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Chapter 5: Problem 12

Suppose a population is growing according to the logistic formula \(N=\frac{500}{1+3 e^{-0.41 t}}\), where \(t\) is measured in years. a. Suppose that today there are 300 individuals in the population. Find a new logistic formula for the population using the same \(K\) and \(r\) values as the formula above but with initial value 300 . b. How long does it take the population to grow from 300 to 400 using the formula in part a?

Short Answer

Expert verified
The new logistic formula is \(N=\frac{500}{1+\frac{2}{3}e^{-0.41t}}\), and it takes about 1.69 years for the population to grow from 300 to 400.

Step by step solution

01

Identify Parameters from Original Formula

The logistic function given is \(N=\frac{500}{1+3e^{-0.41t}}\). Here, the carrying capacity \(K\) is 500 and the initial population \(N_0\) is found when \(t=0\), which gives \(N_0=\frac{500}{1+3}=125\). The growth rate \(r\) is given as 0.41.
02

Adjust Initial Population in Formula

To find a new logistic formula with an initial population of 300, use the form \(N=\frac{K}{1+Ce^{-rt}}\), with \(K=500\), \(r=0.41\). At \(t=0\), set \(N=300\), which gives \(300=\frac{500}{1+C}\). Solving for \(C\), we get \(C=\frac{5}{3}-1=\frac{2}{3}\).
03

New Logistic Formula

The new logistic formula becomes \(N=\frac{500}{1+\frac{2}{3}e^{-0.41t}}\).
04

Determine Time to Reach 400 Individuals

Set the new formula equal to 400 to find the time \(t\): \(400=\frac{500}{1+\frac{2}{3}e^{-0.41t}}\). Rearrange to solve for \(t\): \(1+\frac{2}{3}e^{-0.41t}=\frac{5}{4}\). Solve \(\frac{2}{3}e^{-0.41t}=\frac{1}{4}\), leading to \(e^{-0.41t}=\frac{3}{8}\). Take the natural logarithm to find \(t\): \(t=-\frac{\ln\left(\frac{3}{8}\right)}{0.41}\).
05

Calculate the Time

Calculate \(t\) using \(t=-\frac{\ln\left(\frac{3}{8}\right)}{0.41}\). First, find \(\ln\left(\frac{3}{8}\right)=-0.693\), then solve \(t=\frac{0.693}{0.41}\approx1.69\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Dynamics
Population dynamics refers to the ways in which a population changes over time. These changes can be depicted using various mathematical models. The logistic growth model is frequently used due to its realistic representation of limited growth, which considers environmental constraints.

In population dynamics, key factors include:
  • Birth and death rates: These directly influence how fast a population can grow or decline.
  • Immigration and emigration: These movements can cause population numbers to fluctuate.
  • Environmental limits: These include resources like food and space, which affect how large a population can get.
The logistic growth formula predicts how a population grows towards a balance point called the carrying capacity, before leveling off. In the exercise, this model helps us see how an initial population size will evolve under set conditions.
Carrying Capacity
Carrying capacity, denoted by the symbol \( K \), is a crucial concept in understanding a population's growth potential. It represents the maximum number of individuals that an environment can support without significant negative effects.

In the logistic growth formula \( N = \frac{K}{1+Ce^{-rt}} \), \( K \) is a constant. It impacts how the curve of population growth behaves. When the population size is far from \( K \), growth is rapid. However, as the population nears \( K \), the rate of growth slows down, reflecting constraints from resource limitations.
  • In the exercise, \( K \) is given as 500. This is the limit the population can theoretically reach under stable conditions.
Understanding carrying capacity helps us manage resources better, and make predictions about population sustainability.
Growth Rate
The growth rate is denoted by \( r \) in the logistic growth equation. It's a measure of how quickly a population increases in size. It's a crucial component because it affects the steepness of the population growth curve.

In our formula, the growth rate \( r = 0.41 \) implies a relatively moderate speed of population increase. A higher value would indicate faster growth, pushing the population towards the carrying capacity more swiftly.
  • The exponential part of the formula, \( e^{-rt} \), includes the growth rate, influencing how quickly the initial population can approach \( K \).
  • It helps to project when a population might hit certain milestones, like doubling or reaching a fixed target size.
This rate is critical for predicting future population sizes and understanding the implications of environmental factors and constraints.
Initial Population
The initial population, often denoted as \( N_0 \), is the size of the population at the start of observation or calculation. Changes in initial population can dramatically affect future predictions of population dynamics.

In our exercise, we looked at two different initial populations. Starting with 125 individuals initially, and then recalculating with 300 individuals, demonstrates how different starting conditions alter the trajectory of growth.
  • The adjusted formula for 300 individuals required recalculating the constant \( C \) to fit the new start point, showing the influence of initial conditions on later outcomes.
  • Adjusting for an initial size helps accurately model population growth from a real-world data point.
This parameter is foundational in setting projections for how quickly or slowly a population reaches other milestones, like reaching the carrying capacity or a specific size. Understanding the initial size of a population provides a clearer view of its long-term dynamics.

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