## Stable periodic solutions in a discrete periodic logistic equation.(English)Zbl 1049.39017

Consider the discrete logistic equation $x(n+1)=x(n) \exp \left[r(n)(1-\frac{x(n)}{K(n)})\right],\quad n\in N,\tag{$$*$$}$ where $$x(0)>0,$$ $$\{r(n)\}$$ and $$\{K(n)\}$$ are strictly positive sequences of real numbers defined for $$n\in N=\{0,1,2,\dots\}.$$ In addition there exist positive constants $$r_{*},r^{*},K_{*},$$ and $$K^{*}$$ such that $$0<r_{*}\leq r(n)\leq r^{*},\,0\leq K_{*}\leq K(n)\leq K^{*}$$, $$n\in N.$$ The equation ($$*$$) has been considered by S. Mohamad and K. Gopalsamy [Tohoku Math. J., II. Ser. 52, No. 1, 107–125 (2000; Zbl 0954.39005), correction ibid. 53, No. 4, 629–631 (2001)].
The authors give counterexamples to show that the results obtained by Mohamad and Gopalsamy are false. In addition, they correct some of those results. The authors prove the following: Assume that $$\{r(n)\}$$ and $$\{K(n)\}$$ are positive periodic sequences with a common positive period $$\omega$$. Then
(i) there exists an $$\omega$$-periodic solution for equation ($$*$$)
(ii) for every positive solution $$\{x(n)\}\,$$ of ($$*$$), it is true that $$\lim_{n\rightarrow \infty }(x(n)-\widetilde{x}(n))=0,$$ where $$\{\widetilde{x}(n)\}$$ is a periodic solution of equation ($$*$$) provided that $$\frac{K^{*}}{K_{*}}\exp (r^{*}-1)\leq 2,$$ where $$r^{*}=\max_{n\in N}\{r(n)\}\,,K_{*}=\min_{n\in N}\{K(n)\},$$ and $$K^{*}=\max_{n\in N}\{K(n)\}.$$
Reviewer: Fozi Dannan (Doha)

### MSC:

 39A11 Stability of difference equations (MSC2000) 39A20 Multiplicative and other generalized difference equations

Zbl 0954.39005
Full Text:

### References:

  Mohamad, S.; Gopalsamy, K., Extreme stability and almost periodicity in a discrete logistic equation, Tohoku math. J., 52, 107-125, (2000) · Zbl 0954.39005  Coleman, B.D., Nonautonomous logistic equations as models of the adjustment of populations to environmental changes, Math. biosci., 45, 159-173, (1979) · Zbl 0425.92013  Coleman, B.D.; Hsieh, Y.H.; Knowles, G.P., On the optimal choice of r for a population in a periodic environment, Math. biosci., 46, 71-85, (1979) · Zbl 0429.92022  Gopalsamy, K.; He, X.Z., Dynamics of an almost periodic logistic integrodifferential equation, Methods appl. anal., 2, 38-66, (1995) · Zbl 0835.45004  Zhang, Q.Q.; Zhou, Z., Global attractivity of a nonautonomous discrete logistic model, Hokkaido mathematical journal, 29, 37-44, (2000) · Zbl 0965.39012  So, J.W.-H.; Yu, J.S., Global stability in a logistic equation with piecewise constant arguments, Hokkaido mathematical journal, 24, 269-286, (1995) · Zbl 0833.34075
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.