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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 $$(1)\qquad dx/dt = rx(t)\left\{1-cx(t)-b\sum^\infty_{j=0}c_jx(|t-j|)\right\},$$ 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.).

92D25Population dynamics (general)
37N25Dynamical systems in biology
34K20Stability theory of functional-differential equations
34D23Global stability of ODE
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