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MA2001N Differential Equations
Application Problems 1: Population Dynamics
It is possible to set up a mathematical model of the behaiour of a population!
as its si"e! P ! chan#es $ith time! t % Problems in the #ro$th! or the decay! of
the population! &'t P ! are typically #oerned by one of the follo$in# t$o ode(s:
'i& )he e*ponential model:
P k dt
dP = ! 'A&
$here k is a positie or ne#atie constant%
'ii& )he limited #ro$th model:
P P P r dt
dP m
&' −= ! '+&
$here r is a positie constant and m P is the ma*imum alue that the
population! P ! can attain%
At some initial time! say! 0=t ! the initial population! 0 P ! $ill be ,no$n% )his is
the startin# point for either population #ro$th or decay! dependin# on the
circumstances! as time #oes by% )his #ies rise to $hat is called an initial condition!
$hich is $ritten:
0&0' P P = ! '-&
$here 0 P is a #ien constant% Equation '-& means the follo$in#:
at time! 0=t ! the population is #ien as! 0 P P = %
)his initial condition! in '-&! enables the constant of inte#ration to be determined%
)he constant of inte#ration! it should be remembered! al$ays arises $hen a1st order ode is soled% )he detail of the process $ill become clear shortly! $hen
the questions oerleaf are attempted%
In summary! usin# either 'A& and '-& to#ether! or! alternatiely! '+& and '-& to#ether!
#ies rise to $hat is called an initial alue problem%
No$ try the questions oerleaf%
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MA2001N Differential Equations
Application Problems 1: Population Dynamics 'continued&
.ole the follo$in# initial alue problems:
'1& /or the initial alue problem
P k dt
dP = ! 0&0' P P = ! 0≥t !
in $hich k and 0 P are constants! sho$ that by separatin#
ariables and inte#ratin#! the #eneral solution can be $ritten
kt e A P = !
$here A is a constant of inte#ration%
/urther! sho$ that! after applyin# the initial condition!
kt e P P 0= %
'2& /or the initial alue problem
P P P r dt
dP m
&' −= ! 0&0' P P = ! 0≥t !
in $hich r ! m P and 0
P are constants! sho$ that by separatin#
ariables! usin# partial fractions! inte#ratin# and ta,in# the
e*ponential! the #eneral solution can be obtained from the
e*pression
t rP m me A P
P P −
=
−
!
$here A is a constant of inte#ration%
/urther! sho$ that! after applyin# the initial condition! and
rearran#in#
t rP
m
m
me P P
P P
−
−+
=
&1'10
%
'continues oerleaf&
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MA2001N Differential Equations
Application Problems 1: Population Dynamics 'continued&
.ole the follo$in# initial alue problems 'continued&:
'& .ole the follo$in# initial alue problem for population! P
P dt
dP 2−= ! 00&0' = P ! 0≥t !
sho$in# thatt
e P 2
00 −
= % 'D&
3sin# the ans$er #ien in 'D&! find:
'i& the alue of P at 1=t 4
'ii& the alue of P at 2=t 4
and 'iii& the time it ta,es for the population to be
reduced to 1= P %
'5& .ole the follo$in# initial alue problem for population! P
P P
dt
dP &000!'
000!
1−=
!000!1&0' = P
! 0≥t
!
sho$in# that
t e
P −
+
=
51
000! % 'E&
3sin# the ans$er #ien in 'E&! find:
'i& the alue of P at 1=t 4
and 'ii& the alue of P
at10=t
%
End of Application Problems 1
6666666666666666666666666666666666666666666666666666666666666666666
Ans$ers to Application Problems 1! $here these are required! are:
'&'i& 78 'rounded alue& '5&'i& 2!02 'rounded alue&
'&'ii& 9 'rounded alue& '5&'ii& 5!999 'rounded alue&
'&'iii& 10:% 'to 5 decimal places&
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