Assuming all things in the fictional world are the same to today's world, which real life constraints must be considered in order to increase the credibility in an immortal narrative/story.
To be clear - I mean which biological, ecological, sociological or anthropological constraints would ruin the credibility of an immortal story because the audience suspension of disbelief would be interrupted. Example : Malthusian Catastrophe.
Many thanks to those individuals who have helped to shape this question already in the comments due to an unclear original question.
Information to Consider
Information to aid answerers:
Immortality is to be interpreted as per the accepted definition. A being which can live exceptionally longer than normal or realistic biological standards to the point that normal humans consider them living forever.
Immortality is not affected by disease.
Immortals may end each other's existence through a single mechanism which ensures immortality is destroyed. In Highlander it was spinal severance, in Vampirism is it aortic puncture with a stake etc. For the sake of argument lets just refer to the method of dispatch as Method X.
Immortals pose no real threat to humans other than that which might the result of anger of other normal emotional responses. Immortals do not feed on or otherwise harvest humans.
If two immortals breed then the offspring can grow to teach the age of the father at the conception.
Immortals can engage in intercourse with humans but an immortal cannot be produced although the human offspring may posses stronger biological abilities - stronger immune system etc although not wildly so.
Should the ecosystem be destroyed through a calamitous event (nuclear explosion, catastrophic solar storm, volcano eruption) then it can be realistically assumed the immortal would also perish as regenerative abilities would be overwhelmed.
Immortals heal almost instantly from breaks, standard burns, blood loss etc unless Method X (discussed above) was the mechanism which ensures death.
Examples of real-life constraints discussed this far;
Brain Capacity for Memory is Finite
No true nutrition violates certain laws of Physics since Immortals could not regenerate without nutrition.