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

Liu, Pingzhou; Gopalsamy, K.

Appl. Math. Comput. 101, No. 1, 63-88 (1999).

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.).

Reviewer: G.V.Khmelevskaja-Plotnikova (Namur)

MSC:

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

Keywords:

population model; piecewise constant arguments; positive equilibrium; chaotic behaviour; period doubling bifurcations