You want to know about the population dynamics of the earthworms you plan to put into your compost bin. You get a shipment of 250 worms to put into your bin and you calculate that your bin size can support 2000 worms. The company tells you that they have a per capita birth rate of 0.6 and per capita death rate of 0.2. What will be the instantaneous rate of change in population size over time (include 3 decimal places)

ANSWER: The instantaneous rate change in population of the warm is 0.400 and it will take the worm 7.999yrs to fill the bin.

Explanations: instantaneous change rate of a population is the rate difference between the birth rates and death rate of that population.

STEP 1: FIND THE DAILY BIRTH INCREASE.

birth rate = 0.6

Population = 250

Number of days in a year = 365

Their Daly birth increase will be;

(0.6 ÷ 365) × 250 = 0.411

STEP2: FIND THE DAILY DEATH INCREASE;

Death rate = 0.2

Population = 250

Number of days in a year = 365

Their daily death increase will be;

(0.2 ÷ 365) × 250 = 0.137

STEP 3: THE CHANGE IN THEIR DAILY POPULATION:

daily birth - daily death

0.411 - 0.137 = 0.274

STEP4: FIND THE INSTANTENEOUS CHANGE RATE:

Daily change in population = 0.274

Number of days in a year = 365

Number of population = 250

(365 × 0.274) ÷ 250 = 0.400

Therefore the instantaneous rate of change in the worm population is 0.400

TO CALCULATE HOW LONG IT WILL TAKE THE BIN TO FILL

(0.400 × 2000) ÷ 0.274 = 2919.706 days

Which is 2919.796/365 = 7.999yrs.

Therefore it will take the bin 7.999 years for the worm to fill the bin

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