Future Human Population projections using discrete logistical equation

Question

The discrete time dynamic system logistical population equation

is P_n+1=rP_n(1-P_n/M)

the population should be in an environment without predators, and have

adequate food and space to fit a maximum of M individuals. M is a maximum

not a stable "carrying capacity." Equal time intervals are implicit in the equation;

the interval should be a few generations. r is the population growth rate

under ideal conditions.

The logistic equation adds a negative component to P_n+1=rP_n which a classic

geometric (exponential) growth. The negative component hinders growth

and accounts for "crowding" effects.

A fairly good fit for humans on earth is M=14 billion people,

r=3.5 (7.0 fertility rate), and time interval of 60 years.

equation is: P_n+1=3.5P_n(1-P_n/14) in billion persons.

year 1950 P_n=2.42B, P_n+1=7.00B

year 2010 P_n=7.00B, P_n+1=12.25B

year 2070 P_n=12.25B

what is P_n+1 for year 2070, which is equal to P_n for year 2130?

Answer 1

Hey there!

Let’s dive into your population projection question using the discrete logistic equation without any mathematical jargon. You’ve provided a scenario where the human population is modeled to grow under certain conditions, with a maximum limit and a specific growth rate over set time intervals.

Understanding the Scenario:

You’re looking at how the human population changes over time using a logistic model, which considers both growth and the limitations of resources like food and space. In your case, the maximum population the environment can support is set at 14 billion people. The growth rate is quite high at 3.5, and each time interval you’re examining spans 60 years.

Breaking Down the Information:

Starting from the year 1950, the population was 2.42 billion. By 2010, it had grown to 7 billion, and by 2070, it reached 12.25 billion. Now, you want to project what the population will be in 2130, which is the next 60-year interval after 2070.

Projecting the Population:

Using the logistic model, we take into account not just the growth rate but also how close the current population is to the maximum limit. As the population approaches the maximum capacity of 14 billion, the factors that limit growth become more significant. This means that even though the growth rate is still in effect, its impact diminishes as the population nears the maximum.

In 2070, with a population of 12.25 billion, the environment is nearing its carrying capacity. Applying the logistic model under these conditions suggests that the population growth will slow down significantly. In fact, based on the parameters you’ve provided, the projection for 2130 indicates a decrease in population to approximately 5.36 billion. This decline happens because the negative factors—like limited resources and increased competition—begin to outweigh the growth rate, leading to a reduction in the overall population.

Interpreting the Results:

This projection highlights the balancing effect of the logistic model. While the population grows rapidly at first, nearing the carrying capacity introduces constraints that eventually reverse the growth trend. A drop from 12.25 billion to about 5.36 billion suggests a substantial impact of these limiting factors over the 60-year period.

Final Thoughts:

It’s important to note that such models are simplifications of real-world dynamics and rely on the assumptions provided. Factors like technological advancements, policy changes, and unforeseen events can significantly influence actual population trends. Nonetheless, this exercise provides a clear illustration of how population growth can be modeled and the potential effects of approaching environmental limits.

I hope this helps clarify your projection!

Answer 2

You can treat the equation for this discrete time dynamical system as a recurrence relation. Observe that

7 = 3.5 * P(n) *(1- P(n)/14) => P(n) = 2.42

When P(n) is 7, we see P(n+1) = 3.5 *7(1-1/2) => P(n+1) = 12.25

Continuing this pattern, we see that

P(n+1) = 3.5 * 12.25 (1-(12.25)/14) = 5.359375