How does carrying capacity affect exponential growth?

Carrying capacity is a horizontal asymptote because it is the upper bound of an exponential growth function. If there are no external factors that alter the function's parameters, an exponential growth function representing population growth in a natural environment will not exceed the carrying capacity.

The carrying capacity of a given species is the maximum sustainable population that the species can have in an environment given the resources available to it. It is important to note that this is the maximum sustainable population; under some conditions (such as a large influx of new members from outside the environment or infrequently certain natural cyclic variations), the population may exceed the carrying capacity; however, this will result in a subsequent "population crash" or sharp decline in the population due to a shortage of resources.

For the time being, let's assume that our deer population starts at 1200 specimens. For example, let's say that the carrying capacity of deer in a given environment is 1500. This means that the environment can sustain 1500 deer indefinitely with its available resources.

Assume further that a population of 400 deer from a nearby environment migrates to our initial environment due to external factors (increased predator activity, scarcity of resources). At 1600 deer, we have a carrying capacity of 1500. This means that some of the deer will perish from malnutrition, while others will become easier prey for predators like wolves because of their weakened state, and some may even perish in combat over the resources. Consequently, the population will rapidly decline back below the carrying capacity.

You can find a very basic explanation here: https://tutor.hix.ai There are also graphs and more information.

Carrying capacity influences exponential growth by setting a limit on the population size that can be sustained by the available resources in a given environment. As the population approaches the carrying capacity, the rate of growth slows down and eventually levels off, leading to logistic growth rather than exponential growth.