Are you speaking of genetic algorithms for phylogenies and the models that fit them. What software are you using.
The beauty of the genetic algorithm is that is modeled on natural selection. In this case you are optimizing the "fitness" of the various possible models and then the various tree topologies. The adaptive algorithm does the work of the GA. The initial population may not matter much. But if you have a reasonable one then the search time should be shorter and perhaps will be less like to find a local maxima.
You can apply parallel processing in the way that using several initial populations at be biginning of GA process, therefore, that allow to discover a large areas or different areas in the search space. In this manner, GA may converge fastly and found the optimal solution.
Initial population plays an important step of the algorithms due to directly affect the quality of the solutions. Therefore, there are two procedures for initial population selection. Both procedure has their own merits and demerits.
Randomly selected initial population tends to provide wide solutions as you will have better reach, but it might take more time to converge. While in the case of using heuristics for initial population, there is the chance that your solution will revolve around heuristic solutions, but it will converge rapidly. So there is a trade off using these procedures.
For more details about genetic algorithm, check this title:
"Genetic Algorithm with an Improved Initial Population Technique for Automatic Clustering of Low-Dimensional Data 2018"
The GA performance is highly dependent on the value of it initial solution. Thus, if the initial solution is poor, the algorithm will spend most of its time to improve this solution. However, the GA is considered as iterative local search algorithm and this is a general problem in those types of the algorithms. Therefore, there are an interesting research directions to solve the above-mentioned issue.
Please can you us give more detail about the last point mentined in your reply: " Therefore, there are an interesting research direction that does hybridization between the iterative improvement algorithms and the constructive algorithms to solve the above-mentioned issue."
What are constructive algorithms ? how they work ?
How to perform hybridization between the iterative improvement algorithms and the constructive algorithms ?
I run my genetic problem with numerous numbers for the initial population, but I saw not that much difference in the speed of convergence or the accuracy of the result. the thing is, it mostly affected the run time. I started with 100 initial population up to 1000, may it was already sufficient for my case.