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Problem Description:

The population of the world was about 5.3 billion in 1990. Birth rate in the 1990s ranged from 35 to 40 million per year and death rates ranged from 15 to 20 million per year. Let's assume that the carrying capacity for world population is 100 billion.

Write the logistic differential equation for these data. (Because the initial population is small compared to the carrying capacity, you can take k to be an estimate of the initial relative growth rate.)

My calculation:

Because it's a logistic model and the carrying capacity is 100 billion, I wrote the differential equation as:

$\frac{dy}{dx} = ky(1-\frac{y}{100})$ 100 denotes 100 billions for the carrying capacity.

My question:

How can we know the value of k?

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You have the hint: "you can take $k$ to be an estimate of the initial relative growth rate".

Since, $\text{growth rate} = \text{birth rate} - \text{death rate}$, we get that the growth rate is ranged from $15$ to $35$ million per year. Hence, the average growth rate is $\frac{15+35}{2} = 25$ million per year. Therefore, the initial relative growth rate $k = \frac{25}{5 300} = 0,0047169$.

I think it is what you need.

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