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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&