# Logistic equation as model for a human population

Given the population in 2017,$$P_0=80$$ Million and birth at the same year was $$B=700.000$$, whereas the death rate the same year was $$D=900.000.$$ I need to predict the population in 50 years by assuming the logistic equation $$\frac {dP}{dt}=\mu (1-\frac{P}{K})P - D P$$. I got the solution of the differential equation being $$P(t)=\frac{\mu -D}{\frac{\mu}{K}-exp(-(\mu -D)t)},$$ where $$\mu$$ is the growth rate,$$K$$ is the capacity. It was assumed that birth rate, death rate and population growth is proportional to the total population.

I got the result $$P(50)=0,$$which I dont understand. I might have done a mistake. Can somebody point me out what I did wrong ?

Many thanks.

• What are the values of $\mu$ and $R$ ? – Matti P. Nov 7 '19 at 10:46
• where is $t$ in $P(t)=\frac{\mu -R}{\frac{\mu}{K}-exp(-(\mu -D))},$ ? – Ahmad Bazzi Nov 7 '19 at 10:48
• I did a typo, there is no R. I took for $\mu$ the birth at the initial year, that is mybe wrong. – user249018 Nov 7 '19 at 10:53
• The only way you can get zero out of that expression is if $\mu = D$ ... Which values did you use? – Matti P. Nov 7 '19 at 11:42
• Thanks. $\mu=$Birth at 2017 and $D=$deaths at 2017. But this is a negativ number. The value of the exponential will then be a positiv very big number (kind of infinity). – user249018 Nov 7 '19 at 11:46

Your model should have only two parameters, $$\mu$$ and $$K$$. The last term is for a harvest proportional to the population size, which does not play a role in this scenario.
Thus at $$P=80\cdot 10^6$$ you get $$\mu P=0.7⋅10^6$$ and $$\frac{\mu P^2}K=0.9⋅10^6$$ so that $$\mu=\frac{7}{800}$$ and $$K=\frac{7⋅80}{9}⋅10^6\approx 62.2⋅10^6$$.
Thus the current population is above carrying capacity, it will further shrink by having more deaths than births towards the carrying capacity of $$62\,222\,222$$