At the beginning of a study of the population of ladybugs in a certain area, the

Question

At the beginning of a study of the population of ladybugs in a certain area, the population was estimated to be about 14,000. It was observed that the rate of growth of the population P of ladybugs related to breeding and dying is proportional to that population. Additionally, it was estimated that each year about 2000 ladybugs migrate to neighboring areas. No migration of ladybugs into the studied area was noticed. Set up an initial value problem, a differential equation with an initial condition, to model the population of ladybugs in that area. Do NOT solve this equation.

Explanation / Answer

the rate of change of population at any time 't' in years is related to breeding and dying

=> The rate of change of population = difference in birth and death rates

now the birth and the death rates are proportional to the popuation

Let P(t) be the popuation of the ladybut at any time 't'

=> the birth rate = bP(t) ,b >0

and the death rate = mP(t) , m >0

=> now each year 2000 ladybugs migrate to other areas so if t is the time in years then 2000t ladybugs would migrate to and from the area under study. But e are given that no migration of ladybugs into the studied area was noticed .

=> 2000t ladybugs would be subtracted form the differential equation

Rate of change of popualtion = (birth rate - death rate) - ladybugs migrated from the area under study

dP/dt = bP - mP - 2000t = P(b-m) - 2000t

now the initial condition is at t = 0 years , P(t) = 14000

=> the differential equation model is :

dP/dt = P(b-m) - 2000t , P(0) = 14000

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