Question

Without solving the logistic equation or referring to its solution, explain how you know that if \(y_{0}

Short answer

If \( y_{0} < L \), the population size is increasing because \( \frac{dy}{dt} > 0 \) in the logistic equation.

Step-by-step solution

Understanding the Logistic Equation

The logistic equation is given by \( \frac{dy}{dt} = r y (1 - \frac{y}{L}) \), where \( r \) is the intrinsic growth rate, \( y \) is the population size at time \( t \), and \( L \) is the carrying capacity of the environment, the maximum population size it can support.

Analyzing the Population Growth Rate

To determine whether the population is increasing or decreasing, observe the sign of \( \frac{dy}{dt} \). If \( \frac{dy}{dt} > 0 \), the population is increasing; if \( \frac{dy}{dt} < 0 \), the population is decreasing.

Evaluating the Expression Inside the Equation

For \( y_{0} < L \), substitute this condition into the logistic growth equation: \( \frac{dy}{dt} = r y (1 - \frac{y}{L}) \). Notice that \( 1 - \frac{y}{L} > 0 \). Since \( r > 0 \), the entire expression \( r y (1 - \frac{y}{L}) \) is positive, indicating that \( \frac{dy}{dt} > 0 \) when \( y_{0} < L \).

Drawing a Conclusion

The condition \( y_{0} < L \) and the positive value of \( \frac{dy}{dt} \) mean that the population size is indeed increasing, as \( y \) grows over time towards \( L \).

Key concepts

Population Growth

Population growth is an important concept in understanding how populations change over time. Essentially, it refers to the way a population's size can increase or decrease, depending on various factors. In the case of the logistic equation, population growth is more realistic compared to exponential growth.

When studying population growth using the logistic model, it accounts for resources and environmental limits that can affect how large a population can become. Here’s how it works:

• If the resources are abundant and the population size is well below the carrying capacity, the population tends to grow rapidly.
• As the population size approaches the carrying capacity, the growth rate slows down, because resources become limited.
• This model leads to an S-shaped curve, where the population will start to stabilize as it gets closer to the carrying capacity.

Understanding these basics helps us appreciate how populations naturally tend to balance within certain limits, redistributing populations in an ecosystem in a sustainable manner.

Intrinsic Growth Rate

The intrinsic growth rate, often denoted as \( r \), is a crucial factor in the logistic equation. This rate measures how quickly a population can grow under ideal conditions, without any limitations from resources or environment. Think of it as the base level of growth potential that a population has.

Some points to understand about intrinsic growth rate:

• It is an inherent characteristic of the species in question. Different species have varying intrinsic growth rates depending on their life history traits.
• For example, a species like bacteria, which reproduces quickly, might have a high \( r \), while larger mammals like elephants will have a lower \( r \).
• In the logistic model, \( r \) is assumed to be a constant, which simplifies the model while still providing valuable insights into population dynamics.

Understanding \( r \) helps us predict how quickly a population might grow without external limitations. It is also crucial for applying the logistic model to real-world situations where environmental factors come into play.

Carrying Capacity

Carrying capacity, symbolized by \( L \) in the logistic equation, defines the maximum population size that an environment can sustain indefinitely. It's all about balance and the availability of resources needed to support the population.

Key aspects of carrying capacity include:

• It represents a balance point where resource availability meets the needs of a population.
• This concept acknowledges that environments have limits, as resources such as food, water, and space are finite.
• Carrying capacity can change due to various external factors like climate change, deforestation, or changes in resource availability.
• If a population exceeds \( L \), it may lead to resource depletion, which can cause the population size to decline.

Understanding \( L \) in the logistic equation allows us to see how populations might stabilize or fluctuate based on the environmental carrying capacity. It brings a realistic perspective to population growth, illustrating potential scenarios if environmental conditions alter.