Question

In Exercises \(23-26,\) the logistic equation describes the growth of a population \(P,\) where \(t\) is measured in years. In each case, find (a) the carrying capacity of the population, (b) the size of the population when it is growing the fastest, and (c) the rate at which the population is growing when it is growing the fastest. $$\frac{d P}{d t}=0.0008 P(700-P)$$

Short answer

The carrying capacity of the population is 700. The population is growing fastest when its size is 350, and the rate at which the population is growing fastest is 0.14.

Step-by-step solution

Identify the Carrying Capacity

The carrying capacity, also denoted as \( K \), refers to the value at which the growth rate becomes zero. In other words, when the population stops growing, it has reached its carrying capacity. In the given logistic differential equation, the carrying capacity can be identified as the value of \( P \) that makes the expression \( 700 - P = 0 \). Solving this equation for \( P \), we get \( P = 700 \). So, the carrying capacity of the population is 700.

Find when the Population is Growing Fastest

The population grows fastest when it is halfway to its carrying capacity. In other words, it is the point where \( P \) is equal to half of the carrying capacity. Thus, when the population is growing fastest, the size of the population is \( P = 700 / 2 = 350 \).

Find the Rate of Growth when the Population is Growing Fastest

To find the rate of growth when the population is growing fastest, substitute the value of \( P = 350 \) into the logistic differential equation. That is, \( d P / d t = 0.0008 * P(700 - P) \). Substituting \( P = 350 \), we get \( dP/dt = 0.0008 * 350 * (700 - 350) \), which simplifies to a value of \( 0.14 \). Hence, the rate at which the population is growing fastest is \( 0.14 \).

Key concepts

Carrying Capacity

When studying population dynamics, a key concept is the 'carrying capacity', denoted as \( K \). This term represents the maximum population size that a particular environment can sustain indefinitely given the food, habitat, water, and other necessities available in the environment. The carrying capacity has profound implications for population management and understanding species' survival.

In mathematical terms, the carrying capacity is the value at which the population growth rate equals zero because the population size has reached a point where the environment can no longer support additional individuals. This occurs when resources become limited, preventing further growth. For instance, if a population of animals in a forest grows too large, food resources may become scarce. When the population hits its carrying capacity, it stabilizes, meaning the birth rate equals the death rate and the population growth ceases.

By examining the provided logistic differential equation \( \frac{dP}{dt} = 0.0008P(700 - P) \), it's clear that when \( P \) becomes 700, the growth rate, \( \frac{dP}{dt} \), will be zero. This indicates that 700 is the carrying capacity for this specific population in its environment, where \( P \) represents the population size.

Logistic Differential Equation

The logistic differential equation is a type of differential equation used extensively in biology and ecology to model how populations grow in environments with finite resources, reflecting the concept of carrying capacity. The equation describes how population growth rate changes over time as the population size approaches the carrying capacity of the environment.

The general form of a logistic differential equation is \( \frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right) \), where:\

• \(\frac{dP}{dt}\) is the rate of change of the population size over time,
• \(r\) is the maximum per capita rate of increase,
• \(P\) is the population size,
• \(K\) is the carrying capacity of the environment.

In the logistic model, the growth rate decreases as the population size \(P\) approaches the carrying capacity \(K\). When \(P\) is small relative to \(K\), growth is almost exponential, but as \(P\) grows and resource limitations increase, the growth rate slows and eventually stops when \(P = K\).

In our specific problem, the logistic differential equation provided is \(\frac{dP}{dt} = 0.0008P(700 - P)\), which corresponds to the general model with \(r = 0.0008\) and \(K = 700\). This equation models a situation such that the population grows rapidly at first, and then the growth decelerates as the population approaches the carrying capacity.

Population Growth Rate

Population growth rate is the speed at which the number of individuals in a population increases over a specific time period. This rate can vary widely among different populations and can be influenced by factors such as availability of resources, predation, disease, and climate conditions.

For a population following a logistic growth model, the growth rate is not constant but varies with the size of the population. It is highest when the population is at an intermediate level relative to the carrying capacity. This phenomenon is because when the population is very small, there are many available resources, but the absolute number of births is low. Conversely, when the population is close to its carrying capacity, resource limitations slow the growth.

The fastest rate of growth occurs at exactly half of the carrying capacity, which maximizes the term \((700 - P)\) in the logistic differential equation. Therefore, to determine when the population is growing the fastest, we need to calculate the value of \(P\) that gives the maximum growth rate, which, in our case, is when \(P = 350\). At this point, the calculation for the growth rate would be \(\frac{dP}{dt} = 0.0008 \cdot 350 \cdot (700 - 350) = 0.14\), implying that the population's fastest growth rate is 0.14 per year, or 14% of the population at that time.