In this paper, we consider the adaptive finite element method for population dynamics model with local delay. At first, we propose the fully discrete scheme and the auxiliary problem. Then we derive the upper bound of a posteriori error and a lower bound of the local space error, respectively. The error control theorem is given to guarantee the reliability of the adaptive algorithm. Finally, we prove that the time and space adaptive algorithm will stop in a finite number of steps for any given tolerances. Due to the high efficiency of the adaptive method, our theoretical analysis in this paper has significant meaning in the development of efficient numerical methods for the time delay population model.

population dynamic model, local delay, posteriori error estimate, error control, adaptive finite element method.