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Applied Calculus for the Manageria...

10th Edition
Soo T. Tan
ISBN: 9781285464640

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BuyFindarrow_forward

Applied Calculus for the Manageria...

10th Edition
Soo T. Tan
ISBN: 9781285464640
Textbook Problem

Predator-Prey Model The relationship between the number of rabbits y(t) and the number of foxes x(t) at any time t is given by

C ln y + Dy = A ln xBx + E

where A, B, C, D, and E are constants. This relationship is based on a model by Lotka (1880-1949) and Volterra (1860-1940) for analyzing the ecological balance between two species of animals, one of which is a prey and the other a predator. Use implicit differentiation to find the relationship between the rate of change of the rabbit population in terms of the rate of change of the fox population.

To determine
The relationship between the rate of change of rabbit population in terms of the rate of change of the fox population.

Explanation

Given:

The relationship between the number of rabbits and the number of foxes is,

Clny+Dy=AlnxBx+E (1)

Where, y(t) is the number of rabbits, x(t) is the number of foxes and A,B,C,D and E are constants.

Calculation:

Differentiate equation (1) with respect to x and get the relation in terms of the rate of change of the fox population,

Clny+Dy=AlnxBx+ECddxlny+Dddxy=AddxlnxBddxx+ECydydx+Ddydx=AxB(ddxlny=1ydydxandddxlnx=1x)(Cy+D)dydx=ABxx(LCDisx)

Further solve the above equation

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