Observe the mathematical notation for the curves. J-shaped population growth has a mathematical model that relates population growth directly to the exponential of a certain variable. That is, the J-shaped population growth model should look like P = A*exp(B*x) + C, where "exp" represents the exponential function, A, B and C are constants, and x is the variable of interest. On the other hand, S-shaped population growth has the exponential function in the denominator. In addition, the variable inside the exponential function of an S-shaped population model should be negative. The mathematical form of the S-shaped population model is P = A/(B + C*exp(-D*x)), where the capital letters are constants and x is the variable of interest.
Analyze the graphs of the population models. These models have their corresponding names for a reason. When you graph the J-shaped population model, the curve representing the function will look like a "J," with the population slowly increasing and then suddenly rocketing up without bound. The S-shaped population model looks similar to the J-shaped population model up until a point. Whereas the J-shaped population model's graph continues upward with no bound, the S-shaped population model shows a slowing down of population growth after the sudden burst to yield an "S"-like curve.
Consider the circumstances, as the J-shaped and S-shaped population models represent different situations. The most notable situational difference between these two models is that the J-shaped model signifies a population increase that is not limited for environmental reasons, by the existence of predators or in resources. The S-shaped population model, on the contrary, is describes situations in which the population can only grow to a certain point. Possible reasons are ease of falling prey to predators, border restrictions leading to fights over space and the lack of adequate resources to continue notable growth.