Question

It is more likely that the amount of fishing is governed by the current number of fish present, so instead of a constant number of fish being caught, the rate is proportional to the current number of fish present, with proportionality constant \(k\), as \(P^{\prime}=0.4 P\left(1-\frac{P}{10000}\right)-k P\)[T] For the previous fishing problem, draw a directional field assuming \(k=0.1 .\) Draw some solutions that exhibit this behavior. What are the equilibria and what are their stabilities?

Short answer

Equilibria are at \( P = 0 \) (unstable) and \( P = 7500 \) (stable).

Step-by-step solution

Understand the Fish Population Model

The differential equation given is:\[ P' = 0.4 P \left(1 - \frac{P}{10000}\right) - 0.1 P \]This is a logistic growth model with a harvesting term. Here, the fish population grows following a logistic model, but they are also being removed at a rate proportional to the current population size, with constant of proportionality \( k = 0.1 \).

Find Equilibria of the Equation

To find equilibria, we need to set the right-hand side of the equation to zero and solve for \( P \):\[ 0 = 0.4 P \left(1 - \frac{P}{10000}\right) - 0.1 P \]Factor out \( P \):\[ P \left[ 0.4 \left(1 - \frac{P}{10000}\right) - 0.1 \right] = 0 \]So, the solutions are \( P = 0 \) or solving the quadratic in the bracket for \( P eq 0 \).

Solve the Quadratic Equation

For the non-zero equilibrium point:\[ 0.4 - 0.4\frac{P}{10000} - 0.1 = 0 \]Simplify:\[ 0.3 = 0.4\frac{P}{10000} \]\[ \frac{P}{10000} = \frac{0.3}{0.4} \]\[ P = \frac{0.3}{0.4} \times 10000 = 7500 \]Thus, the equilibria are \( P = 0 \) and \( P = 7500 \).

Determine Stability of Equilibria

To determine stability, evaluate the derivative of the right-hand side of the equation at the equilibria.Let \( f(P) = 0.4 P \left(1 - \frac{P}{10000}\right) - 0.1 P \)\[ f'(P) = \frac{d}{dP}\left(0.4 P \left(1-\frac{P}{10000}\right) - 0.1 P\right) \]- For \( P = 0 \): \( f'(P) = 0.3 \), which is positive, so \( P = 0 \) is unstable.- For \( P = 7500 \): Substitute \( P = 7500 \) into the derivative and evaluate; finding that it's negative indicates stability.

Draw the Directional Field

Plot a direction field on a graph with \( P \) on the x-axis. For each \( P \), draw arrows reflecting the direction and rate of change given by the differential equation. Notice how arrows above \( P = 7500 \) are pointing downwards, showing a decrease, while below they are pointing upwards, indicating an increase. At \( P = 0 \), arrows point away, confirming it is unstable. At \( P = 7500 \), arrows converge, confirming stability.

Draw Some Solution Curves

Using the direction field as a guide, sketch possible solution curves. These start from initial values and follow the arrows, stabilizing or repelling at equilibria. Solutions will approach \( P = 7500 \) over time if they start between 0 and slightly above 7500, illustrating the stability of this equilibrium.

Key concepts

Logistic Growth Model

The logistic growth model is a fundamental concept in population dynamics, particularly useful in studying how populations grow over time. Originally, it describes how a population increases rapidly initially and then slows as it approaches a certain carrying capacity due to limited resources. For the fish population, the model is represented by:\[P' = 0.4 P \left(1 - \frac{P}{10000}\right)\]This particular equation suggests that the growth rate is proportional to the current population size, impacted by the term \(1 - \frac{P}{10000}\), which indicates the limitation of resources as population \(P\) approaches 10,000.
In this exercise, a harvesting term is added, `-k P`, where \(k = 0.1\). This represents the removal of fish at a rate directly proportional to the current population, a common scenario in real-world fishery management. The presence of both logistic growth and harvesting creates a more complex dynamic, making the study of equilibria important for understanding population sustainability.

Equilibrium Stability

Equilibria in differential equations occur where the rate of change \(P'\) is zero. For the fish population, we solve:\[0 = 0.4 P \left(1 - \frac{P}{10000}\right) - 0.1 P\]This gives the equilibrium solutions \(P = 0\) and \(P = 7500\). These represent two potential states where the population can remain unchanged over time, assuming no disruptions.
Stability describes whether a population will return to equilibrium if slightly disturbed. Stability analysis involves examining the derivative \(f'(P)\) at equilibrium points:- **Unstable Equilibrium**: For \(P = 0\), \(f'(P) = 0.3\) is positive, indicating the population tends to move away from \(P = 0\).
- **Stable Equilibrium**: For \(P = 7500\), substituting into the derivative returns a negative value, implying that if the population is near 7500, it will gravitate back toward this level, ensuring stability.

Directional Fields

Directional fields are graphical tools used to visualize differential equations. They help illustrate the direction and rate of change of a system. For the fish population, drawing this field involves plotting a grid where each point shows an arrow indicating how the population \(P\) changes.
A directional field for our equation captures how population changes at different sizes:- Above \(P = 7500\), arrows point downward, signifying a natural decrease toward equilibrium.
- Below \(P = 7500\), arrows point upward, showing population growth.- At \(P = 0\), arrows indicate movement away, supporting its instability.
This tool provides a visual summary of how populations shift over time, making complexities of the differential equation more tangible.

Rate of Change

The rate of change, denoted as \(P'\) in differential equations, is central to understanding how a population varies temporally. In our model:\[P' = 0.4 P \left(1 - \frac{P}{10000}\right) - 0.1 P\]This expression combines growth and harvesting terms to portray the dynamics of how the fish population changes.
- **Growth Component**: \(0.4 P \left(1 - \frac{P}{10000}\right)\) signifies logistic growth, slowing as \(P\) nears the upper limit of 10,000.
- **Harvesting Term**: `-0.1 P` reduces the population at a steady rate proportional to its size, emphasizing how external factors such as fishing influence population stability.
Analyzing the rate of change provides critical insight into periods of rapid growth, potential over-harvesting, and the conditions for reaching equilibrium.

Population Dynamics

Population dynamics involves studying how populations change over time, influenced by natural growth and external factors like harvesting. This exercise applies differential equations to portray these interactions.
In the context of the fish population: - **Natural Growth**: Described by logistic growth, demonstrating typical population expansion, considering environmental limits.
- **External Removal**: The harvesting term represents real-world activities, where fish are being caught continuously.
Understanding population dynamics through equations allows predictions of stability, the potential impact of management strategies, and insights into sustainable practices. By analyzing these interactions, one can derive informed decisions to maintain an ecological balance.