If you are given an unlimited growth model in the form:

$\frac {dP(t)}{dt} = k P(t)$

Obviously the population growth would never be unlimited, but let's presume for the moment that we are introducing a species into an environment where there is the possibility for unlimited growth, at least on for a given time -- i.e. invasive species.

$k$ is some rate of growth of the population at time $t$, denoted by $P(t)$

What are some feasible values of $k$? In other words, if a number is way above or way below $k$, where would I know that the research I am reading is preposterously off-base?

I am sure it is different for different types of animals, including mammals, birds, bacteria, etc.

  • $\begingroup$ The feasible value for K depends on the growth rate and is variable. The good thing with using mathematical models is that you can provide a range of values for a parameter and observe the effect on the system. You can get an idea of the feasible values based on literature evidence. You first decide what that invading species might be. $\endgroup$
    Apr 9, 2013 at 9:05

1 Answer 1


A solid limit: k must be greater than zero. Unless you're talking about some cannibalistic species or something that isn't suited to the model at all.

As long as the species is productive in the new environment: k is greater than 1. The population is probably growing or again you probably won't be using an exponential growth model.

As mentioned before you would need to know the species for more information. But if you look at generation times and litter sizes:

  • Some bacterial generation times (from here) range from 10 to 2000 minutes (33 hours). So that is $k=2$ per generation time. Per day you're looking at a lower bound of 2 per day and an upper bound of $k=2^{14}=10^{43}$ per day.
  • Mice are something like 12 week generation time and a litter of 10 giving something like $k=10^{10}$ per year.
  • Elephants are one young every 25 years. So $k=16$ per century or so.

Of course this is all based on gross assumptions. But you're looking for guidelines for a unrealistic model so hopefully they'll do.

  • $\begingroup$ Why k greater than 1? $\endgroup$
    – Did
    Apr 21, 2015 at 23:42
  • $\begingroup$ If k is between 0 and 1, the population is dying out. Of course that is possible but the question seemed to be asking about growing populations, not dying populations. $\endgroup$
    – timcdlucas
    Apr 22, 2015 at 9:03
  • $\begingroup$ "If k is between 0 and 1, the population is dying out." Certainly not, if k is between 0 and 1, the population grows exponentially. Likewise, k being negative does not mean cannibalism, only some death rate greater than the birth rate. $\endgroup$
    – Did
    Apr 22, 2015 at 9:24
  • $\begingroup$ Hm. Yes you're right. I'll point out that this answer is 2 years old and I haven't thought about it since. I suggest you edit my answer or add your own as I've got it wrong. $\endgroup$
    – timcdlucas
    Apr 22, 2015 at 10:36

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