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Generalised logistic function
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<h2 id="definition">Definition</h2> <p>Richards's curve has the following form: </p> Y ( t ) = A + K − A ( C + Q e − B t ) 1 / ν {\displaystyle Y(t)=A+{K-A \over (C+Qe^{-Bt})^{1/\nu }}} <p>where Y {\displaystyle Y} = weight, height, size etc., and t {\displaystyle t} = time. It has six parameters: </p> <ul><li> A {\displaystyle A} : the left horizontal asymptote;</li> <li> K {\displaystyle K} : the right horizontal asymptote when C = 1 {\displaystyle C=1} . If A = 0 {\displaystyle A=0} and C = 1 {\displaystyle C=1} then K {\displaystyle K} is called the <a href="/facts/Carrying_capacity/7KNgGmIv">carrying capacity</a>;</li> <li> B {\displaystyle B} : the growth rate;</li> <li> ν > 0 {\displaystyle \nu >0} : affects near which asymptote maximum growth occurs.</li> <li> Q {\displaystyle Q} : is related to the value Y ( 0 ) {\displaystyle Y(0)} </li> <li> C {\displaystyle C} : typically takes a value of 1. Otherwise, the upper asymptote is A + K − A C 1 / ν {\displaystyle A+{K-A \over C^{\,1/\nu }}} </li></ul> <p>The equation can also be written: </p> Y ( t ) = A + K − A ( C + e − B ( t − M ) ) 1 / ν {\displaystyle Y(t)=A+{K-A \over (C+e^{-B(t-M)})^{1/\nu }}} <p>where M {\displaystyle M} can be thought of as a starting time, at which Y ( M ) = A + K − A ( C + 1 ) 1 / ν {\displaystyle Y(M)=A+{K-A \over (C+1)^{1/\nu }}} . Including both Q {\displaystyle Q} and M {\displaystyle M} can be convenient: </p> Y ( t ) = A + K − A ( C + Q e − B ( t − M ) ) 1 / ν {\displaystyle Y(t)=A+{K-A \over (C+Qe^{-B(t-M)})^{1/\nu }}} <p>this representation simplifies the setting of both a starting time and the value of Y {\displaystyle Y} at that time. </p><p>The <a href="/facts/Logistic_function/IaQ254t7">logistic function</a>, with maximum growth rate at time M {\displaystyle M} , is the case where Q = ν = 1 {\displaystyle Q=\nu =1} . </p> <h2 id="generalised-logistic-differential-equation">Generalised logistic differential equation</h2> <p>A particular case of the generalised logistic function is: </p> Y ( t ) = K ( 1 + Q e − α ν ( t − t 0 ) ) 1 / ν {\displaystyle Y(t)={K \over (1+Qe^{-\alpha \nu (t-t_{0})})^{1/\nu }}} <p>which is the solution of the Richards's differential equation (RDE): </p> Y ′ ( t ) = α ( 1 − ( Y K ) ν ) Y {\displaystyle Y^{\prime }(t)=\alpha \left(1-\left({\frac {Y}{K}}\right)^{\nu }\right)Y} <p>with initial condition </p> Y ( t 0 ) = Y 0 {\displaystyle Y(t_{0})=Y_{0}} <p>where </p> Q = − 1 + ( K Y 0 ) ν {\displaystyle Q=-1+\left({\frac {K}{Y_{0}}}\right)^{\nu }} <p>provided that ν > 0 {\displaystyle \nu >0} and α > 0 {\displaystyle \alpha >0} </p><p>The classical logistic differential equation is a particular case of the above equation, with ν = 1 {\displaystyle \nu =1} , whereas the <a href="/facts/Gompertz_curve/9nYLlKvx">Gompertz curve</a> can be recovered in the limit ν → 0 + {\displaystyle \nu \rightarrow 0^{+}} provided that: </p> α = O ( 1 ν ) {\displaystyle \alpha =O\left({\frac {1}{\nu }}\right)} <p>In fact, for small ν {\displaystyle \nu } it is </p> Y ′ ( t ) = Y r 1 − exp ( ν ln ( Y K ) ) ν ≈ r Y ln ( Y K ) {\displaystyle Y^{\prime }(t)=Yr{\frac {1-\exp \left(\nu \ln \left({\frac {Y}{K}}\right)\right)}{\nu }}\approx rY\ln \left({\frac {Y}{K}}\right)} <p>The RDE models many growth phenomena, arising in fields such as oncology and epidemiology. </p> <h2 id="gradient-of-generalized-logistic-function">Gradient of generalized logistic function</h2> <p>When estimating parameters from data, it is often necessary to compute the partial derivatives of the logistic function with respect to parameters at a given data point t {\displaystyle t} (see<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>). For the case where C = 1 {\displaystyle C=1} , </p> ∂ Y ∂ A = 1 − ( 1 + Q e − B ( t − M ) ) − 1 / ν ∂ Y ∂ K = ( 1 + Q e − B ( t − M ) ) − 1 / ν ∂ Y ∂ B = ( K − A ) ( t − M ) Q e − B ( t − M ) ν ( 1 + Q e − B ( t − M ) ) 1 ν + 1 ∂ Y ∂ ν = ( K − A ) ln ( 1 + Q e − B ( t − M ) ) ν 2 ( 1 + Q e − B ( t − M ) ) 1 ν ∂ Y ∂ Q = − ( K − A ) e − B ( t − M ) ν ( 1 + Q e − B ( t − M ) ) 1 ν + 1 ∂ Y ∂ M = − ( K − A ) Q B e − B ( t − M ) ν ( 1 + Q e − B ( t − M ) ) 1 ν + 1 {\displaystyle {\begin{aligned}\\{\frac {\partial Y}{\partial A}}&=1-(1+Qe^{-B(t-M)})^{-1/\nu }\\\\{\frac {\partial Y}{\partial K}}&=(1+Qe^{-B(t-M)})^{-1/\nu }\\\\{\frac {\partial Y}{\partial B}}&={\frac {(K-A)(t-M)Qe^{-B(t-M)}}{\nu (1+Qe^{-B(t-M)})^{{\frac {1}{\nu }}+1}}}\\\\{\frac {\partial Y}{\partial \nu }}&={\frac {(K-A)\ln(1+Qe^{-B(t-M)})}{\nu ^{2}(1+Qe^{-B(t-M)})^{\frac {1}{\nu }}}}\\\\{\frac {\partial Y}{\partial Q}}&=-{\frac {(K-A)e^{-B(t-M)}}{\nu (1+Qe^{-B(t-M)})^{{\frac {1}{\nu }}+1}}}\\\\{\frac {\partial Y}{\partial M}}&=-{\frac {(K-A)QBe^{-B(t-M)}}{\nu (1+Qe^{-B(t-M)})^{{\frac {1}{\nu }}+1}}}\\\end{aligned}}} <h2 id="special-cases">Special cases</h2> <p>The following functions are specific cases of Richards's curves: </p> <ul><li><a href="/facts/Logistic_function/IaQ254t7">Logistic function</a></li> <li><a href="/facts/Gompertz_curve/9nYLlKvx">Gompertz curve</a></li> <li><a href="/facts/Von_Bertalanffy_function/I54juBT2">Von Bertalanffy function</a></li> <li>Monomolecular curve</li></ul> <h2 id="footnotes">Footnotes</h2> <ul><li>Richards, F. J. (1959). "A Flexible Growth Function for Empirical Use". <i><a href="/facts/Journal_of_Experimental_Botany/cflIP92G">Journal of Experimental Botany</a></i>. 10 (2): 290–300. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1093%2Fjxb%2F10.2.290">10.1093/jxb/10.2.290</a>.</li> <li>Pella, J. S.; Tomlinson, P. K. (1969). "A Generalised Stock-Production Model". <i>Bull. Inter-Am. Trop. Tuna Comm</i>. 13: 421–496.</li> <li>Lei, Y. C.; Zhang, S. Y. (2004). <a href="https://doi.org/10.15388%2FNA.2004.9.1.15171">"Features and Partial Derivatives of Bertalanffy–Richards Growth Model in Forestry"</a>. <i>Nonlinear Analysis: Modelling and Control</i>. 9 (1): 65–73. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.15388%2FNA.2004.9.1.15171">10.15388/NA.2004.9.1.15171</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Fekedulegn, Desta; Mairitin P. Mac Siurtain; Jim J. Colbert (1999). "Parameter Estimation of Nonlinear Growth Models in Forestry" (PDF). Silva Fennica. 33 (4): 327–336. doi:10.14214/sf.653. Archived from the original (PDF) on 2011-09-29. Retrieved 2011-05-31. <a href="https://web.archive.org/web/20110929005929/http://www.metla.fi/silvafennica/full/sf33/sf334327.pdf" target="_blank">https://web.archive.org/web/20110929005929/http://www.metla.fi/silvafennica/full/sf33/sf334327.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>