Question

Logistic growth Scientists often use the logistic growth function \(P(t)=\frac{P_{0} K}{P_{0}+\left(K-P_{0}\right) e^{-r_{d}}}\) to model population growth, where \(P_{0}\) is the initial population at time \(t=0, K\) is the carrying capacity, and \(r_{0}\) is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model. World population (part 1 ) The population of the world reached 6 billion in \(1999(t=0)\). Assume Earth's carrying capacity is 15 billion and the base growth rate is \(r_{0}=0.025\) per year. a. Write a logistic growth function for the world's population (in billions) and graph your equation on the interval \(0 \leq t \leq 200\) using a graphing utility. b. What will the population be in the year 2020? When will it reach 12 billion?

Short answer

Answer 1: The world population in 2020 will be approximately 7.34 billion people. Question 2: In which year will the world population reach 12 billion? Answer 2: The world population will reach 12 billion around the year 2122 (approximately 122.6 years from 1999).

Step-by-step solution

Write the logistic growth function

Given the logistic growth equation: $$ P(t)=\frac{P_{0} K}{P_{0}+\left(K-P_{0}\right) e^{-r_{d}}} $$ Substitute the given values: Initial population, \(P_{0}=6\) billion Carrying capacity, \(K=15\) billion Base growth rate, \(r_{0}=0.025\) per year The logistic growth function for the world's population (in billions) is: $$ P(t)=\frac{6 \cdot 15}{6 + \left(15 - 6\right) e^{-0.025t}} $$

Find the population in 2020

To find the population in 2020, we need to find the value of \(P(t)\) when \(t=21\) (since the initial time is 1999). Substitute \(t=21\) in the logistic growth function: $$ P(21) = \frac{6 \cdot 15}{6 + \left(15 - 6\right) e^{-0.025(21)}} $$ Calculate the value of \(P(21)\): $$ P(21) \approx 7.34 \, billion $$ The population in 2020 will be approximately 7.34 billion people.

Find the time when the population reaches 12 billion

Now, we want to find the time when the world population reaches 12 billion. Set \(P(t)=12\) and solve for \(t\). $$ 12 = \frac{6 \cdot 15}{6 + \left(15 - 6\right) e^{-0.025t}} $$ To solve for \(t\), first isolate the exponential term: $$ \frac{12(6+\left(15-6\right)e^{-0.025t})}{6\cdot15} = 1 $$ $$ e^{-0.025t} = \frac{6\cdot15-12(6)}{12(9)} $$ Take the natural log of both sides and solve for \(t\): $$ -0.025t = \ln\left(\frac{90-72}{108}\right) $$ $$ t \approx 122.6 \, years $$ The world population will reach 12 billion approximately 122.6 years from 1999, which corresponds to the year 2121.6.

Key concepts

Population Growth

Population growth describes the increase in the number of individuals in a population over time. This concept is essential for understanding how populations expand and the factors influencing this growth. In the context of logistic growth, population growth is not constant. Instead, it starts rapidly and then slows as it faces limitations such as resources, space, and environmental conditions.

In the logistic growth model, initial growth is exponential, meaning it starts off strong because there are abundant resources and minimal competition. Over time, as the population approaches its carrying capacity, the growth rate decreases. This change results in a more sustainable and stable population size. Understanding this curve is crucial for managing species' populations in conservation efforts and urban planning.

Carrying Capacity

Carrying capacity refers to the maximum population size that an environment can sustain indefinitely. This is determined by the availability of resources such as food, water, and habitat within that environment. The carrying capacity sets a ceiling for population growth in the logistic model.

Once a population reaches its carrying capacity, the growth rate slows down, and the birth rates will come to equal the death rates. This balance implies that the available resources can just support the current population size without leading to environmental degradation. It's an important concept when considering the long-term sustainability of a population.

Growth Rate

The growth rate in logistic growth refers to how quickly a population is increasing during a specific time period. In the logistic growth equation, this is represented by the rate parameter, often denoted as \( r_0 \).

In a logistical context, the growth rate starts high because resources are plentiful, but it decreases as resources become scarce. This rate reflects the population's ability to grow under ideal conditions versus the more realistic conditions constrained by the carrying capacity. Understanding growth rates can help predict how quickly a population is expected to grow or shrink over time. It is also crucial for creating sustainable population policies.

Sigmoid Curve

The sigmoid curve, often referred to as an S-curve, is a graphical representation of logistic growth. It demonstrates how the population grows rapidly at first, followed by a slowing growth rate as it nears the carrying capacity.

Initially, the curve appears to be a steep exponential line due to the high growth rate. As the population grows and resources become less available, the curve starts to bend and eventually flattens as the carrying capacity is reached. This flattening indicates the stabilization of the population.

The sigmoid curve is helpful for visualizing how environmental limits on resources affect population growth. It is a valuable tool in both ecology and economics to forecast how populations might change under different conditions.