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Richards Growth Equation

The Richards growth equation is a generalization of the logistic equation with a parameter that changes the shape of the growth curve (Richards 1959, Tsoularis and Wallace 2002). A form for population growth is

N^'=rN[1-(N/K)^nu],

where r>0 is a rate parameter, K>0 is the carrying capacity, and nu>0 is the shape parameter. For the initial condition N(0)=N_0>0, the solution for t>=0 is

N(t)=K[1+[(K/(N_0))^nu-1]e^(-nurt)]^(-1/nu).

The substitution U=(K/N)^nu reduces the equation to the linear first-order ordinary differential equation U^'=-nur(U-1). An increasing solution has its inflection point at population K(1+nu)^(-1/nu) if it reaches that level.

For nu=1 the equation is the logistic equation. With kappa=nur held fixed and nu->0^+, the solution tends to the Gompertz curve Kexp[-ln(K/N_0)e^(-kappat)].

See also

Gompertz Curve, Logistic Equation, Population Growth

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References

Richards, F. J. "A Flexible Growth Function for Empirical Use." J. Exp. Bot. 10, 290-301, 1959. https://doi.org/10.1093/jxb/10.2.290.Tsoularis, A. and Wallace, J. "Analysis of Logistic Growth Models." Math. Biosci. 179, 21-55, 2002. https://doi.org/10.1016/S0025-5564(02)00096-2.

Cite this as:

Weisstein, Eric W. "Richards Growth Equation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RichardsGrowthEquation.html

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