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] Use software or a calculator to draw directional fields for \(k=0.4 .\) What are the nonnegative equilibria and their stabilities?

Short answer

The only nonnegative equilibrium is \( P = 0 \), and it is stable.

Step-by-step solution

Understand the Problem

We have a differential equation given by \( P' = 0.4P(1 - \frac{P}{10000}) - kP \). This equation models the population dynamics of fish where the growth rate of the fish population depends on the current population size. We are tasked with analyzing the system for a specific value of \( k \) and finding equilibria.

Set k to 0.4 and Simplify

Substitute \( k = 0.4 \) into the equation to get: \[ P' = 0.4P(1 - \frac{P}{10000}) - 0.4P \]. Simplifying gives: \[ P' = 0.4P - 0.00004P^2 - 0.4P = -0.00004P^2 \].

Find the Equilibria

To find the equilibria, set \( P' = 0 \). So we solve \( -0.00004P^2 = 0 \), which gives \( P = 0 \). Thus, the only nonnegative equilibrium point is \( P = 0 \).

Analyze Stability

To analyze the stability of \( P = 0 \), consider the equation \( P' = -0.00004P^2 \). Since \( P' \leq 0 \) for any \( P \geq 0 \), the equilibrium at \( P = 0 \) is stable, meaning any initial population will remain at or decrease to zero.

Direction Field Analysis

A direction field can be used to visualize the behavior of solutions over time. Using software to plot \( P' = -0.00004P^2 \) will show that solutions decrease towards zero for any initial positive \( P \). This visually confirms the stability of \( P = 0 \).

Key concepts

directional fields

Directional fields are a powerful tool when studying differential equations. They offer a visual representation of how solutions to a differential equation behave over time. When dealing with the given problem, we have the differential equation \( P' = -0.00004P^2 \).

A directional field is plotted by drawing small arrows at various points in the plane. These arrows indicate the direction in which the solution moves, allowing us to visualize the trajectory of solutions without solving the equation analytically.

By plotting the directional field for our equation, we can confirm that regardless of our initial population \( P \), the direction of movement is towards zero. This visual helps us see that over time, the fish population will decrease.

Using software or a calculator, you can plot this directional field and observe that all arrows point downwards, emphasizing that solutions approach zero as time progresses. This serves as a practical way to interpret the dynamic behavior of the system.

equilibrium points

Equilibrium points are crucial in understanding differential equations. They represent states where the system doesn't change, providing key insights into its long-term behavior.

For the equation \( P' = -0.00004P^2 \), finding the equilibrium involves solving for \( P' = 0 \). In this case, we get \( P = 0 \) as the only nonnegative equilibrium point.

An equilibrium point occurs when the rate of change is zero. Here, when \( P \) reaches zero, the population doesn't increase or decrease. It is important to identify these points as they help us understand where the system might settle over time.

Recognizing equilibria allows us to predict the future behavior of the system and understand under what conditions the population reaches a static state.

stability analysis

Stability analysis tells us about the behavior of solutions near an equilibrium point and whether they tend to remain close or move away. For our equation, \( P' = -0.00004P^2 \), the stability analysis shows that the equilibrium point \( P = 0 \) is stable.

To determine stability, we check whether small deviations from the equilibrium point grow or shrink over time. Here, since \( P' \) is non-positive (i.e., \( P' \leq 0 \) for any nonnegative \( P \)), it indicates that any initial population size \( P \geq 0 \) will not increase but will decrease towards zero.

A stable equilibrium means that the population will naturally return to it after small perturbations. This means that despite any small increase in the population, the dynamics will inevitably lead it back to zero.

This analysis is essential for understanding the resilience of the population under study and how disturbances will impact its size over time.