# What would be the growth rate in the following scenario?

In my textbook the exponential growth rate is described as

$$W'= W e^{rt}.$$

Here $$W'$$ is the final size, $$W$$ initial size, $$r$$ growth rate and $$t$$ time.

I wanted to have a clear mathematical understanding. So far according to what I understand, if the size is growing by 10% per unit of time, then the growth rate in the above function should be

$$r=\ln(1.1).$$

Am I right? Considering infinite resources.

• So why don't we just write it as W λ^t? I have some faint idea that it's to represent that it's a derivative of itself and that it's change in value depends upon itself but I am not sure how that works exactly. Commented Sep 5 at 5:20
• I've been doing some brainstorming to figure out why we write this function in "e" form, so please tell me if I'm right. It's just to make the calculations of rate a bit more standard (as the rate is changing because it depends on it's initial value, even though the percentage change is same). Without e we would have to find the proportionality factor for every individual percent change to figure out what the rate was. Right?? Commented Sep 5 at 6:46

My comments disappeared, so I'll explain here:

There are two forms of the exponential growth model, in discrete time or continuous time.

A discrete-time model is most appropriate for a population that grows at discrete times in regular intervals, such as an animal that reproduces during a specific breeding season. We only consider t = 0, 1, 2, etc., and not the times in-between. The discrete-time exponential growth model is derived from N(t + 1) = λ N(t), where N(t + 1) is the population 1 time step after N(t) - that is, the population at any one time step is λ times the population at the previous time step. From this, we get N(t) = N(0) λ^t.

A continuous-time model is appropriate when growth happens continuously, without seasonality, etc. - perhaps a human population - or when time steps are negligible relative to the time period of interest. The continuous-time exponential growth model is dN(t)/dt = r N(t) - that is, the population grows at a rate of r times the population. By solving the differential equation, we get N(t) = N(0) e^(rt). Euler's constant comes up because of its unique property, d[e^x]/dx = e^x, so dN(t)/dt = d[N(0) e^(rt)]/dx = r N(0) e^(rt) = r N(t)

These two models are equivalent when r = ln(λ). (Note that the symbols r and λ are commonly, but far from always used for these values.) So whether we choose to express the model in as N(t) = N(0) λ^t or N(t) = N(0) e^(rt) comes down to whether it is more intuitive to think about the growth rate in discrete time (λ) or continuous time (r).

Generally you can rearrange the exponential growth equation to get

r = ln(W'/W) / t.

So indeed if time t = 1 and W'/W = 1.1, then your growth rate would be

r = ln(1.1).

For example in microbiology frequently also the doubling time tD is used, thereby W'/W = 2 and growth rate r = ln(2) / tD.