Final answer:
Zero population growth for a prey population occurs when the intrinsic rate of increase, r, is zero, meaning that birth rates equal death rates or the population size has reached the environment's carrying capacity, N = K. Logistic growth models this scenario with an S-shaped curve showing initial exponential growth that slows down and stops at the carrying capacity.
Step-by-step explanation:
The net population growth rate for the prey population will be zero when the intrinsic rate of increase, r, equals zero. In other words, when the prey population is neither increasing nor decreasing in size, it is said to be in a state of zero population growth. This condition can occur either when birth rates equal death rates, or when the population size reaches the carrying capacity of the environment (N = K) which is the maximum population size that a particular environment can sustain.
When considering logistic growth, the initial exponential growth phase happens because resources are plentiful relative to the small population size. As the population size increases and approaches the carrying capacity, growth rate decreases and eventually levels off, resulting in zero population growth. This S-shaped curve is termed logistic growth and is more realistic than infinite exponential growth because it accounts for the limitation of resources as described by the logistic growth model.