The question asks for the correct equation representing exponential population growth.

Exponential Growth Equation Formula

Exponential population growth occurs when a population increases at a rate proportional to its current size. This is often modeled using the following equation:

$P_{t} = P_{0}e^{rt}$

Understanding the Variables

• $P_{t}$: Population size at time $t$.
• $P_{0}$: Initial population size (at time $t=0$).
• $e$: The base of the natural logarithm (approximately 2.718).
• $r$: The intrinsic rate of natural increase (per capita growth rate).
• $t$: Time elapsed.

Identifying the Correct Equation

This formula models continuous growth where the rate of increase depends directly on the population size. Comparing this standard form to the options provided:

• Option 1: $P_{t-1}=P_{0}e^{rt}$ is an incorrect formulation.
• Option 2: $P_{t}=P_{t-1}e^{rt}$ represents growth from one discrete time step to the next, not continuous exponential growth from $t=0$.
• Option 3: $P_{0}=P_{t}e^{rt}$ incorrectly places the initial population on the left side and implies a decrease.
• Option 4: $P_{t}=P_{0}e^{rt}$ correctly represents the exponential population growth model, relating the future population ($P_t$) to the initial population ($P_0$) based on the growth rate ($r$) and time ($t$).

Therefore, the correct equation for exponential population growth is $P_{t}=P_{0}e^{rt}$.