why natural phenomena follow exponential growth?

[PG1995]

Hi :smile:


1: In the context of the definition given below, what does the word "shallower" mean? Here is given the **broken link removed**.

exponential
Something is said to increase or decrease exponentially if its rate of change must be expressed using exponents. A graph of such a rate would appear not as a straight line, but as a curve that continually becomes steeper or shallower.

The American Heritage Science Dictionary

2: I have read that many phenomena in nature follow exponential growth. Why is so? Could you please explain how this connection is established between nature and mathematical world? Obviously, if bacterial follows exponential growth, then it is not under any obligation to do so, it does it unknowingly. Besides exponential growth, what other type of growth the natural phenomena most often follow? Please don't use more math to explain math. Thank you.

Regards
PG

 

[FvM]

1. The definition refers to increase or decrease exponentially. Following these two types of exponential functions, shallower is the opposite to steeper.

2. To give a simple example of exponential growth, consider a bacteria population that doubles once an hour. Calculate the number series for illustration. In this case the mathematical doubling operation can be easily related to the natural behaviour of a population not restricted by limited resources.

 

[betwixt]

Expanding on FvM's explanation, do not hink in terms of each unit increasing exponentially because it doesn't. The growth of each unit can remain constant, but each growth cycle increases the number of units, each with its own growth capability.

Brian.

 

[FvM]

I forgot to mention, that the cell division rate is the foundation of exponential growth in this case, it rules until constrained by limited enviromental resources.

 

[PG1995]

Many thanks, FvM, betwixt.

@betwixt - I also owe you lots of thanks for helping me with many other threads too and you helped me really wonderfully by keeping things simple. I'm sorry I couldn't offer my thanks in a timely manner. Forgive me for that. Actually I did try to send you a PM but you have disabled the feature, I believe.


1: I have checked the definitions of "shallow" in Merriam Webster and American Heritage and nowhere it suggests that "shallow" in any way carries meaning opposite to "steep". Please help me with it.

exponential
Something is said to increase or decrease exponentially if its rate of change must be expressed using exponents. A graph of such a rate would appear not as a straight line, but as a curve that continually becomes steeper or shallower.

The American Heritage Science Dictionary

2: It is said that human population growth rate is also exponential. But I don't see it. At least where I live some people have one kid, some two, some as many as five, and so on. I think people don't really care much about exponential growth! What's your opinion on this?

Thank you.

 

[betwixt]

Sorry PG, I got so many people sending me questions that should really be shared knowledge on here that I disabled private messages. You can still reach me through the "contact me" section of my web site at ATV Projects Home Page but it might take me a few days to reply.

In answer to your questions:

1. For "shallower" think in terms of "less steep". What FvM is telling you is that as well as exponential increase where the graph line increases in gradient, you can have the opposite where the gradient becomes less. An exponential decrease looks like an exponential increase with the graph rotated 90 degree clockwise.

2. Some people have too many kids here too.:roll: Looking at the World as a whole and over a long period, where food and water supplies are scarce, the population tends to fall. Large population areas also tend to suffer more from disease than isolated communities. These all work against exponential growth. It isn't an exact mathematical population increase, it would be in ideal conditions but in reality the growth is constrained by limited resources and natural decay.

Where in the World are you PG ?

Brian.

 

[FvM]

1. I suggest to sketch the two types of expontial functions, increasing y = e^x and decreasing y = e^-x. It's simply said, that the former becomes continually steeper and the latter shallower. I think, the function graph clarifies why the terms can be considered opposite.

2. Population growth is a statistical quantity, no matter if people care about it. Population growth isn't exactly exponentially, the rate decreased in the last decades (the rate itself would be constant for an exponential growth, but it isn't). See: Population growth - Wikipedia, the free encyclopedia

P.S.: An increasing exponential function will be turned into a decreasing my mirroring along the y-axis, as the above formulas suggest.