FOUR. Consider the general predator-prey system x = ax-bxy where all constants a

Question

FOUR. Consider the general predator-prey system x = ax-bxy where all constants are positive. y = -cy+dxy Find the equilibrium points. Find the Jacobian of the system at the equilibrium point with positive coordinates and verify that the eigenvalues are purely imaginary. What does Theorem 7.23 say about this equilibrium? Let v- ba an equilibrium of v = f(v).If there exists a Lyapunov function v-, then v- is stable. If there a strict Lyapunov function for v-, then v- is asymptotically stable. Verify that E(x, y) = dx - c In x + by - a In y is a Lyapunov function for the equilibrium in the first quadrant. Show in fact that E(t) = 0 so E is conserved along a solution.

Explanation / Answer

the equlibrium points are x'=0 and y'=0 we ahve y=b/a as equlibrium point and x=c/d as equlibrium point fro jacobian we have x'=u and y'=v calculate we ahve a-by -bx dy dx-c as jacobian matrix byu substuiting we have y=b/a as equlibrium point and x=c/d as equlibrium point we ahve a^2-b^2/a -bc/d db/a o so total det value is bcd/a

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