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 $P^{\prime }=0.4P(1-\frac{P}{10000})-kP$Solve this equation, assuming a value of $k=0.05$ and an initial condition of 5000 fish.

Short answer

The fish population will eventually stabilize at 8750 fish.

Step-by-step solution

Understanding the Equation

The given differential equation models the rate of change of a fish population. It includes natural growth and a proportional harvest rate. The equation is: $P^{\prime }=0.4P(1-\frac{P}{10000})-kP$.

Substitute the Proportionality Constant

Substitute the given value of $k=0.05$ into the equation:$$P^{\prime }=0.4P(1-\frac{P}{10000})-0.05P$$.

Simplify the Equation

Factor out $P$ from each term: $$P^{\prime }=P(0.4-\frac{0.4P}{10000}-0.05)$$.

Combine the Coefficients

Combine $0.4$ and $-0.05$ to simplify the equation: $$P^{\prime }=P(0.35-\frac{0.4P}{10000})$$.

Solve the Separable Equation for Equilibrium

Find equilibrium solutions by setting $P^{\prime }=0$: $$0=P(0.35-\frac{0.4P}{10000})$$. By setting $0.35=\frac{0.4P}{10000}$ and solving for $P$, we find equilibrium points.

Find Equilibrium Points

Solve $0.35-\frac{0.4P}{10000}=0$: $$0.35=\frac{0.4P}{10000}\Rightarrow P=\frac{0.35\times 10000}{0.4}=8750$$.Thus, the equilibrium points are $P=0$ and $P=8750$.

Solve Initial Condition

With $P(0)=5000$, compare with equilibrium points to determine long-term behavior. Since 5000 is less than 8750, the population will converge to the equilibrium point of $8750$ over time.

Key concepts

Population Dynamics

Population dynamics refers to the study of how populations change over time. This involves analyzing how they grow, shrink, and maintain stability. The fish population problem showcases population dynamics through differential equations. In this case, the equation expresses how fish numbers change due to natural growth and harvesting. The aim is to understand and predict changes over time.

The given problem is a classic example where different forces influence population size:

• Natural Growth: This part is represented by the logistic growth model, which shows how populations grow rapidly then level off as they approach a maximum sustainable size (often called the carrying capacity).
• Harvest Rate: The rate at which fish are caught is proportional to the number of fish present, which means more fish leads to higher harvests.

Understanding these factors is crucial for managing resources and achieving sustainability, ensuring the fish population remains stable over the long term.

Rate of Change

The rate of change in population dynamics is a measure of how fast the population size alters over time. It is fundamentally represented in our differential equation by the term $P^{\prime }$, which emphasizes how the fish population is changing at any given moment.

In the provided model:

• The rate of natural growth is modeled by $0.4P(1-\frac{P}{10000})$. Here, the term $(1-\frac{P}{10000})$ simulates an environment where resources are limited, causing growth to slow as the population nears the carrying capacity of 10,000 fish.
• Harvest rate impacts the population negatively by removing individuals. It is modeled by $-kP$, where $k$ represents the proportionality constant determining the share of the population harvested at any time. For $k=0.05$, the impact is 5% of the population size.

This balance between natural growth and harvest rate dictates the fish population dynamics and allows us to predict its future states.

Equilibrium Solutions

An equilibrium solution in the context of population dynamics refers to a state where the population size remains constant over time. This occurs when the rate of change $P^{\prime }$ equals zero, indicating balance between the forces of growth and decline.

Analyzing the differential equation:

• The equilibrium points are derived by setting $P^{\prime }=0$ and solving the resulting algebraic equation.
• This yields solutions that correspond to potential stable states: in our case, $P=0$ and $P=8750$.

A population of zero represents extinction, while 8750 is a non-zero equilibrium indicating a stable population size when growth perfectly offsets losses from harvests. Understanding equilibrium solutions is crucial for ecological management, helping apply strategies to maintain or adjust population stability effectively.