**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 :

**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:

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:

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 .