Problem 14 - Calculus for Biology and Medicine

Question

(a) Use the stability criterion to characterize the stability of the equilibria of $$x_{t+1}=\frac{10 x_{t}^{2}}{9+x_{t}^{2}}, \quad t=0,1,2, \ldots$$ (b) Use cobwebbing to decide the limit \(x_{t}\) converges to as \(t \rightarrow \infty\) if (i) \(x_{0}=0.5\) and (ii) \(x_{0}=3\).

Step-by-Step Solution

Answer

(a) Equilibrium at 0 is stable, \(\pm3\) unstable. (b) For \(x_0 = 0.5\), \(x_t\) converges to 0; for \(x_0 = 3\), diverges.

Step 1: Defining the Equilibrium Points

To find the equilibrium points, we need to solve the equation \(x_{t+1} = x_{t}\). This gives us the equation \(\frac{10x_{t}^{2}}{9+x_{t}^{2}} = x_{t}\). By rearranging and simplifying this equation, we find that the equilibrium points are solutions to the equation \(10x_{t}^{2} = x_{t}(9 + x_{t}^{2})\), which simplifies to \(x_{t}^{3} - 9x_{t} = 0\). Factoring out \(x_{t}\), we have \(x_{t}(x_{t}^{2} - 9) = 0\), leading to the equilibrium points \(x_{t} = 0, \pm 3\).

Step 2: Analyzing Stability via Derivative Test

To determine the stability of each equilibrium point, calculate the derivative of \(f(x) = \frac{10x^{2}}{9+x^{2}}\) with respect to \(x\). The derivative is \(f'(x) = \frac{20x(9+x^2)-20x^3}{(9 + x^{2})^2} = \frac{180x}{(9+x^2)^2}\). Assess stability by evaluating \(f'(x)\) at the equilibrium points. If \(|f'(x)| < 1\), the equilibrium is stable; if \(|f'(x)| > 1\), it is unstable. For \(x = 0\), \(f'(0) = 0\), indicating stability. For \(x = 3\), \(|f'(3)| > 1\) and for \(x = -3\), \(|f'(-3)| > 1\), indicating instability.

Step 3: Using Cobwebbing for Convergence - Initial Condition i

For \(x_{0} = 0.5\), draw the line \(y = x\), the curve \(y = \frac{10x^2}{9+x^2}\), and utilize cobwebbing. Start from \(x = 0.5\) on the x-axis, draw vertically to the curve, project horizontally to the line \(y = x\), repeat the steps. The visual pattern reveals convergence towards the equilibrium point \(x = 0\).

Step 4: Using Cobwebbing for Convergence - Initial Condition ii

For \(x_{0} = 3\), perform similar steps as before with cobwebbing. Start from \(x = 3\), proceed vertically and horizontally as before. Here, observe whether the pattern moves towards a cycle or diverges, indicating that the sequence moves away from \(x = 0\) and does not settle around the equilibrium, as expected due to the unstable nature of \(x = 3\).

Key Concepts

Equilibrium Points

Equilibrium points, in the context of dynamical systems, are the values where the system remains constant over time. For the equation given in the exercise, \[ x_{t+1} = \frac{10x_{t}^{2}}{9+x_{t}^{2}} \] the equilibrium points are found by setting the equation equal to its predecessor, i.e., \(x_{t+1} = x_{t}\). This simplifies into the equation: \[ 10x_{t}^{2} = x_{t}(9 + x_{t}^{2}) \] which ultimately gives us the cubic equation \[ x_{t}^{3} - 9x_{t} = 0. \] By factoring, \[ x_{t}(x_{t}^{2} - 9) = 0, \] we obtain the solutions \( x_{t} = 0, \pm 3 \). These points indicate where the system could potentially settle into equilibrium if not perturbed.

• Equilibrium at \(x_{t} = 0\): This is where the system doesn’t grow or decline but remains at zero.
• Equilibrium at \(x_{t} = \pm 3\): These are symmetric points, indicating potential stability or instability depending on further analysis.

Cobwebbing Method

The cobwebbing method is a visual tool for understanding how sequences evolve over time, particularly in one-dimensional maps. This graphical method helps students and mathematicians determine the behavior of iterations without computing numerous aperiodic steps numerically. It involves plotting both the functional curve (such as \(y = \frac{10x^2}{9+x^2}\)) and the line \(y = x\) on a graph. By starting at a given point \(x_0\), you move vertically to the curve, then horizontally to the line \(y = x\), repeating the process.

• For \(x_0 = 0.5\): Starting at this point on the x-axis, following the cobwebbing steps leads to convergence towards the equilibrium point \(x = 0\), demonstrating stable behavior.
• For \(x_0 = 3\): Beginning at \(x_0 = 3\), the cobwebbing process indicates diverging behavior, avoiding settling near zero. This reflects the instability of the equilibrium point \(x = 3\).

Cobwebbing helps visualize these iterations and provides insight into the system's overall dynamical behavior, making it an invaluable method in stability analysis.

Dynamical Systems

Dynamical systems are mathematical concepts used to describe systems, often in physics and engineering, that evolve over time according to a specific set of rules. They consist of repeated applications of a transformation rule, which is typically a mathematical function. In the given exercise, the transformation rule is \[ x_{t+1} = \frac{10x_{t}^{2}}{9+x_{t}^{2}}. \] Dynamical systems can exhibit varying behavior, from stable equilibria to chaotic patterns, based on such a rule. When analyzing dynamical systems:

• **Stability Analysis**: It helps determine whether small perturbations around an equilibrium will die out (stable) or grow (unstable).
• **Behavior at Equilibria**: Involves understanding long-term trends, which could approach points (convergence) or cycles (bounded).
• **Visual Tools**: Methods like cobwebbing are crucial for visualizing and understanding the complex behavior without requiring extensive calculations.

By examining such concepts, students can gain insights into how initial conditions affect the long-term evolution of dynamical systems and predict future states effectively.