# Stpm Mathematics T Assignment 2013 Gmc

STPM MATHEMATICS (T) – ASSIGNMENT B: MATHEMATICAL MODELLING (2013)STEPHEN, P. Y. BONG (APRIL 2013) Page

2

of

11

ASSIGNMENT PROBLEMS

The population of Aedes mosquitoes which carry the Dengue virus can be modeled by adifferential equation which describes the rate of growth of the population. The populationgrowth rate

dd

P t

is given by

d1d

P P rP t k

= −

, where

r

is a positive constant and

k

is thecarrying capacity.1.

When the population is small, the relative growth rate is almost constant. What do youunderstand by the term relative growth rate?2.

(

a

) Show that if the population does not exceed its carrying capacity, then the populationis increasing.(

b

) Show that if the population exceeds its carrying capacity, then the population isdecreasing.3.

(

a

) Suppose that the initial population is

P

0

. Discuss, by considering the sign of

dd

P t

, therelationship between

P

and

k

if

P

0

is less than

k

and if

P

0

is greater than

k

.(

b

)Determine the value of

P

for constant growth, increasing growth and declining growth.4.

Determine the maximum value of

dd

P t

and interpret the meaning of this maximum value.5.

Express

P

in terms of

r

,

k

and

t

. Using different initial population sizes

P

0

,

r

, and

k

, plot,on the same axes, a few graphs to show the behavior of

P

versus

t

.6.

The population sizes of the mosquitoes in a certain area at different times, in days, aregiven in the table below.

Time Number of mosquitoes Time Number of mosquitoes

0 49 245 70135 77 252 71263 125 322 77691 196 371 791105 240 392 794126 316 406 796140 371 441 798182 534 504 799203 603 539 800224 658 567 800It is interesting to determine the carrying capacity and the growth rate based on the above datato control the population of Aedes mosquitoes. Using different values of

P

and

t

, plot

∆

P

/

∆

t

against

P

and hence obtain the approximate values for

r

and

k

.

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