Changes to determination, and personalities in general, are influenced by a number of factors: squad's overall personality, mentoring group, rare off the field events, complacency, and the greeting when a player joins the club which acts as a one time mentoring session. Disciplining a player also gives a chance to increase his determination, along with work rate.
The squad's overall personality is heavily influenced by the team leaders so if they have poor or average personalities, there is a high likelihood that they are going to drag the rest of your player's personalities down to their level, particularly younger players who are more susceptible to being influenced, for good or bad. Mentoring groups have the same effect but on a smaller scale and a more controlled fashion as you can pick and choose who influences who. Off the field events are something that you have no control over and can give your players big increased to their personality stats or they give big drops, basically ruining your player forever. Complacency is similar to off the field events but more common though you can take a few steps to lessen the chances of it happening (which I'm not going to get into here). The greeting, as mentioned above, is a one time mini-mentoring session which builds a rapport between the greeter and new player if you have chosen the right player to do it or they can instantly dislike each other if you pick a greeter who has too different of a personality from the new player.
Lastly, there is discipline. You are able to have a special interaction with your players for approximately one week after a match if the player has received a 6.5 or lower match rating in a match. This includes reserve team and youth team matches. This interaction allows you to give them a warning or fine them. If you are successful in convincing the player that they need to improve, they will receive a point in determination, work rate, or both. This is an extremely powerful tool that will boost your player's stats very quickly.