Asymptotically Periodic Logistic Equation

It was known to all that P. F. Verhulst and A. L. J. Quetelet who proposed the classical Logistic model in 1838, and then T. G. Hallam and C. E. Clark mended the classical Logistic model in 1993. And after that, many experts and scholars had studied the intrinsic growth rate and carrying capability, which were two important indexes about Logistic equation. Nevertheless, the Logistic equation was not far from precise, but after a hunderd years, it was still an important and widely applied model in ecology. Especially in recent score years, in autonomous and nonautonomous systems, the neutral, finite-delay, infinite-delay, diffusive, discrete, discrete-delay, generalized, difference and other style models were discussed on optimal harvesting, osillation, stability, global stability, and other asmptotic behaviors of Logistic equation. Reference were searched in[l-6], [10-18], [20-31], [34]. The periodic and almost periodic function Logistic equation, which are widely studied recently, the results about the above are found on many kinds of magazines. The article [1] has a unique periodic solution, which is globally asmptotic stable periodic solution with period T. The expression is obtained in [1]. And we get global attractivity in the article [2], [3].Within the author's knowledge, recently, the results are few about the asmptotic Logistic equation. The articles, [32-33] discuss the Logistic equation, in which the two important indexes r(t) and K(t) are not only dependent on t. but also asmptotically tend to periodic function. In article [14], about the three-species competition system, under moderate conditions, all asmptotic periodic solutions tend to a unique periodic solution on a given system. Therefore, we need more concrete function to better discribe the variation of some biological populations in our real world. The two important indexes r(t) and K(t) of Logistic equation are asmptotic periodic function, and discussed by author. The asmptotic solution of the asmptotic periodic Logistic function is obtained. Some good properties, such as, existence, uniqueness, global attractivity are discussed. The properties of asmptotic periodic function and the asmptotic periodic function space are studied, accordingly, we have, the asmptotic periodic function space is Banach space. And then, multiple-dimention asmptotic periodic function space is still Banach space.The arrangement of the article is: in the second section the properties of existence, uniqueness, global attractivity are discussed; in the third section, the properties of asmptotic periodic function are proved; in the fourth and the fifth section, on one dimention and multiple-dimention, asmptotic periodic function space are Banach space.