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 \(k=0.6\). What are the equilibria and their stabilities?

Short answer

The equilibrium at \( P=0 \) is asymptotically stable; fish population will decline to zero.

Step-by-step solution

Understand the Differential Equation

The differential equation is given as \( P' = 0.4P \left(1 - \frac{P}{10000}\right) - kP \). This is a logistic-type equation with harvesting, where \( P \) represents the population of fish, and \( k \) is a constant related to the rate of harvest.

Substitute Constant Value

Substitute \( k = 0.6 \) into the differential equation, resulting in: \( P' = 0.4P \left(1 - \frac{P}{10000}\right) - 0.6P \).

Simplify the Equation

The equation simplifies to: \( P' = 0.4P - 0.00004P^2 - 0.6P \), which simplifies further to \( P' = (-0.2)P - 0.00004P^2 \).

Find Equilibria

To find equilibria, set \( P' = 0 \) and solve: \( (-0.2 - 0.00004P)P = 0 \). This results in \( P = 0 \) and \( P = -\frac{0.2}{0.00004} \) which is not possible since it is negative and \( P \) represents population.

Directional Field Analysis

Plot the directional field or phase portrait for the equation using software or graphing tool, focusing on the positive values of \( P \). This helps visualize how solutions behave for different initial conditions.

Analyze Stability of Equilibria

The feasible equilibrium is only \( P = 0 \). Check the sign of \( P' \) for values of \( P<0 \) and \( P>0 \). For \( P>0 \), \( P' < 0 \), indicating that \( P=0 \) is asymptotically stable. The fish population tends to decrease to zero.

Key concepts

Logistic Growth

In the context of differential equations, logistic growth models the change in population over time where there's a natural limit to growth. The logistic model incorporates a carrying capacity, which restricts how large the population can grow. This carrying capacity is often influenced by factors like available resources or environment sustainability.
Logistic growth is typically expressed as:

• The population change given by: \( P' = rP \left(1 - \frac{P}{K}\right) \)
• Where \( r \) is the intrinsic growth rate, \( P \) is the population, and \( K \) is the carrying capacity.

In our exercise, we have an additional term \(-kP\) representing a harvesting rate proportional to the current population. Here the differential equation is formed by combining natural growth and harvesting effects. Essentially, when the harvesting effect (negative term) is small, the population tends towards the carrying capacity. But, substantial harvesting can drastically lower population sustainability. This is captured beautifully in logistic models.

Equilibrium Analysis

Equilibrium points in a differential equation are where the population remains constant, meaning the derivative or rate of change \( P' \) is zero. To find these equilibria, we solve the equation \( P' = 0 \).
In our example, setting \( P' = 0 \): solves \( (-0.2P - 0.00004P^2) = 0 \). This gives two solutions:

• \( P = 0 \)
• \( P = -\frac{0.2}{0.00004} \), which is not feasible physically as it results in a negative value.

Since the population cannot be negative, \( P = 0 \) is the only meaningful equilibrium. Equilibria are useful in understanding the long-term behavior of populations. At equilibrium, the population is stable and any change from this point will remain as long as other factors remain constant.

Stability Analysis

Stability analysis involves understanding how small perturbations or changes in population affect its progression over time. In simpler terms, it determines whether an equilibrium is stable (returns to equilibrium after disturbance) or unstable (moves away from equilibrium).
For the logistic-harvesting equation given, we focused on the equilibrium \( P = 0 \). By checking the derivative sign \( P' \) for populations \( P > 0 \) and \( P < 0 \):

• For \( P > 0 \), we found \( P' < 0 \), suggesting that any positive population will decrease, heading back to zero.

Thus, \( P = 0 \) is asymptotically stable, meaning any small initial population moves toward extinction without interventions.
This stability analysis is crucial for understanding long-term dynamics, particularly for resource management, where it highlights the potential need for intervention to prevent unwanted outcomes like extinction in real-world scenarios.