Problem 15 - Calculus for Biology and Medicine

Question

Assume that $$ \begin{array}{l} \frac{d N}{d t}=N-4 P N \\ \frac{d P}{d t}=2 P N-3 P \end{array} $$ (a) Show that this system has two equilibria: the trivial equilibrium \((0,0)\), and a nontrivial one in which both species have positive densities. (b) Use the eigenvalue approach to show that the trivial equilibrium is unstable. (c) Determine the eigenvalues corresponding to the nontrivial equilibrium. Does your analysis allow you to infer anything about the stability of this equilibrium? (d) Use a graphing calculator to sketch curves in the \(N-P\) plane. Also, sketch solution curves of the prey and the predator densities as functions of time.

Step-by-Step Solution

Answer

The system has equilibria at \((0,0)\) and \((\frac{3}{2},\frac{1}{4})\). The trivial equilibrium is unstable; the nontrivial has stable spiral characteristics.

Step 1: Identify Equilibria (Part a)

An equilibrium point occurs when \( \frac{dN}{dt} = 0 \) and \( \frac{dP}{dt} = 0 \). From the equations: - \( N - 4PN = 0 \) gives \( N(1 - 4P) = 0 \), leading to \( N = 0 \) or \( P = \frac{1}{4} \). - \( 2PN - 3P = 0 \) gives \( P(2N - 3) = 0 \), leading to \( P = 0 \) or \( N = \frac{3}{2} \). This implies the equilibria are \((0,0)\) and \((\frac{3}{2}, \frac{1}{4})\).

Step 2: Analyze Stability at Trivial Equilibrium (Part b)

The Jacobian matrix is used to analyze stability. Calculate the Jacobian matrix:\[ J = \begin{bmatrix} \frac{\partial}{\partial N}(N - 4PN) & \frac{\partial}{\partial P}(N - 4PN) \ \frac{\partial}{\partial N}(2PN - 3P) & \frac{\partial}{\partial P}(2PN - 3P) \end{bmatrix} \]Calculate the derivatives: - For \( \frac{dN}{dt} \): \(1-4P\) and \(-4N\). - For \( \frac{dP}{dt} \): \(2P\) and \(2N-3\). At \((0,0)\), \[ J = \begin{bmatrix} 1 & 0 \ 0 & -3 \end{bmatrix} \]The eigenvalues of \(J\) are \(1\) and \(-3\). The eigenvalue \(1 > 0\) implies instability at \((0,0)\).

Step 3: Determine Eigenvalues for Nontrivial Equilibrium (Part c)

Evaluate the Jacobian at \((\frac{3}{2}, \frac{1}{4})\): \[ J = \begin{bmatrix} 0 & -\frac{3}{2} \ 1/2 & 0 \end{bmatrix} \]The characteristic equation, derived from \( \text{det}(J - \lambda I) = 0\), gives: \[ \left| \begin{matrix} -\lambda & -\frac{3}{2} \ \frac{1}{2} & -\lambda \end{matrix} \right| = \lambda^2 - \frac{3}{4} = 0 \]Solving, the eigenvalues are \( \lambda = \pm \frac{\sqrt{3}}{2}i \), implying purely imaginary eigenvalues and indicating a center or possible stable/unstable spiral.

Step 4: Sketch Graphs (Part d)

Using a graphing calculator, input the system to generate phase and solution curves:1. Phase Plane: Plot trajectories in the \( N-P \) plane showing cyclical orbits around \((\frac{3}{2}, \frac{1}{4})\).2. Time Series: Graph \( N(t) \) and \( P(t) \) with time as the x-axis, exhibiting oscillatory behavior reflecting predator-prey dynamics.

Key Concepts

Jacobian Matrix

The Jacobian matrix is a fundamental concept in the study of dynamical systems, often instrumental in assessing the behavior near equilibrium points. It contains partial derivatives of the system's functions with respect to its variables. In simple terms, it is like a multi-variable version of a derivative.
When you have a system of equations like our predator-prey model, the Jacobian helps you understand how small changes in population densities can affect the rate of change of each species.

• For our given system, the Jacobian matrix is calculated by taking partial derivatives of the equations:
• The first equation \( \frac{dN}{dt} = N - 4PN \) yields derivatives \( 1-4P \) with respect to \( N \) and \(-4N \) with respect to \( P \).
• The second equation \( \frac{dP}{dt} = 2PN - 3P \) yields \(2P\) with respect to \(N\) and \(2N-3\) with respect to \(P\).

At the trivial equilibrium point \((0,0)\), the Jacobian matrix simplifies to \\[J = \begin{bmatrix} 1 & 0 \ 0 & -3 \end{bmatrix}\] which you can think of as a snapshot describing how each species influences the other at this specific point, where there is no interaction, reflecting their growth or decay rates.

Eigenvalue Stability

Eigenvalue stability analysis involves examining the eigenvalues of the Jacobian matrix to infer the nature of equilibrium points. Each eigenvalue tells a story about the dynamic behavior near an equilibrium. Specifically, this involves evaluating whether small disturbances grow, shrink, or remain neutral when the system is slightly perturbed.
How do you determine stability from eigenvalues?

• If all eigenvalues have negative real parts, the equilibrium is stable.
• If any eigenvalue has a positive real part, the equilibrium is unstable.
• Complex eigenvalues with zero real parts might suggest cyclical behavior (such as centers in a phase plane).

In our example, the trivial equilibrium has eigenvalues \(1\) and \(-3\). Here, since one eigenvalue is positive \((1)\), it indicates an unstable equilibrium \((0,0)\). Even a slight population increase of prey \(N\) or predators \(P\) leads to further growth away from the point.

Predator-Prey Model

The predator-prey model is a classic interaction model in ecology, describing how two species interact based on their population rates. Our equations illustrate a simple yet powerful aspect of these dynamics: - Prey population \(N\) grows naturally but is reduced by predation.- Predator population \(P\) depends on the presence of prey for sustenance but decreases naturally in their absence.
The specific equations we used, \[ \frac{dN}{dt} = N - 4PN \qquad \frac{dP}{dt} = 2PN - 3P \], express these concepts mathematically. The coefficients "4" and "2" relate to interaction strength between species, while "3" describes the natural rate of predator death.
This model exhibits biological insight:

• The balance created by these interactions leads to cyclical dynamics.
• The species do not stay constant but fluctuate, representing real-world cycles of booms and busts.

Phase Plane Analysis

Phase plane analysis is a vital tool when studying systems of differential equations. It provides a visual way to understand how two variables interact over time. In our predator-prey model, the phase plane is plotted with prey \(N\) on one axis and predators \(P\) on the other.
By plotting trajectory lines representing the system's behavior over time, you can visually observe the tendency of populations to fluctuate cyclically or move towards or away from equilibria.

• The orbits around nontrivial equilibrium points indicate possible closed loops.
• Such closed loops are typical of predator-prey models, illustrating repeating cycles of rise and fall in population densities.

Through phase plane analysis, one can recognize patterns such as spirals or circles, and determine stability and potential oscillations by seeing how trajectories align or divert from equilibria. This visualization is more than just lines, it is the narrative of how these species coexist and how they possibly reach a balance in their eco-system.