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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.

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  • $\begingroup$ What are the values of $\mu$ and $R$ ? $\endgroup$ – Matti P. Nov 7 at 10:46
  • $\begingroup$ where is $t$ in $P(t)=\frac{\mu -R}{\frac{\mu}{K}-exp(-(\mu -D))},$ ? $\endgroup$ – Ahmad Bazzi Nov 7 at 10:48
  • $\begingroup$ I did a typo, there is no R. I took for $\mu$ the birth at the initial year, that is mybe wrong. $\endgroup$ – user249018 Nov 7 at 10:53
  • $\begingroup$ The only way you can get zero out of that expression is if $\mu = D$ ... Which values did you use? $\endgroup$ – Matti P. Nov 7 at 11:42
  • $\begingroup$ 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). $\endgroup$ – user249018 Nov 7 at 11:46
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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 $62\,222\,222$

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