In the study of the ecological dynamics, the researchers always assume the factors f(t) of the circumstances vary periodically according to the changes of the seasons. But as the sunlight and other factors of this year may be different from that year, so the variation of f(t) is not rigidly periodic, that is, f(t + T ) = w(t)f(t) with w(f) 6≠ 1, which is called weighted periodic function in our previous works. Here this case is tried on the Logistic population-evolution model and it gives a very interesting result: in case the inherent increasing rate and the interspecific competition rate vary in a weighted periodic manner, the evolution of the population will show itself asymptotic weighted periodicity and the weight is just the reciprocal of that for the interspecific competition rate. It gives a good explanation to the ecological phenomenon that more fierce competition implies more rapid decreasing of the population.