Logistic differential equation

Figure 6.1 Average plankton population density as a function of the Damkohler number for logistic growth with non-uniform carrying capacity of the form K(x,y) = Kq + (5sin(27rx) sin(27ry) and chaotic mixing in the time-periodic sine-flow of Eq. (2.66). The continuous line represents results from the solution of the full partial differential equation with diffusion (Pe 104) and stars ( ) show the time-averaged plankton populations calculated from the non-diffusive Lagrangian representation.

This very interesting equation will be the main thread running through this introduction to the theory of chaos. Indeed, Verhulst s logistic equation is a sqtarable differential equation of the first order. For an initial population of Xo, the solution obtained by direct integration is ... 

Verhults s logistic equation is an analytically soluble non-linear differential equation. Indeed, besides the linear term kKX, it contains the quadratic (harmonic) term - kX ... 

Under these conditions, the evolution of the mechanism of synthesis of cAMP signals is governed by the three-variable system (5.12), to which are added four differential equations for the variation of parameters Rj, a, k, and these parameters will be taken as proportional to a variable X, whose time evolution is itself governed by the logistic equation ... 

The differential equation form of the logistic model of cell growth is basically a two-parameter (k and X ) representation of cell growth that is largely empirical in character. There have been numerous attempts to extend and interpret the model so as to identify parameters in other variations or extensions of the model. One approach commonly described in the literature begins by expressing the Monod equation (13.1.13) for the rate of substrate consumption as... 

When Equation 28.25 is compared to the monoexponential or logistic expressions, similarities and differences are noted. First, the linear dependence between dP/dt and P t) can be reproduced. With appropriate values for constants 1/ I, and j. (l/p /), this can approximate the differential equation that generates the monoexponential fit. Similarly, the solution can reproduce a curvilinear fit in the PPP, similar to the logistic fit, with the appropriate parametric limit (1/p j). While the EoM is different... 

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