by the rules of power series, we have:

We are ready to evaluate the convergence. The following table gives a relationship between and .

Case is the easiest. If then for any . Population is fixed since the beginning, so in some sense we already converged from the start to .

In case we have . If we have no clue we can take one number and see what happens. Let's try . We have the following series (assuming is close to ):

In the following picture you can see the series graphically:

Figure 1: Series with .

You can see where we are going. We can immediately compute from the expression above (remember that for any we have that ):

What we conclude is that for a value of between and population grows, slower at each step, and eventually reaches a steady state level (different for different values of ).

As for case , with , we use the same strategy, namely plugging numbers for the specific value . The series looks like this:

As before, I plotted the series:

Figure 2: Series with .

As you can see the series here goes up and down, it is not monotonic in its growth, contrary to the previous case. However, we can see where it converges to:

Interestingly, notice that the term has in both case no role in determining the convergence, only shaped by .