Problem 5 - Advanced Engineering Mathematics

Question

(a) If a constant number \(h\) of fish are harvested from a fishery per unit time, then a model for the population \(P(t)\) of the fishery at time \(t\) is given by $$ \frac{d P}{d t}=P(a-b P)-h, \quad P(0)=P_{0}, $$ where \(a, b, h\), and \(P_{0}\) are positive constants. Suppose \(a=5, b=1\), and \(h=4\). Since the \(\mathrm{DE}\) is autonomous, use the phase portrait concept of Section \(2.1\) to sketch representative solution curves corresponding to the cases \(P_{0}>4,1

Step-by-Step Solution

Answer

The fish population approaches 4 for \( P_0 > 1 \) and becomes extinct for \( P_0 < 1 \). Population does not generally reach extinction in finite time for \( P_0 > 1 \).

Step 1: Analyze the Differential Equation

The given differential equation for the fish population is \( \frac{dP}{dt} = P(5 - P) - 4 \). This is a first-order nonlinear ordinary differential equation. The equation is autonomous, meaning it involves \( P \) and its derivatives but not \( t \) directly. Our goal is to determine where the rate of change is zero and how it changes with respect to \( P \).

Step 2: Determine Critical Points for Phase Portrait

To find the critical points (where \( \frac{dP}{dt} = 0 \)), solve \( P(5 - P) - 4 = 0 \). Simplifying gives the quadratic equation \( P^2 - 5P + 4 = 0 \), which factors into \((P-4)(P-1)=0\). Therefore, the critical points are \( P = 4 \) and \( P = 1 \).

Step 3: Analyze Stability of Critical Points

Examine the sign of \( \frac{dP}{dt} \) in the intervals \( P > 4 \), \( 1 < P < 4 \), and \( 0 < P < 1 \). For \( P > 4 \), \( \frac{dP}{dt} < 0 \) indicating population decrease. For \( 1 < P < 4 \), \( \frac{dP}{dt} > 0 \) indicating population increase. And for \( 0 < P < 1 \), \( \frac{dP}{dt} < 0 \) again indicating a decrease. Hence, \( P = 4 \) is stable and \( P = 1 \) is unstable.

Step 4: Solve the Initial Value Problem (IVP)

Separate the variables to solve \( \frac{dP}{dt} = P(5 - P) - 4 \). This requires integration, but a direct substitution solution may not be straightforward. Instead, check the phase portrait results and potential explicit methods for approximate solutions if needed, acknowledging potential limitations. Use direct numerical or graphical solutions for detailed insights.

Step 5: Verify Phase Portrait with Graphing Utility

Plot \( P(t) \) for different initial values: \( P_0 > 4 \), \( 1 < P_0 < 4 \), and \( 0 < P_0 < 1 \) using a graphing utility. Observe behavior: for \( P_0 > 4 \), population approaches \( P = 4 \); for \( 1 < P_0 < 4 \), population converges to \( P = 4 \); for \( 0 < P_0 < 1 \), population falls to extinction or near zero.

Step 6: Determine Extinction Time

Use the phase plane or numerical methods to determine if finite-time extinction occurs for each \( P_0 \). The analysis indicates that finite-time extinction does not generally occur due to stability but will depend on exact parameterization and critical points like \( P = 4 \) acting as an upper stable population threshold. Verify these points through the time-scale of model or small population simulations.

Key Concepts

Phase Portrait Analysis

Phase portrait analysis is a visual way of understanding autonomous differential equations. It involves plotting solution curves and identifying behaviors of the solutions over time based on initial conditions. This method helps us understand the dynamics of a system without solving the equation analytically.

For the fish population model given by \[\frac{dP}{dt} = P(5 - P) - 4,\]we look at the derivative \(\frac{dP}{dt}\) as a function of \(P\). By finding where this derivative equals zero, known as critical points, we determine where the population stops changing.

• Critical points are found by solving \(P(5 - P) - 4 = 0\).
• This results in two critical points: \(P = 1\) and \(P = 4\).

Next, we consider the intervals defined by these critical points: \(P > 4\), \(1 < P < 4\), and \(0 < P < 1\). The signs of \(\frac{dP}{dt}\) in these intervals indicate whether the population is increasing or decreasing:

• For \(P > 4\), \(\frac{dP}{dt} < 0\), meaning the population decreases.
• For \(1 < P < 4\), \(\frac{dP}{dt} > 0\), indicating an increase.
• For \(0 < P < 1\), \(\frac{dP}{dt} < 0\), suggesting a decrease.

With this information, plotting a phase portrait involves sketching these solution behaviors. The phase portrait visually represents the dynamics where population tends toward \(P = 4\) stability, showcasing overall population trends.

Stability of Critical Points

Understanding the stability of critical points is crucial in predicting long-term behavior in dynamic systems like our fish population model. A critical point can be stable, meaning the system tends to return to this point over time, or unstable, where small perturbations lead the system away.

In our model:

• \(P = 4\) is a stable critical point. This means if the population is disturbed from this point, it will eventually return or tend towards this value.
• \(P = 1\) is an unstable critical point. Here, any deviation from the point leads to population moving away, either towards \(P < 1\) or converging upwards past \(P = 1\).

By understanding these points:

• An initial population \(P_0 > 4\) tends to decrease towards \(P = 4\).
• For \(1 < P_0 < 4\), population increases up to \(P = 4\), indicating long-term stability at \(P = 4\).
• When \(0 < P_0 < 1\), it decreases, potentially to extinction if not managed properly.

Analyzing stability helps predict whether a population will stabilize over time or risk extinction, making it an essential aspect of ecological models.

Initial Value Problem (IVP)

Solving an initial value problem (IVP) is about finding a particular solution to a differential equation given an initial condition. The goal is to determine a specific behavior of the system at a given starting point.
For our fishery population model:\[\frac{dP}{dt} = P(5 - P) - 4, \quad P(0) = P_0\]We must integrate this equation to find \(P(t)\), the population at any time \(t\), starting from \(P_0\).

Steps for solving include:

• Separating variables in the differential equation.
• Integrate both sides to solve for \(P(t)\).
• Apply the initial condition \(P(0) = P_0\) to determine the constant of integration.

Despite some algebraic complexity, solutions can often be verified using numerical methods or graphing utilities. This allows us to visualize the population trajectory based on different \(P_0\):

• For \(P_0 > 4\), verifying that the population approaches \(P = 4\).
• For \(1 < P_0 < 4\), confirming the population gradually stabilizes at \(P = 4\).
• For \(0 < P_0 < 1\), observing the population tends towards extinction.

The results align with the phase portrait predictions, offering insights into practical applications like fishery management strategies.