Global stability and chaos in a population model with piecewise constant arguments. (English) Zbl 0954.92020

Sufficient conditions are obtained for the global stability of the positive equilibrium of the equation \[ (1)\qquad dx/dt = rx(t)\left\{1-cx(t)-b\sum^\infty_{j=0}c_jx(|t-j|)\right\}, \] where \(r>0\) , \(c>0\) , \(d_j\) ( \(j=0,1,2,\cdots\) ) are nonnegative and \(\sum\limits^\infty_{j=0}d_j<\infty\) . Here \(|\cdot|\) denotes the greatest integer function. This equation can be considered as a semi-discretization of the delay differential equation \[ dx/dt=x(t)\left\{r-cx(t)-\sum^\infty_{j=0}d_jx(t-\tau_j)\right\}. \] For a special case of equation (1), one shows the complexity of its behaviour for certain regions in the parameter space. There is a very good connection with many other publications on this subject (23 ref.).


92D25 Population dynamics (general)
37N25 Dynamical systems in biology
34K20 Stability theory of functional-differential equations
34D23 Global stability of solutions to ordinary differential equations
Full Text: DOI