Logistic growth describes somes phenomena (for example, population growth, bubble markets, tumor growth, etc) but the current models don't consider an abrupt fall ("crash"). Tsoularis et al (1997) suggested the "Generalized Logistic Growh" with four or five parameters but none of the linked equations (Blumberg, Bertalanffy, etc) simulate this situations.
Logistic growth equation has the original form dN/dt=aN(1-N); see for example http://mathworld.wolfram.com/LogisticEquation.html . It has analytical solution as smooth function. If you want to model crashes, you should go to other equations with this critical property, maybe combining it with logistic.
Catastrophic behavior can emerge in Burgers equation; see for example “Gradient catastrophes and sawtooth solutions for a generalized Burgers equation on an interval” by Samokhin; http://www.sciencedirect.com/science/article/pii/S0393044014000965
I guess that critical behaviour often emerges in a system of nonlinear differential equations. I can suggest to look at my paper: https://www.researchgate.net/publication/278110040_Low-Dimensional_Nonlinear_Dynamical_Systems_as_a_Tool_for_Socio-Economic_Modelling
Conference Paper Low-Dimensional Nonlinear Dynamical Systems as a Tool for So...
I'm glad to your feedback. It's interesting to know that coupled population equations generate crashes. But I'm looking for a simple equation for a special condition that a population (N) consume the resources diminishing your carrying capacity (K), or in other words, where carrying capacity is function own population (K is K(N)).
In about 1980 Werner Mende developed a generalization to the logistic growth model: dN/dt=N^a*(K-N)^b with positive real exponents. If a=b=1 the logistic growth model is obtained. If b>1 then the growth is less than exponentially near the limit point K, if b
Thanks Sr. Jansen. I'm very curious to test this equation but the related articles are not open. Reading the Peschel-Mende's abstracts, they found a diferencial equation that integrates Lotka-Volterra phenomena in a simple diferencial format (if I'm sure, Evolon rises from this). Where I find the analytical solution to the Evolon? Thanks!
If you combine the simple logistic curve f(x)=A / ( 1 + exp(-k*(x-xo) ) ),
2 times i think you get what you need, that means you need the point (xo_a) where you have a sudden increase, and also a point (xo_b) where you have a sudden decrease , then you will have your crush that you need.
The total function would be just a simple linear combination of 2 logistic curves. That is
F(x)= A/( 1 + exp(-k*(x-xo_a) ) ) + A/( 1 + exp(-k*(x-xo_b) ) ).
Where A is the 'Amplitude of your curves', and k is the steepness of the curve.
The overall result will be like a curved box-like function.
Also the overall differential equation it will be F'(x)=F(x)(1-F(x), where F(x) is the previous function.
See https://en.wikipedia.org/wiki/Logistic_function
Interesting! Let's plot this equation. It can rise some features in this special case when a growth phenomena falls suddenly. Thanks!
Now, I found an old overview article. The attachment contains a copy. It is contained in:
@proceedings{Wartburg-84,
editor={Ebeling, W. and M. Peschel},
title={Lotka-Volterra Approach to Cooperation and Competition in
Dynamical Systems},
booktitle={Lotka-Volterra Approach to Cooperation and Competition in
Dynamical Systems},
publisher={Akademie-Verlag},
address={Berlin},
series={Mathematical Research},
volume=23,
year=1984,
confaddr={Eisenach},
confyear=1984,
confmonth=mar,
}
A correction to my first answer is in order: the differential equation reads dN/dt = c N^a (K-N^b). Factor c cannot be removed since it is negative for decaying processes (as in your case). And I misplaced exponent b.
Anyway, explicit solutions do not exist. But the non-exponential growth/decay can be estimated as follows (for sake of simplicity, discussion is for the start value N=0). Factor c(K-N^b) can be assumed constant near N=0. Then the ODE can be solved by classical means: N(t)=d(t-t_0)^(1/(1-b)) where d collects all occurring factors and t_0 is the integration constant. So, the exponent of (t-t_0) is positive for b in interval (0,1) and N becomes 0 in finite time t_0.
my solution is the attached article: "geometria naturale" , in that , there is my natural original solution for a natural growth generation
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