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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

$\left(1\right)\phantom{\rule{2.em}{0ex}}dx/dt=rx\left(t\right)\left\{1-cx\left(t\right)-b\sum _{j=0}^{\infty }{c}_{j}x\left(|t-j|\right)\right\},$

where $r>0$ , $c>0$ , ${d}_{j}$$j=0,1,2,\cdots$ ) are nonnegative and $\sum _{j=0}^{\infty }{d}_{j}<\infty$ . Here $|·|$ denotes the greatest integer function. This equation can be considered as a semi-discretization of the delay differential equation

$dx/dt=x\left(t\right)\left\{r-cx\left(t\right)-\sum _{j=0}^{\infty }{d}_{j}x\left(t-{\tau }_{j}\right)\right\}·$

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.).

##### MSC:
 92D25 Population dynamics (general) 37N25 Dynamical systems in biology 34K20 Stability theory of functional-differential equations 34D23 Global stability of ODE