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To solve for the equilibrium point \(x^*\) of the given discrete-time dynamical system (DTDS):

\[
x_{t+1} = 0.3x_t + 9
\]

The equilibrium point \(x^*\) occurs when the population remains constant, i.e., when \(x_{t+1} = x_t = x^*\). To find \(x^*\), we substitute \(x_{t+1} = x^*\) and \(x_t = x^*\) into the equation:

\[
x^* = 0.3x^* + 9
\]

Now, solve for \(x^*\):

\[
x^* - 0.3x^* = 9
\]
\[
0.7x^* = 9
\]
\[
x^* = \frac{9}{0.7}
\]
\[
x^* = 12.86
\]

Thus, the equilibrium population is approximately \(12.86\) red deer per square kilometer.

Would you like more details on any part of this process? 

Here are some related questions:
1. How is the concept of equilibrium useful in population modeling?
2. What does the equilibrium point represent in the context of this problem?
3. How would changes in the constants (e.g., 0.3 or 9) affect the equilibrium?
4. What are the stability conditions for this DTDS?
5. How would the population evolve over time if the initial population is far from the equilibrium?

**Tip**: To verify the equilibrium, try substituting back into the original equation to ensure consistency.

In a forest in Alberta, every year 70% of the population of red deer either die of natural causes or are eaten by predators. In the meantime, there are 9 new red deer per square kilometer. The discrete-time dynamical system (DTDS) that gives the population of red deer per square kilometer each year is: 
 x_{t+1} = 0.3x_t + 9. 
 Find the equilibrium point x^* of this linear DTDS.

Explore how to find the equilibrium point in a linear discrete-time dynamical system (DTDS) for the population of red deer in Alberta. The problem involves calculating the equilibrium using algebraic methods, resulting in an approximate population of 12.86 red deer per square kilometer. Ideal for high school students.

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