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Example
Paper 1 – Predator-Prey Models
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Let x and y be the populations , in thousands , of tiger and lion in a forest respectively . The changes in the populations can be modelled by the coupled differential equations �dx
�( 4 � yx )
� dt �� . dy
�( 2x�5 ) y
�� dt
( a ) State the equilibrium points for the two populations .
The initial populations of tiger and lion are 1200 and 1000 respectively .
[ 2 ]
( b ) ( i ) Express d y dx in terms of x and y .
( ii ) Hence , find the value of d y dx when t � 0.
( iii ) State , at the beginning , whether the population of tiger is increasing or decreasing .
[ 4 ]
Solution
( a ) ( 0 , 0 ) , ( 2.5 , 4 ) A2
d y ( 2x �5 ) y ( b ) ( i ) �
A1 d x ( 4 � y) x
[ 2 ]
( ii ) d y ( 2 ( 1.2 ) �5 )( 1 )
� d ( 4 �1 )( 1.2 ) x t � 0 dy
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�� A1 dx �
18 t 0
( M1 ) for substitution
( iii ) The population of tiger is increasing at the beginning .
A1 [ 4 ] www . seprodstore . com
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