5 . Let x and y be the populations , in thousands , of brown bears and giant pandas in a national park respectively . The changes in the populations can be
modelled by the coupled differential equations
�dx �6x
�0
� dt �
.
�dy � 5y
� �x
�� dt
( a ) When x � 5, state the range of values of y such that d y
25 dt � . [ 1 ]
The system can be expressed by a matrix equation
2� 2 matrix , and
�dx �
� dt
� X � � � and dy � � � dt
� be the eigenvalues of M , where �1 � �2.
X � MX , where M is a
�x
� X � � � are two 2� 1 matrices . Let �
1 and �
2
�y
�
( b ) Find det ( M��I ) , giving the answer in terms of � .
( c ) Hence , write down the values of �
1 and �
2
.
Let v
1 and v
2 be the eigenvectors of M corresponding to �
1 and �
2 respectively .
( d ) Write down v
1 and v
2
.
The initial populations of brown bear and giant panda are 22000 and 5000 respectively .
[ 2 ]
[ 2 ]
[ 2 ]
( e ) Find the particular solution of
( i ) x ;
( ii ) y .
( f ) Hence , state the long-term behaviour for the population of
[ 5 ]
( i ) brown bear ;
( ii ) giant panda . [ 2 ]
© SE Production Limited 18 All Rights Reserved 2021