What does the population growth model equation mean? dN/dt=rN

1 Answer
Feb 10, 2016

The equation #(dN)/dt = rN#means that rate change of the population is proportional to the size of the population, where r is the proportionality constant.


This is a rather simple and impractical equation because it signifies an Exponential Population Growth. If you are familiar to the Future Value of a compounded interest rate, #FV = PV (1+r)^n#.
dN/dt = rN : a differential equation describing the population growth
where N is the population size, r is the growth rate, and t is time.
#N(t) = N_0e^(rt) # : the solution of the differential equation for exponential growth. The equation grows exponential and you know population does not grow exponentially, as a result we have have a more reasonable model called "The Logistic Equation". The Logistic model sets limit to the growth. Why? Well a control space like a nation, a savanna, or the plane carry a finite amount of resources and cannot support exponential populations growth in perpetuity.
#(dN/dt) = rN(1 - N/K)# : The logistic differential equation, has
N as the population size, r is growth rate, K is carrying capacity.
This equation forces, populations to converge to the carrying capacity. The speed at which the populations approach K is related to the growth rate r.