View source for Pearl-Verhulst logistic process

''Verhulst–Pearl logistic process''

In many biological situations the finiteness of available resource means that an isolated population cannot grow without limit, so one may suppose that the net individual growth rate, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p1100901.png" />, is a decreasing function of the population size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p1100902.png" />. The simplest assumption to make is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p1100903.png" />, whence the deterministic rate of increase

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p1100904.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

This defines the Verhulst–Pearl logistic equation, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p1100905.png" /> denotes the intrinsic rate of natural increase for growth with unlimited resources and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p1100906.png" /> is the carrying capacity. Integrating (a1) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p1100907.png" /> yields the solution

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p1100908.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

though a neater representation is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p1100909.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p11009010.png" /> is the time taken for the population to reach half its maximum size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p11009011.png" />. The population therefore initially rises exponentially, followed by a roughly linear phase, before growth tails off towards the asymptote <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p11009012.png" />.

Though P.F. Verhulst [[#References|[a1]]] introduced the logistic curve in 1838, its use was virtually ignored until R. Pearl and L.J. Reed [[#References|[a2]]] rediscovered it empirically in 1920, closely followed by a rational explanation by A.J. Lotka [[#References|[a3]]]; further early references are contained in [[#References|[a4]]]. A full review is provided in [[#References|[a5]]], including a discussion of several logistic-type data sets. Although for many populations the simple linear argument cannot hold, population growth often closely follows the logistic curve (e.g., the population of the USA from 1790 to 1920) even when the underlying assumptions are violated. This has wrongly given it the credence of a universal law.

In contrast, some data sets show an initial logistic rise towards the asymptote <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p11009013.png" />, but then fluctuate about it thereafter. Such behaviour can be described by the associated stochastic process with birth and death rates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p11009014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p11009015.png" />, respectively (cf. also [[Birth-and-death process|Birth-and-death process]]), for non-negative constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p11009016.png" />. The corresponding deterministic equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p11009017.png" /> reduces to (a1), provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p11009018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p11009019.png" />. Conditional on extinction not having occurred, the equilibrium probabilities (see [[#References|[a5]]]) are given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p11009020.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p11009021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p11009022.png" />. Stochastic simulation is straightforward, since for independent uniformly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p11009023.png" />-distributed random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p11009024.png" />, the next event is a birth if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p11009025.png" /> and is a death if not, whilst the inter-event time is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p11009026.png" />. Studies show that unless <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p11009027.png" /> is small, extinction is only likely to occur in the medium term if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p11009028.png" />, i.e. if the individual birth rate increases with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110090/p11009029.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.F. Verhulst,   "Notice sur la loi que la population suit dans son accroissement"  ''Corr. Math. et Phys. publ. par A. Quetelet'' , '''X'''  (1838)  pp. 113–21</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Pearl,   L.J. Reed,   "On the rate of growth of the population of the United States since 1790 and its mathematical representation"  ''Proc. Nat. Acad. Sci.'' , '''6'''  (1920)  pp. 275–88</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.J. Lotka,   "Elements of physical biology" , Williams and Wilkins  (1925)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Pearl,   "Introduction of medical biometry and statistics" , Saunders  (1930)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E. Renshaw,   "Modelling biological populations in space and time" , Cambridge Univ. Press  (1991)</TD></TR></table>