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

Question

Connor Anderson · Accepted Answer

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.

Josephine Allen · Answer

undefined The population growth model equation $ \frac{{dN}}{{dt}} = rN $ represents the rate of change of a population (dN/dt) over time (dt) as proportional to the size of the population (N) multiplied by the growth rate (r). In simpler terms, it shows how the population size changes over time based on the growth rate, where an increase in the population is directly proportional to the current population size.