Show that if the population exceeds its carrying capacity, then the population is decreasing.

Question

The population of Aedes mosquitoes which carry the Dengue virus can be modeled by a differential equation which describes the rate of growth of the population. The population growth rate dP/dt is given by dP/dt = rP(1-P/k) , where r is a positive constant and k is the carrying capacity.

Answer 1

When dP/dt is zero it is not increasing or decreasing, and when dP/dt<0 it is decreasing. Otherwise it is increasing and dP/dt>0.

So the turning point of neither increase or decrease is dP/dt=0 and rP(1-P/k)=0. Since rP cannot be zero or negative for an existing population P, 1-P/k=0 so P=k, the carrying capacity. If P<k, then 1-P/k>0 and the population increases because dP/dt>0. If P>k, 1-P/k<0 and dP/dt<0 and the population decreases. So if P exceeds its carrying capacity the population decreases.