Question

The following problems consider the logistic equation with an added term for depletion, either through death or emigration.[T] The population of trout in a pond is given by \(P^{\prime}=0.4 P\left(1-\frac{P}{10000}\right)-400\), where 400 trout are caught per year. Use your calculator or computer software to draw a directional field and draw a few sample solutions. What do you expect for the behavior?

Short answer

The trout population stabilizes around 1125 due to fishing pressure.

Step-by-step solution

Understand the given logistic equation

The logistic equation provided is: \[ P^{\prime} = 0.4 P \left(1-\frac{P}{10000}\right) - 400 \] This equation models the rate of change of the population of trout in a pond, with the term \(0.4 P\left(1-\frac{P}{10000}\right)\) representing natural growth and the term \(-400\) representing the depletion due to 400 trout being caught per year.

Determine equilibrium points

To find equilibrium points, set \(P^{\prime} = 0\):\[ 0 = 0.4 P \left(1-\frac{P}{10000}\right) - 400 \]Solving \(0.4 P - \frac{0.4P^2}{10000} - 400 = 0\) gives us a quadratic equation:\[ 0.4 P - 0.00004P^2 = 400 \]Rearrange to: \[ 0.00004 P^2 - 0.4 P + 400 = 0 \] We will solve this using the quadratic formula.

Apply the quadratic formula

For the quadratic equation \( ax^2 + bx + c = 0 \), the roots can be found using:\[ P = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] In our case, \(a = 0.00004\), \(b = -0.4\), and \(c = 400\):\[ P = \frac{0.4 \pm \sqrt{(-0.4)^2 - 4 \cdot 0.00004 \cdot 400}}{0.00008} \] Calculate the discriminant and solve for \(P\).

Solve for the roots

Calculate the discriminant:\[ (-0.4)^2 - 4 \cdot 0.00004 \cdot 400 = 0.16 - 0.064 = 0.096 \]Solve:\[ P = \frac{0.4 \pm \sqrt{0.096}}{0.00008} \]Calculate the roots for the equilibrium points.

Interpret equilibrium points and draw conclusions

Calculate:\[ \sqrt{0.096} \approx 0.31 \]So the roots are:\[ P = \frac{0.4 + 0.31}{0.00008} \approx 9125 \]and\[ P = \frac{0.4 - 0.31}{0.00008} \approx 1125 \]Hence, we have equilibrium points at approximately 1125 and 9125. The behavior of these points indicates one stable equilibrium near 1125 and one unstable near 9125.

Draw directional field and sample solutions

Using a calculator or computer software, plot the direction field for the differential equation. Overlay sample solution curves for initial populations such as 500, 1000, 2000, and 9000. Observe how these solutions approach or diverge from the equilibrium points at 1125 and 9125.

Examine long-term behavior

The stable equilibrium point is around 1125, indicating that the trout population will eventually decrease or increase towards this value from a wide range of starting populations. This suggests a long-term population stabilization at 1125 due to the constant fishing pressure of 400 trout per year.

Key concepts

Trout Population Modeling

Trout population modeling is a fascinating study of how trout numbers in a pond change over time under various influences. In our given example, we use a logistic equation that considers both the natural growth of trout and the impact of fishing. Natural growth of the population is typically reflected by a logistic model, which assumes growth decreases as the population approaches its carrying capacity. In this context, the maximum population the pond can sustain is 10,000 trout. However, by considering the yearly catch of 400 trout, we include a depletion factor, making our model even more realistic. By introducing this factor, we simulate more accurate population dynamics, which helps in understanding and predicting future trout population sizes in the pond.

Differential Equations

Differential equations are essential tools in modeling dynamic systems, like our trout population. These equations describe how a quantity changes over time, given initial conditions and external influences. In the trout population model example, the differential equation is expressed as:

• The term \(0.4 P\left(1-\frac{P}{10000}\right)\) represents logistic growth, where \(0.4\) is the growth rate constant.
• The constant \(-400\) denotes the annual trout catch rate.

By solving this equation, we can predict how population sizes change based on current numbers and rates of change, helping ecologists and wildlife managers make informed decisions about conservation strategies.

Equilibrium Points

Equilibrium points are crucial in determining the long-term behavior of a population under a given model. They represent the stable and unstable states a system can reach over time. In the trout population scenario, we identify these points by setting the rate of change to zero, yielding two key results: one stable and one unstable.When solving the equation \(P^{\prime} = 0\), we find two equilibrium points around 1125 and 9125 trout. The primary distinction here is stability:

• 1125 trout is seen as a stable equilibrium, where changes in population naturally return to this number.
• 9125 trout is identified as unstable, meaning small changes in population size are likely to move the population away from this point.

These insights allow biologists to predict population trends and devise strategies to maintain ecological balance.

Directional Field

Directional fields are graphical representations that show the direction in which a system changes over time. They provide valuable visual insight into the behavior of differential equations, like our trout population model. By plotting the direction field of the equation, we see arrows that indicate the speed and direction of population change for various initial sizes:

• Populations starting below 1125 tend to increase and move toward stability.
• Populations above 9125 show a tendency to decrease.
• Initial values between these equilibria display intriguing dynamics, moving towards the stable equilibrium at 1125.

With direction fields, we gain an intuitive grasp of how different initial populations evolve under the influence of both natural growth and fishing pressure.

Population Dynamics

Population dynamics is the study of how and why populations change over time. It encompasses factors such as birth rates, death rates, immigration, and resource availability. In the trout population model, these dynamics are shaped by natural growth constraints and fishing activities. Understanding these elements:

• Natural growth tends to increase population size up to a point of saturation, known as the carrying capacity.
• External pressures, such as fishing, affect overall trends, often reducing population to a lower stable level.

By analyzing these dynamics, we can predict trends and implement measures to ensure sustainable population levels, maintaining balance within the ecological system of the pond.