Need help with book example - modeling \"field mice and owl\" as a differential

Question

Need help with book example - modeling "field mice and owl" as a differential equation.

Hello. I'm currently reading the 1st chapter of Elementary Differential Equations and Boundary Value Problems (10th edition). I'm reading an example that models a differential equation based on the population of field mice and the rate at which owls consume the mice.

At first, they start with the following diff eq:
dp/dt = rp , where R is the growth rate and P is the population of mice.

Next, they assume that the growth rate is 0.5/month and that the owls kill 15 field mice per day.

Finally, they reach the following differential equation

dp/dt = 0.5p - 450

My question is how they calculated 450, what it represents, and how you can tell that it should be a negative number. I'm not sure the book explains this very well.

Explanation / Answer

dp/dt represents the rate of change of the population of the mice.

t is time in months

now we are given that the growth rate of the mice is .5/month

now the owls kill 15 mice every day

and we know that we are talking in terms of per month

so lets say its a 30 day month so in 1 month the owls will kill = 15*30 = 450 mice

now to get the rate of change of population of mice per month we need to subtract the number of mice killed by the owls in 1 month that is = 450 mice

hence the final differential equation becomes :

dP/dt = rp - 450

dP/dt = .5p - 450

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