Editing Logistic function (section)

==== Time-varying carrying capacity ====
Since the environmental conditions influence the carrying capacity, as a consequence it can be time-varying, with <math>K(t) > 0</math>, leading to the following mathematical model:

<math display="block">\frac{dP}{dt} = rP \cdot \left(1 - \frac{P}{K(t)}\right).</math>

A particularly important case is that of carrying capacity that varies periodically with period <math>T</math>:

<math display="block">K(t + T) = K(t).</math>

It can be shown<ref>{{Cite journal |last1=Griffiths |first1=Graham |last2=Schiesser |first2=William |date=2009 |title=Linear and nonlinear waves |journal=Scholarpedia |language=en |volume=4 |issue=7 |page=4308 |doi=10.4249/scholarpedia.4308 |bibcode=2009SchpJ...4.4308G |issn=1941-6016|doi-access=free }}</ref> that in such a case, independently from the initial value <math>P(0) > 0</math>, <math>P(t)</math> will tend to a unique periodic solution <math>P_*(t)</math>, whose period is <math>T</math>.

A typical value of <math>T</math> is one year: In such case <math>K(t)</math> may reflect periodical variations of weather conditions.

Another interesting generalization is to consider that the carrying capacity <math>K(t)</math> is a function of the population at an earlier time, capturing a delay in the way population modifies its environment. This leads to a logistic delay equation,<ref name="delay carrying">{{Cite journal | last1 = Yukalov | first1 = V. I. | last2 = Yukalova | first2 = E. P. | last3 = Sornette | first3 = D. | s2cid = 14456352 | doi = 10.1016/j.physd.2009.05.011 | title = Punctuated evolution due to delayed carrying capacity | journal = Physica D: Nonlinear Phenomena | volume = 238 | issue = 17 | pages = 1752–1767 | year = 2009 | arxiv = 0901.4714 | bibcode = 2009PhyD..238.1752Y }}</ref> which has a very rich behavior, with bistability in some parameter range, as well as a monotonic decay to zero, smooth exponential growth, punctuated unlimited growth (i.e., multiple S-shapes), punctuated growth or alternation to a stationary level, oscillatory approach to a stationary level, sustainable oscillations, finite-time singularities as well as finite-time death.