Question

Another model for a growth function for a limited population is given by the Gompertz function, which is a solution to the differential equation

d P/d t=c ln(K/P) P
where c is a constant and K is the carrying capacity.
Solve the differential equation with a constant c=0.15, carrying capacity K=2000, and initial population P₀=500.

Answer 1

The Gompertz function, a model of population growth accounting for carrying capacity, can be solved by separating variables and integrating the given equation with specified constants.

Step-by-step explanation:

The question pertains to the solution of a differential equation known as the Gompertz function, which is used to model population growth considering a carrying capacity. Given the differential equation dP/dt = c ln(K/P) P with c = 0.15, carrying capacity K = 2000, and initial population P₀ = 500, we can solve this differential equation by separating variables and integrating.

Let's start the process by reorganizing terms:

\(dP/P = c ln(K/P) dt\)

Integrate both sides, which results in the equation of the Gompertz function.

Apply the initial condition to solve for the constant of integration.

Obtaining the population function P(t) then allows us to understand how the population changes over time in comparison to the carrying capacity. This helps us visualize the logistic growth, where initially, when the population is small, it grows nearly exponentially. As it approaches the carrying capacity, the growth rate slows down, finally leveling off as per the logistic model.