Let ?, ?, ?, and ? all be positive constants. Let X and Y be the populations of

Question

Let ?, ?, ?, and ? all be positive constants. Let X and Y be the populations

of two di?erent species. Let t be time. Consider the population model

dX

dt = ??X ? ?XY

dY

dt = ?Y ? ?XY

a) According to this model, what happens to species X in the long term if

species Y is extinct? Explain.

b) According to this model, what happens to species Y in the long term if

species X is extinct? Explain.

c) If both species are present, does the presence of species Y serve to increase

or decrease the population of species X? Explain.

d) If both species are present, does the presence of species X serve to

increase or decrease the population of species Y ? Explain.

Explanation / Answer

Let ?, ?, ?, and ? all be positive constants. Let X and Y be the populations

of two di?erent species. Let t be time. Consider the population model

dX

dt = ??X ? ?XY

dY

dt = ?Y ? ?XY

X ' = AX - BXY...................................................1.........................

Y ' = CY - DXY ...................................................2...................

a) According to this model, what happens to species X in the long term if

species Y is extinct? Explain.

Y TENDS TO ZERO .....FROM EQN.1 WE THEN GET X ' = AX ....

DX/DT = AX .......DX/X = ADT .....INTEGRATING .....LN[X]-LN[K]=AT ...

WHERE K IS A CONSTANT ...

X=K*[E^(AT)]...................SINCE A IS GIVEN TO BE POSITIVE ,X WILL INCREASE INDEFINITELY WITH T ...AS T TENDS TO INFINITY , X TENDS TO INFINITY ..

b) According to this model, what happens to species Y in the long term if

species X is extinct? Explain.

X TENDS  TO ZERO.....FROM EQN.2 WE THEN GET Y' = CY...

DY/DT= CY .......DY/Y = CDT .....INTEGRATING .....LN[Y]-LN[K']=CT

WHERE K IS A CONSTANT ...

Y=K'[E^CT]...................SINCE C IS GIVEN TO BE POSITIVE ,Y WILL INCREASE INDEFINITELY WITH T ...AS T TENDS TO INFINITY , Y TENDS TO INFINITY ..

c) If both species are present, does the presence of species Y serve to increase

or decrease the population of species X? Explain.

LET US CONSIDER THE SIMPLE CASE OF A=B=C=D .....WE GET ..

X'=X-XY.....Y'=Y-XY ...........SUBTRACTING ....

X'-Y'=X-Y ............X'-X=Y'-Y.......= CONSTANT ......

C.F. OF BOTH EQNS ARE ..

X=K1*[E^T].................Y=K2*[E^T]

SO WE GET ...

X'=X[1-K2*{E^T}]............Y'=Y[1-K1*{E^T}]

SO THE PRESENCE OF EITHER SPECIES RESULTS IN DECREASED RATE OF INCREASE OF OTHER SPECIES

d) If both species are present, does the presence of species X serve to

increase or decrease the population of species Y ? Explain.

SAME AS ABOVE

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