Reflection 1- Predicting Limits

In looking at the difference equation (x_n+1=(lambda)xn, x_0), I've noticed a correlation with how the sequences behave long term and the value of lambda. When lambda equals 1, the long term outcome is no growth, yet no decline. If the value falls below 1 in the form of a decimal/fraction of one, the outcome is a decrease, with lower values indicating faster decreases. Anything above 1 indicates growth, with higher values indicating more extreme cases.

As an example, if lambda was a 0.5, a population of 100 deer would decrease to 50 by x_1. By x_2, the population would only be 25, assuming the rate stayed the same.

Alternatively, with the same example in mind but with a different lambda, such as 1.5, the population would increase by one and a half times per generation. Therefore, by x_1 (with x_0 population still being 100), the population will be 150.

Additionally, the limits of these equations can depend a lot on what x_0 is compared to what the lambda value is. For example, the higher the original population is, the lower lambda would have to be to decrease the population to its limit of zero. However, this equation doesn't have a limit for increase; therefore, the original population could be literally any number, as this equation assumes infinite growth is possible (whereas in the real world, limits such as carrying capacities exist, and therefore the carrying capacity would have to be implemented into the equation to add a limit of increase).

EDIT: After actually doing the activity this time, a few more rules were noticed. This includes the fact that a lambda less than 1 when dealing with a negative X_0 still results in a limit of zero, so technically the population is increasing in this instance, though it will never go beyond zero. However, a lambda greater than 1 with a negative X_0 will result in the decrease of the population infinitely. A negative lambda greater than -1 (-1<x<0) with a positive X_0 results in the graph "pinching" inward to the limit of zero from both a negative and positive side. A negative lambda less than -1 results in the population decreasing indefinitely. These rules were added to the list below.

The ultimate list of  "rules" that I mentioned are as follows:
1. When lambda = 1, there is no growth or decrease.
2. When lambda > 1, the population is growing.
3. When lambda < 1, the population is decreasing.
4. The limit of a function with a lambda < 1 is 0.
5. The limit of a function with a lambda > 1 is infinite (DNE).
6. The size of the population at x_0 can determine how fast a population grows/decreases in proportion to lambda.
7. A lambda less than 1 with a negative x_0 results in a limit of zero.
8. A lambda greater than 1 with a negative x_0 does not have a limit as it decreases.
9. A lambda greater than -1 with a positive x_0 causes the graph to "pinch" inward to the limit of zero from a positive and negative end.
10. A negative lambda less than -1 causes a population to decrease infinitely.