Determine when the following steady states are stable: (a) \(x_{n+1}=r x_{n}\left(1-x_{n}\right), \quad \bar{x}=0\) (b) \(x_{n+1}=-x_{n}^{2}\left(1-x_{n}\right), \quad \bar{x}=(1+\sqrt{5}) / 2\) (c) \(x_{n+1}=1 /\left(2+x_{n}\right), \quad \bar{x}=\sqrt{2}-1\) (d) \(x_{n+1}=x_{n} \ln x_{n,}^{2}, \quad \bar{x}=e^{1 / 2}\) Sketch the functions \(f(x)\) given in this problem. Use the cobwebbing method to sketch the approximate behavior of solutions to the equations from some initial starting value of \(x_{0}\). -Problems preceded by an asterisk \((\%)\) are especially challenging.

Understanding how to calculate derivatives is crucial for finding the stability of steady states. A derivative tells us how a function is changing at a particular point. It's the slope of the tangent line at a given point on the function curve. To determine stability, we'll compute the derivative of the given function at the steady state.

For example, in the case of the function given in part (a) of the exercise, we have:
Given: \(x_{n+1} = r x_{n}(1 - x_{n})\).
We need to find \(f'(x)\):
\( f'(x) = \frac{d}{dx}[r x (1 - x)] = r(1 - 2x) \).
We then evaluate this derivative at the steady state \(\bar{x} = 0\): \( f'(0) = r \). Stability is determined by checking if the absolute value of the derivative at the steady state is less than 1:
\[ |f'(\bar{x})| < 1 \]
Hence, \(|r| < 1\). This process applies similarly to parts (b), (c), and (d) with their respective functions.

The cobwebbing method is a visual tool used in analyzing the behavior of iterated functions, particularly useful for understanding difference equations. By drawing a cobweb diagram, we can see how the sequence evolves over time, giving insight into the stability of steady states.

Here’s how you use the cobwebbing method:

• First, plot the function \(f(x)\) and the line \(y = x\).
• Start with an initial value \(x_0\).
• Draw a vertical line from \(x_0\) to the curve \(f(x)\).
• From this intersection point, draw a horizontal line to the line \(y = x\).
• Repeat the process: vertical line to the curve, horizontal line to \(y = x\).

This back-and-forth movement between the function and the line \(y = x\) will illustrate how the system behaves, converging to or diverging from a steady state.

Mathematical biology uses mathematical equations and models to describe and analyze biological systems. Difference equations often model populations where generations do not overlap.

For instance, the function from part (a) of the exercise: \(x_{n+1} = r x_n (1 - x_n)\) is a well-known logistic map used in population dynamics. Here:

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