Nonlinear Dyn (2010) 61: 861–862 DOI 10.1007/s11071-010-9693-7

R E P LY

Response to the rebuttal by Eduardo Liz S.H. Saker

Received: 6 March 2010 / Accepted: 9 March 2010 / Published online: 23 March 2010 © Springer Science+Business Media B.V. 2010

This is in response to the rebuttal [8] of the paper “Compatibility of local and global stability conditions for some discrete population models” [12]. Regarding the novelty of the results, I did not claim that the results are new, since the conditions of local stability existed before the work of Prof. Liz and my work. The same question has been posed by Cull in 1986 in one of his interesting papers in this area. The question was: Does local stability imply global stability? I concentrated my work on the papers that used the Lyapunov function as well as on the equation xn+1 = f (xn ). In the first version of the paper, I have indicated to the readers the paper of Prof. Liz and Example 2.3. But since the introduction was 10 pages, the referees suggested deleting some pages. So, according to their advice, I deleted some of these pages. In fact, I stated in my paper that Prof. Liz used the enveloping principle and proved that local stability implies global stability of the Hassell Maynard Smith model, and that our aim is to prove the same results by a different approach. In the last paragraph of the paper before the aim, I deleted some of the results, not only the result of Liz but also the results of Cull [2–4], Rubi-Massegú

S.H. Saker () Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia e-mail: [email protected] S.H. Saker e-mail: [email protected]

and Manosa [11], Camouzis and Ladas [1], and Kulenovi´c and Merino [6, 7]. I presented the following: Cull [2–4] introduced a new approach based on an enveloping principle and proved that the enveloping by a linear fractional function is sufficient for global stability. He then applied his results to some different models and proved that local stability implies enveloping and hence global stability. However, this proof is not easy for complicated models and the parameter in the enveloping function must be adjusted for each particular model. One of the motivates for writing this paper is to develop a criterion for global stability that is easier to use. In Sect. 2, as a special case, I applied my results to two different models considered in [4] to illustrate the main results and to show that there exists a fractional enveloping function for a specific model, but the model is not globally stable. For related results of discrete models (I mean the results that used the enveloping function), I refer the reader to the results by Liz [9], Rubi-Massegú and Manosa [11]; and for higher order difference equations, I refer the reader to the papers by Camouzis and Ladas [1] and Kulenovi´c and Merino [6, 7], and the references cited therein. Note [10] considered the delay difference equation xn+1 = xn F (xn−k ), which is different from the equation xn+1 = f (xn ) that I have studied. In fact, I have presented a brief summary of the related results obtained for this equation. In fact, I did not ignore the note by Liz and his contribution in this direction cannot be ignored. I have discussed some of the results in this direction with Liz when I met him in Poland last

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year. The main result in [10] was to give a sharp global stability condition; as a special case when k = 0, he deduced that the global stability condition is the local stability condition and this has been done by employing a different method that depends on the Schwarzian derivative. The Lyapunov method that I have applied has been applied by Goh [5, Example 3.14.6, p. 110]. I found that this method is effective and I did not check the existence of other critical values, because Goh did not do this. Therefore, I did not expect the existence of other critical values except the equilibrium. I have given some different examples about the application of this method, and by trial and error I have shown that outside a small neighborhood of the equilibrium, ΔV is negative and then the function V (x) attains it minimum at the equilibrium while the function ΔV (x) attains its maximum at the equilibrium. The results illustrated by the graphs of the iterations, time series, and the bifurcation diagrams imply that the results are not completely wrong and this weak condition needs only an improvement. For example, it is enough to prove βx that the term (αx + 1+x γ + x − 2x) is increasing. This term appears in   βx + x − 2x ΔV (x) = αx + 1 + xγ   βx − x . × αx + 1 + xγ I have shown that, for different values of the parameter γ (which I have considered as the bifurcation parameter), the population will exhibit variations in dynamics. I will close this note by quotes attributed to Von Neumann, Hilbert, and Cayley. Von Neuman said, “In mathematics you don’t understand things. You just

S.H. Saker

get used to them”. Hilbert said, “One can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it”. Cayley said, “As for everything else, so for a mathematical theory: beauty can be perceived but not explained”.

References 1. Camouzis, E., Ladas, G.: When does local asymptotic stability imply global attractivity in rational equations? J. Differ. Equ. Appl. 12, 863–885 (2006) 2. Cull, P.: Local and global stability for population models. Biol. Cybern. 54, 141–149 (1986) 3. Cull, P.: Stability in one-dimensional models. Sci. Math. Jpn. Online 8, 349–357 (2003) 4. Cull, P.: Population models: Stability in one dimension. Bull. Math. Biol. 69, 989–1017 (2007) 5. Goh, B.S.: Management and Analysis of Biological Populations. Elsevier, Amsterdam (1979) 6. Kulenovi´c, M.R.S., Merino, O.: A global attractivity result for maps with invariant boxes. Discrete Contin. Dyn. Syst. Ser. B 6, 97–110 (2006) 7. Kulenovi´c, M.R.S., Merino, O.: Global bifurcations for competitive systems in the plane. Discrete Contin. Dyn. Syst. Ser. B 12, 133–149 (2009) 8. Liz, E.: Rebuttal of “Compatibility of local and global stability conditions for some discrete population models,” by S.H. Saker. Nonlinear Dyn. (2010). doi:10.1007/ s11071-010-9692-8 9. Liz, E.: Local stability implies global stability in some onedimensional discrete single-species models. Discrete Contin. Dyn. Syst. Ser. B 7, 191–199 (2007) 10. Liz, E.: A sharp global stability result for a discrete population model. J. Math. Anal. Appl. 330, 740–743 (2007) 11. Rubi-Massegú, J., Manosa, V.: On the enveloping method and the existence of global Lyapunov functions. J. Differ. Equ. Appl. 13, 1029–1035 (2007) 12. Saker, S.H.: Compatibility of local and global stability conditions for some discrete population models. Nonlinear Dyn. 59, 375–396 (2010)