For each of the following determine if the set of Lotka-Volterra equations repre

Question

For each of the following determine if the set of Lotka-Volterra equations represents a Cooperative System, a Competitive System, or a Predatory-Prey System. Be sure to justify your answer. (a) The model given by the equations dx/t = 0.15x - 0.0002x^2 - 0.006xy and dy/dt = 0.2y - 0.00008y62 - 0.0002y (b) The model given by the equations dx/dt = 0.8y + 0.004xy and dy/dt = 0.1x + .00001xy - 0.00003x^2 (c) The model given by the equations dx/dt = 0.05xy - 0.0004x^2 - 0.002x and dy/dt = 0.008x^2 + 0.0012x - 0.00015xy

Explanation / Answer

In orderto determine whether a given system is cooperative , competetive or predator-prey system , we shall look at the coeffcients of xy in both dx/dt and dy/dt.

a) In this set of differential equations the coefficent of xy in dx/dt is -0.0006.

and the coefficient of xy in dy/dt is -0.0002.

Since the both the coefficients are negative .

So this system is competetive system.

b) Here coeffcient of xy in dx/dt is 0.004 .

The coefficient of xy in dy/dt is 0.00001.

Since coefficients are of positive sign .

So system is cooperative .

c) Here for dx/dt , the coefficient of 'xy' is 0.005.

And the coefficient of xy in dy/dt is -0.00015.

Both the coefficients are of opposite signs .

So the given system is a predator prey system.

The one with positive coeffcients of 'xy' i.e. dx/dt determines a predator And predator will be 'x'.

The one with negative coefficients of 'xy' i.e. dy/dt determines a pray and pray will be 'y'.

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