Running after an animal or from one, and running for play are different forms of human experience corresponding to different pragmatic contexts.
Wolf is wolf, whether it is running after someone, or it is only a product of someone's imagination, or it is displayed in a cage in the zoo, or it is in the process of becoming extinct.
I can't say in what way he could tell, but when I had set off and had gone right down the street, I heard something, and there he was, running after me without his cap.
You mean to make it known about Marya Timofyevna," Pyotr Stepanovitch muttered, running after him, and, as though not thinking of what he was doing he clutched at his shoulder.
Lizaveta Nikolaevna, this is such cowardice," cried Pyotr Stepanovitch, running after her.
They afterwards had a good race stark naked, running after me and grinning.
The angry brute slowly and deliberately got up, but, once on her legs, she made a dead set at me, running after me.
A day or two afterwards these young rascals came running after me again in the same way; but they were chased by an adult Moor, who gave them a good thrashing.
The above list will hopefully provide you with a few useful examples demonstrating the appropriate usage of "running after" in a variety of sentences. We hope that you will now be able to make sentences using this group of words.