One reason for saying running instead of ran is to make running parallel with (have the same form as) traveling and lecturing.
Here is a simpler example:
Maude divides her time between earning money and spending money.
Maude's time is divided between two activities. Since the words for both activities use the same verb conjugation, it's easy to see that they play the same role in relation to Maude's divided time.
2. Naming an action abstractly, without making a new assertion
Parallelism is helpful but not required. Even disregarding parallelism, you would still have to say running, not ran, because there is another principle at work here. Here are some simpler examples to illustrate it:
In 2008, Goodall spent all her time playing with chimps.
Currently, Goodall spends all her time playing with chimps.
Next year, Goodall will spend all her time playing with chimps.
The verb spend is the main verb of each sentence, so it has tense and it agrees with Goodall (singular, so in the present tense you say spends).
The verb play is a complement of spend; it does not make an assertion of its own the way spend does. It's just part of the assertion made by spend. I think it's best to understand playing as a verb "playing" the role of a noun in regard to spend. So, the action of playing is named abstractly, not with a tense. Even though Goodall is playing, she is not the grammatical subject of playing in the sentences above. The verb play must become a gerund to indicate that it's serving as the complement of spend.
Note that the form of play is dictated by the customs for the word spend, just as the form of traveling, lecturing, and running is dictated by the customs for divide … between. In other circumstances, you could use play as a noun without adding -ing, or it might be necessary to use an infinitive. It's hard to state a simple, useful, general rule for when each form is required.
The word ran would be inappropriate in your original example because it has tense (specifically, past tense). A verb can only have tense when it's making an assertion; the tense tells what time the assertion applies to. When a verb is a complement to a main verb, only the main verb has tense. The complement verb (run) takes an abstract form (running) that indicates that it is only a part of what the main verb (divides) is saying about the main subject, not making a new assertion of its own.