Three innovations are proposed for dynamically varying the population size during the run of the genetic programming (GP) system. These are related to what is called Dynamic Population Variation (DPV), where the size of the population is dynamically varied using a heuristic feedback mechanism during the execution of the GP with the aim of reducing the computational effort compared with Standard Genetic Programming (SGP). Firstly, previously developed population variation pivot functions are controlled by four newly proposed characteristic measures. Secondly, a new gradient based pivot function is added to this dynamic population variation method in conjunction with the four proposed measures. Thirdly, a formula for population variations that is independent of special constants is introduced and evaluated. The efficacy of these innovations is examined using a comprehensive range of standard representative problems. It is shown that the new ideas do have the capacity to provide solutions at a lower computational cost compared with standard genetic programming and previously reported algorithms such as the plague operator and the static population variation schemes previously introduced by the authors.