Problem 7 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

For the following systems, find all of the equilibrium points \(\left(x_{e}, y_{e}\right)\). a. For each equilibrium point with \(x_{e}>0\) and \(y_{e}>0,\) determine the stability of the system at that equilibrium point. b. Draw a phase plane and in each region of the phase plane bounded by null lines, draw a vector pointing from a point \(\left(x_{n}, y_{n}\right)\) toward \(\left(x_{n+1}, y_{n+1}\right)\). \(\begin{aligned} \text { a. } & x_{n+1}-x_{n}=0.1 * x_{n} *\left(1-0.5 y_{n}-x_{n}\right) \\ & y_{n+1}-y_{n}=0.05 * y_{n} *\left(1+0.2 x_{n}-y_{n}\right) \\ \text { b. } & x_{n+1}-x_{n}=0.05 * x_{n} *\left(1-0.2 y_{n}-x_{n}\right) \\ & y_{n+1}-y_{n}=0.1 * y_{n} *\left(1-0.5 x_{n}-y_{n}\right) \\ \text { c. } \quad & x_{n+1}-x_{n}=0.2 * x_{n} *\left(1+0.4 y_{n}-x_{n}\right) \\ & y_{n+1}-y_{n}=0.4 * y_{n} *\left(1+0.8 x_{n}-y_{n}\right) \\ \text { d. } \quad & x_{n+1}-x_{n}=0.05 * x_{n} *\left(1-0.2 y_{n}-x_{n}\right) \\ & y_{n+1}-y_{n}=0.1 * y_{n} *\left(1-y_{n}\right) \end{aligned}\)

Step-by-Step Solution

Answer

Find equilibrium points by setting differential equations to zero. Analyze stability using Jacobians. Draw phase plane with vector fields.

Step 1: Understanding Equilibrium Points

Equilibrium points occur where the rates of change become zero, meaning where \(x_{n+1} - x_n = 0\) and \(y_{n+1} - y_n = 0\). This implies for each equation \(x_{n}f(x_n, y_n) = 0\) and \(y_{n}g(x_n, y_n) = 0\). We set these equal to zero to solve for the equilibrium points.

Step 2: Solve Equilibrium for System a

For system a: \( x_{n+1}-x_{n}=0.1x_{n}(1-0.5y_{n}-x_{n}) = 0 \) and \( y_{n+1}-y_{n}=0.05y_{n}(1+0.2x_{n}-y_{n})=0 \). Set each expression equal to zero and solve for \(x_{n}\) and \(y_{n}\). From \(0.1x_{n}(1-0.5y_{n}-x_{n}) = 0\), the solutions are \(x_{n} = 0\), \(1 - 0.5y_{n} - x_{n} = 0\). From \(0.05y_{n}(1+0.2x_{n}-y_{n})=0\), solutions are \(y_{n} = 0\), \(1 + 0.2x_{n} - y_{n} = 0\). Solve these equations to find the possible equilibrium points.

Step 3: Analyze Stability for Positive Equilibrium Points

Examine for stability any equilibrium points where \( x_{e} > 0 \) and \( y_{e} > 0 \). Calculate the Jacobian matrix for the system at these points and evaluate the eigenvalues. An equilibrium point is stable if all eigenvalues have negative real parts.

Step 4: Phase Plane Diagram

Draw a set of axes for \( x \) and \( y \), plot the nullclines determined from setting the right hand sides of the equations to zero, and delineate the regions they form. In each region, select a representative point, calculate the vector field at that point, and draw the direction of the vector field.

Step 5: Repeat for Systems b, c, and d

For each of the remaining systems (b, c, and d), repeat Steps 2 through 4. Solve for the equilibrium points, analyze their stability, and develop a phase plane diagram.

Key Concepts

Phase Plane Analysis

Phase plane analysis is a way to visualize the behavior of a system of differential equations. This technique allows us to understand how the system's state evolves over time.
In a phase plane, each axis represents one of the state variables, often named as \( x \) and \( y \). By plotting nullclines—curves where either \( \dot{x} = 0 \) or \( \dot{y} = 0 \)—we can identify regions within the plane where the direction of movement changes. Each point in this plane can be visualized as a vector whose direction and magnitude describe the rate and direction of change at that particular point.
To create meaningful insights, one must:

• Identify equilibrium points where the system does not change—where the derivatives \( \frac{dx}{dt} = 0 \) and \( \frac{dy}{dt} = 0 \).
• Draw vectors indicating the system's tendency to move away or towards these points within different regions of the plotted plane.
• Analyze trajectories showing long-term behavior of the system from various starting points.

By observing these dynamics, you can predict how an initial configuration will evolve over time and identify stable and unstable regions within the phase plane.

Stability Analysis

Stability analysis involves understanding whether an equilibrium point in a system is stable or unstable. An equilibrium point is stable if small deviations from this point decay over time, bringing the system back to equilibrium.
Conversely, it is unstable if small disturbances from the point lead to trajectories that diverge further from the equilibrium. Stability is rigorously analyzed by examining the nature of the eigenvalues of the system's Jacobian matrix.
Here are the steps for analyzing stability:

• Compute the Jacobian matrix at an equilibrium point. This involves taking partial derivatives of the system's equations.
• Determine the eigenvalues of this Jacobian matrix.
• Judge stability based on these eigenvalues:
  • If all eigenvalues have negative real parts, the equilibrium is locally stable (as disturbances lead to decay back to equilibrium).
  • If any eigenvalue has a positive real part, the equilibrium is unstable (as disturbances grow).
  • If eigenvalues have zero real parts, further analysis is required as this can indicate a center or other complex dynamical behavior.

This method provides crucial insights into the behavior of dynamic systems at specific points and helps in designing systems with desired stability characteristics.

Jacobian Matrix

The Jacobian matrix is a critical tool in analyzing the local behavior of dynamical systems near equilibrium points.
It is a matrix of all first-order partial derivatives of a vector-valued function, giving us insights into how changes in input affect the outputnear a given point.
In the context of stability analysis:

• Construct the Jacobian matrix by computing partial derivatives \( \frac{\partial f}{\partial x} \), \( \frac{\partial f}{\partial y} \), \( \frac{\partial g}{\partial x} \), and \( \frac{\partial g}{\partial y} \) for functions describing the system, where \( f \) and \( g \) are the components of the dynamic system.
• The resulting matrix provides a linear approximation of the system near equilibrium points.
• This linearized model is used to determine the system’s response to small perturbations through its eigenvalues—which represent the expansion or contraction rates along each dimension.

The Jacobian essentially tells us how the system behaves near its equilibria, thus serving as a foundation for understanding more complex nonlinear dynamics.