Final answer:
The function P(t) = aeᵢt best represents the total number of a species, both in the wild and in captivity, with an exponential growth model, where a is the initial population size and eᵢt shows the growth over time.
Step-by-step explanation:
To represent the total number of a species, both in the wild and in captivity, assuming all those born in captivity are still alive today and considering a long lifespan for the species, the function must accommodate the exponential growth pattern. The most suitable function for the given scenario is an exponential growth function. Out of the given choices, option D P(t) = a(1 + e⁻ᵢt) does not represent an exponential growth pattern because of the linear addition with the constant a. Option B involves a fraction and also does not represent pure exponential growth. Option A P(t) = eᵢt, represents the exponential growth with the omission of an initial population size, which is necessary for calculating the current population. Therefore, the most appropriate function for this purpose is option C, P(t) = aeᵢt, which includes a coefficient a representing the initial population size, and an exponent kt to demonstrate the continuous exponential growth over time t, with k as the growth rate constant. This aligns with the general exponential growth formula P = Poert, where Po is the initial population and r is the growth rate.