Dynamische Systeme: Steuerbarkeit und chaotisches Verhalten

Stability in predator – prey models and discretization of a modified Volterra – Lotka model

“…(see [2]). In the case c 22 5 0 we have shown in [1] that the fixed point solution (3.5) of the system (3.4) is an attractor, if the step size h 4 0 is chosen sufficiently small. The question is how small h has to be chosen.…”

“…We consider a predator -prey model which is described by the following system of differential equations In [1] we have shown that this solution is stable. In this way we obtain the following system of difference equations: In the case c 22 = 0 (where (3.3) is satisfied) this fixed point, however, is repelling for every choice of h 4 0 and hence not stable (see [1]). If we replace the system (3.4) with…”

“…as a unique fixed point solution. This fixed point, however, is repelling for every choice of h and hence not stable (see [1]). So the time-discrete model is not a good approximation of the time-continuous model, which has the fixed point solution as a stable steady state solution.…”

“…In Section 3 we replace the Volterra -Lotka model by the system (3.1) where the additional coefficient c 22 is non-positive. If c 22 5 0 and the condition (3.3) is satisfied, then the system (3.1) has a steady state solution which is asymptotically stable (see [1]) and the discrete system (3.4) has this solution as an attractive fixed point, if the step size is sufficiently small (also see [1]). So the time-discrete system (3.4) can be considered as a good approximation of the system (3.1), if the time step size is sufficiently small.…”

Book Reviews

“…The book is a continuation of the series of books authored or coauthored by Professor Krabs (Krabs 1997(Krabs , 1998(Krabs , 2005Krabs and Pickl 2003). It comprises four chapters and an appendix.…”