Question

Stocking a Lake with Fish \(\quad\) A lake is stocked with 500 fish, and the fish population \(P\) increases according to the logistic curve \(P=\frac{10,000}{1+19 e^{-t / 5}}, \quad t \geq 0\) where \(t\) is the time (in months).

Short answer

The logistic function models the growth of a fish population in a lake over time, starting with 500 fish and having a maximum carrying capacity of 10,000 fish. Initially, the growth is rapid, but as the fish population approaches the carrying capacity, the growth slows down and eventually nearly stops.

Step-by-step solution

Understanding the Logistic Function

The given logistic function is \(P=\frac{10,000}{1+19 e^{-t / 5}}\), where \(P\) is the fish population and \(t\) is the time (in months). The function tells us that the fish population increases over time t, following a logistic growth pattern with an upper limit or carrying capacity of 10,000 fish, as denoted by the 10,000 in the numerator of the function. The denominator of the function represents the rate of growth of the fish population. Initially, when \(t=0\), the fish population is \(P=\frac{10,000}{1+19}\), which is approximately 500 fish. As \(t\) increases, the term \(e^{-t / 5}\) gets smaller, making the entire term \(1+19 e^{-t / 5}\) smaller, thus increasing the value of \(P\).

Analyzing the Growth Over Time

For small values of \(t\), the term \(19 e^{-t / 5}\) is relatively large, which means the denominator of the function is large and \(P\) is small. However, as \(t\) increases, \(19 e^{-t / 5}\) decreases rapidly, causing the denominator to get closer to 1, and the population \(P\) to approach its maximum capacity of 10,000. This reflects how the growth of the fish population slows down as it reaches its carrying capacity.

Graphing the Function

When graphing the function, we can see that the fish population starts at around 500, and as time progresses, the population increases and starts to approach the carrying capacity of 10,000. Initially, the growth rate is exponential, but as the population approaches the carrying capacity, the growth slows down and the curve starts to flatten.

Key concepts

Carrying Capacity

In biology and ecology, **carrying capacity** refers to the maximum number of individuals in a population that an environment can support sustainably. This means it's the highest population size that can exist without causing harm to the environment or exhausting resources. In the context of the lake stocked with fish, the carrying capacity is 10,000 fish.
This figure is derived from the logistic function's numerator, indicating the maximum population limit. As population increases, factors such as limited food supply, habitat space, and environmental conditions come into play, gradually slowing down growth. This concept is crucial as it demonstrates how populations interact with their ecosystems. As resources get scarcer, growth eventually plateaus, reflecting nature's balance. Understanding carrying capacity helps predict how changes in resources or conditions impact populations. It's an essential factor in conservation and ecological management practices.

Exponential Growth

**Exponential growth** occurs when the growth rate of a population is proportional to its current size, leading to faster and faster increases. In the initial phase of the fish population in the lake, this kind of growth is noticeable. When time, denoted as \(t\), is small, the term \(19e^{-t / 5}\) indicates a slower decrease since the exponential term diminishes gradually at first. This causes the denominator of the logistic function to be large, keeping the population \(P\) small relative to its maximum capacity, thus allowing for rapid growth.
The early stages of the graph will reflect a steep incline, typical of exponential growth, where resources seem abundant, and there are fewer environmental restraints. This continues until the population size significantly infringes on the available resources. However, this isn't sustainable indefinitely, as exponential growth is limited by factors leading to a gradual slowdown as hinted by the logistic model in place, marking the transition to another growth stage.

Population Dynamics

**Population dynamics** refers to the changes in population size over time and the factors that influence these changes, including the aspects of birth rates, death rates, immigration, and emigration. The logistic growth model used in the exercise illustrates key concepts in population dynamics, particularly how populations grow in an environment with limited resources.
Initially, as the fish population increases in the lake, it follows almost an exponential growth pattern due to available resources. However, as time progresses and the number of fish approaches the carrying capacity of 10,000, the growth rate slows down, illustrating regulatory feedback inherent in population dynamics.
Various biological and environmental factors can also influence these dynamics, such as competition for resources, predation, disease, and resource availability. In managed environments, understanding these dynamics is critical for maintaining balance, as it helps predict potential population crashes or booms, informing conservation and resource-management strategies.