Problem 34 - Biocalculus Calculus for the Life Sciences

Question

Jellyfish locomotion Jellyfish move by contracting an elastic part of their body, called a bell, that creates a high-pressure jet of water. When the contractive force stops, the bell then springs back to its natural shape. Jellyfish locomotion has been modeled using a second-order linear differential equation having the form \(m x^{\prime \prime}(t)+b x^{\prime}(t)+k x(t)=0\) where \(x(t)\) is the displacement of the bell at time \(t, m\) is the mass of the bell (in grams), \(b\) is a measure of the friction between the bell and the water (in units of \(N / m \cdot s ),\) and \(k\) is a measure of the stiffness of the bell (in units of \(N / m )\) Suppose that \(m=100 \mathrm{g}, b=0.1 \mathrm{N} / \mathrm{m} \cdot \mathrm{s},\) and \(k=1 \mathrm{N} / \mathrm{m}\) (a) Define the new variables \(z_{1}(t)=x(t)\) and \(z_{2}(t)=x^{\prime}(t),\) and show that the model can be expressed as a system of two first-order linear differential equations. (b) Construct the phase plane, including the nullclines, for the equations from part (a).

Step-by-Step Solution

Answer

Express the system as: \(z_{1}^{\prime} = z_{2}, z_{2}^{\prime} = -0.001z_{2} - 0.01z_{1}\). Nullclines: \(z_{2}=0\) and \(z_{2}=-10z_{1}\).

Step 1: Define New Variables

We are given the second-order differential equation: \(m x^{\prime \prime}(t) + b x^{\prime}(t) + k x(t) = 0\) and need to express it using the new variables \(z_{1}(t) = x(t)\) and \(z_{2}(t) = x^{\prime}(t)\). First, note that because \(z_{1}(t) = x(t)\), it follows directly that \(z_{2}(t) = x^{\prime}(t)\). Thus, by differentiating \(z_{1}(t)\) with respect to \(t\), we get \(z_{1}^{\prime}(t) = z_{2}(t)\).

Step 2: Reformulate Original Equation

Substitute \(z_{2}(t)\) for \(x^{\prime}(t)\) to get the derivative of \(z_{2}(t)\): \(x^{\prime \prime}(t) = z_{2}^{\prime}(t)\). Substitute these into the original equation: \(m z_{2}^{\prime}(t) + bz_{2}(t) + k z_{1}(t) = 0\). Rearrange this to get \(z_{2}^{\prime}(t) = -\frac{b}{m}z_{2}(t) - \frac{k}{m}z_{1}(t)\).

Step 3: Express as a System of First-Order Equations

We now have the system of equations: \(z_{1}^{\prime}(t) = z_{2}(t)\) and \(z_{2}^{\prime}(t) = -\frac{b}{m}z_{2}(t) - \frac{k}{m}z_{1}(t)\). Substituting the given values, \(m = 100\), \(b = 0.1\), and \(k = 1\), we have: \(z_{1}^{\prime}(t) = z_{2}(t)\) and \(z_{2}^{\prime}(t) = -0.001z_{2}(t) - 0.01z_{1}(t)\).

Step 4: Identify the Nullclines

To construct the phase plane, determine the nullclines where the derivatives are zero. For \(z_{1}^{\prime}(t) = 0\), \(z_{2}(t) = 0\). For \(z_{2}^{\prime}(t) = 0\), solve \(-0.001z_{2}(t) - 0.01z_{1}(t) = 0\), which simplifies to \(z_{2}(t) = -10z_{1}(t)\). These are the nullclines that intersect at the origin.

Step 5: Sketch the Phase Plane

Plot the nullclines: the line \(z_{2} = 0\) is the horizontal axis, and \(z_{2} = -10z_{1}\) is a line through the origin with slope \(-10\). These lines divide the plane into regions where the vector field can be analyzed to show the direction of motion and equilibrium points.

Key Concepts

Jellyfish Locomotion

Jellyfish are fascinating creatures that move through the water using a unique method called jet propulsion. This involves the contraction and relaxation of a part of their body known as the bell. The contraction creates a high-pressure jet, propelling the jellyfish forward. This bell then returns to its original shape, allowing the process to repeat.
Understanding this movement can be approached through modeling the motion of the bell as a system governed by differential equations. A second-order linear differential equation is used to capture the dynamics of this process. Such equations contain terms that depend on the displacement of the bell, its velocity, and acceleration, helping us understand how different forces work together to create movement. For jellyfish locomotion, in particular, these equations allow scientists to predict how changes in stiffness or water resistance affect swimming efficiency.

Second-Order Linear Differential Equations

Second-order linear differential equations are incredibly useful in modeling various physical systems, including jellyfish locomotion. The general form of such an equation is given by:
\[ m x''(t) + b x'(t) + k x(t) = 0 \] Here, the variable \(x(t)\) represents the displacement at time \(t\). The terms \(m\), \(b\), and \(k\) represent the mass, friction, and stiffness within the system, respectively.

• The mass \(m\) influences the acceleration of the bell.
• The damping coefficient \(b\) accounts for the resistance the water exerts.
• The spring constant \(k\) measures the bell's stiffness.

These elements work together to determine how the jellyfish's bell moves when contracting and relaxing. Solving these equations helps predict the movement patterns observed in jellyfish, revealing the intricate details of their swimming mechanics.

Phase Plane Analysis

Phase plane analysis is a powerful tool for visualizing and understanding the behavior of differential equation systems like the one modeling jellyfish locomotion. By converting the second-order differential equation into a system of first-order equations, we can plot the phase plane with axes \(z_1\) and \(z_2\), where \(z_1(t) = x(t)\) and \(z_2(t) = x'(t)\).

• Nullclines in the phase plane indicate points where the rates of change, or derivatives, are zero.
• The intersections of these nullclines often mark equilibrium points of the system.
• This analysis allows us to determine stability and visualize the flow of the system's states over time.

For jellyfish locomotion, analyzing the phase plane helps illustrate how the jellyfish bell transitions between states of contraction and relaxation, highlighting stable and unstable points in its motion.

First-Order Linear Differential Equations

To simplify complex differential systems, such as those involving jellyfish movement, we often express them in terms of first-order linear differential equations. By defining new variables like \(z_1(t) = x(t)\) and \(z_2(t) = x'(t)\), a second-order equation can be transformed into two first-order equations:

• \(z_1'(t) = z_2(t)\)
• \(z_2'(t) = -\frac{b}{m}z_2(t) - \frac{k}{m}z_1(t)\)

This simplification makes the equations easier to work with numerically and conceptually, aiding in the analysis and simulation of the system's dynamics. These equations can be analyzed collectively to monitor the system's evolution over time, predict future states, and understand how different variables influence the system’s behavior.