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HW7_solns

# HW7_solns - 8.1.11 u = a(1 u uv 2 There is a xed point at(1...

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8.1.11 ˙ u = a (1 - u ) - uv 2 ˙ v = uv 2 - ( a + k ) v There is a fixed point at (1 , 0) for all a and k . Nullclines curves are given by u = a a + v 2 and u = a + k v Figure 1: Nullcline Curves as the parameters are varied These curves intersect when a a + v 2 = a + k v (0.1) solving for v v = a ± radicalbig a 2 - 4 a ( a + k ) 2 2( a + k ) The discriminant defines the number of solutions If a 2 - 4 a ( a + k ) 2 > 0 2 solutions If a 2 - 4 a ( a + k ) 2 < 0 No solutions If a 2 - 4 a ( a + k ) 2 = 0 One solution To determine the bifurcation point analytically, use the tangency condition ∂v parenleftbigg a a + v 2 parenrightbigg = ∂v parenleftbigg a + k v parenrightbigg 2 av 3 = ( a + k )( a + v 2 ) 2 Filling this back into equation 0.1 a = v 2 k = - a ± a 2 So the bifurcation occurs along a bifurcation curve defined by k = - a ± a 2 . Now to determine the stability of the fixed points 1
D f ( u * , v * ) = bracketleftbigg - a - ( v * ) 2 - 2 u * v * ( v * ) 2 2 u * v * - ( a + k ) bracketrightbigg For the fixed point at (1 , 0) the eigenvalues are - a and - ( a + k ), which corresponds to a stable node.

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HW7_solns - 8.1.11 u = a(1 u uv 2 There is a xed point at(1...

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