Stabilization in a Finite Time to a Stationary State

2002 ◽  
pp. 69-117
Author(s):  
S. N. Antontsev ◽  
J. I. Díaz ◽  
S. Shmarev
Keyword(s):  
Stationary State ◽  
Finite Time

scholarly journals Population Dynamics with Nonlinear Delayed Carrying Capacity

2014 ◽  
Vol 24 (02) ◽  
pp. 1450021 ◽  
Author(s):  
V. I. Yukalov ◽  
E. P. Yukalova ◽  
D. Sornette
Keyword(s):  
Population Dynamics ◽  
Stationary State ◽  
Carrying Capacity ◽  
Finite Time ◽  
Evolution Equations ◽  
Critical Time ◽  
Early Time ◽  
Nonlinear Functional ◽  
Time Dynamics ◽  
Time Singularities

We consider a class of evolution equations describing the population dynamics in the presence of a carrying capacity depending on the population with delay. In an earlier work, we presented an exhaustive classification of the logistic equation where the carrying capacity is linearly dependent on the population with a time delay, which we refer to as the "linear delayed carrying capacity" model. Here, we generalize it to the case of a nonlinear delayed carrying capacity. The nonlinear functional form of the carrying capacity characterizes the delayed feedback of the evolving population on the capacity of their surrounding by either creating additional means for survival or destroying the available resources. The previously studied linear approximation for the capacity assumed weak feedback, while the nonlinear form is applicable to arbitrarily strong feedback. The nonlinearity essentially changes the behavior of solutions to the evolution equation, as compared to the linear case. All admissible dynamical regimes are analyzed, which can be of the following types: punctuated unbounded growth, punctuated increase or punctuated degradation to a stationary state, convergence to a stationary state with sharp reversals of plateaus, oscillatory attenuation, everlasting fluctuations, everlasting up–down plateau reversals, and divergence in finite time. The theorem is proved that, for the case characterizing the evolution under gain and competition, solutions are always bounded, if the feedback is destructive. We find that even a small noise level profoundly affects the position of the finite-time singularities. Finally, we demonstrate the feasibility of predicting the critical time of solutions having finite-time singularities from the knowledge of a simple quadratic approximation of the early time dynamics.

The Rollercoaster Ride to the Stationary State - Thoughts on the Current Crisis

2008 ◽  
Author(s):  
Viktor Ukrainetz ◽  
Alechov Voknomatra Chudachin
Keyword(s):  
Stationary State ◽  
Current Crisis
Sign in / Sign up

Export Citation Format

Share Document