Verhulst equations

Exercise. For the following Malthus-Verhulst equation with random coefficient,... 

In other cases the kinetics of tumors growth is described by equation of auto-catalysis (in biology it is logistic Verhulst equation) ... 

Technology growth rates according to the logistics curve may be estimated from the Verhulst equation,... 

Now consider one-step processes. As an example we take the Malthus-Verhulst problem mentioned in VI.9, because it has been the subject of some discussion. The M-equation for the population growth is... 

The analysis of ordinary differential equation (ODE) systems with small parameters e (with 0 < generally referred to as perturbation analysis or perturbation theory. Perturbation theory has been the subject of many fundamental research contributions (Fenichel 1979, Ladde and Siljak 1983), finding applications in many areas, including linear and nonlinear control systems, fluid mechanics, and reaction engineering (see, e.g., Kokotovic et al. 1986, Kevorkian and Cole 1996, Verhulst 2005). The main concepts of perturbation theory are presented below, following closely the developments in (Kokotovic et al. 1986). 

Verhulst model contains two parameters - p and a. Moreover, the equation contains initial size of population N0. 

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 ... 

Note that the reduction of the 18 steps of the FKN mechanisms to the 5 kinetic equations of the Oregonator does not allow the correct stoechiometry to be obtained. We note the analogy of the first step of the process B (A+X -> 2X+Z) with the "chemical" interpretation of Verhulst s logistic equation (A + X 2X+Z) described in section 2. 

APPENDIX C TRANSFORMATION OF VERHULST S ANALYTICAL EQUATION INTO MAY S NUMERICAL EQUATION... 

COMMENT. Despite the fact that the Malthusian model seems to fit the (admittedly crude) population data it has been much criticized, An alternative rate equation that takes into amount the carrying capacity K of the planet is due to Verhulst (1836), This rate equation is... 

Verhulst, F. Nonhnear Differential Equations and Dynamical Systems. Springer, Berlin... 

This is known as Verhulst-Pearl s equation, and it can be used in place of the normal non-substrate-limited growth of a population... 

Figure 18.9 The Verhulst model in which the population c(t) grows until it saturates, according to Equations (1 8.32) and (18.33). Source JD Murray, Mathematical Biology, 2nd edition, Springer-Verlag, Berlin, 1 993.

Equation (4.84) is the well-known logistic equation of Verhulst and Pearl [4.1-5] with the solution... 

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