Explanation Given: The Lotka-Voltera model is x ′ = r 1 K 1 x ( K 1 − x − α 12 y ) and y ′ = r 2 K 2 y ( K 2 − y − α 21 x ) . Calculation: The equations of Lotka-Voltera model is given as follows: x ′ = r 1 K 1 x ( K 1 − x − α 12 y ) … … ( 1 ) y ′ = r 2 K 2 y ( K 2 − y − α 21 x ) … … ( 2 ) Consider P ( x , y ) = x ′ and substitute in equation ( 1 ) . P ( x , y ) = r 1 K 1 x ( K 1 − x − α 12 y ) … … ( 3 ) Consider Q ( x , y ) = y ′ and substitute in equation ( 2 ) . Q ( x , y ) = r 2 K 2 y ( K 2 − y − α 21 x ) … … ( 4 ) Differentiate equation ( 3 ) partially with respect to x . ∂ P ∂ x = r 1 K 1 ( K 1 − 2 x − α 12 y ) Differentiate equation ( 3 ) partially with respect to y . ∂ P ∂ y = − r 1 K 1 α 12 x Differentiate equation ( 4 ) partially with respect to x . ∂ Q ∂ x = − r 2 K 2 α 21 y Differentiate the both sides of the equation ( 4 ) with respect to y . ∂ Q ∂ y = r 2 K 2 ( K 2 − 2 y − α 21 x ) The Jacobian matrix is given as follows: g ′ ( x , y ) = ( ∂ P ∂ x ∂ P ∂ y ∂ Q ∂ x ∂ Q ∂ y ) Substitute the values of ∂ P ∂ x , ∂ P ∂ y , ∂ Q ∂ x and ∂ Q ∂ y in above equation. g ′ ( x , y ) = ( r 1 K 1 ( K 1 − 2 x − α 12 y ) − r 1 K 1 α 12 x − r 2 K 2 α 21 y r 2 K 2 ( K 2 − 2 y − α 21 x ) ) … … ( 5 ) At the critical point ( 0 , 0 ) the Jacobian matrix of equation ( 5 ) is given as follows: g ′ ( 0 , 0 ) = ( r 1 0 0 r 2 ) The determinant of the above matrix is calculated as follows: Δ = | r 1 0 0 r 2 | = r 1 ( r 2 ) − 0 = r 1 r 2 Consider the trace point τ and τ is equal to the sum of the diagonal elements of the above matrix. τ = r 1 + r 2 Calculate the value of τ 2 − 4 Δ as follows: τ 2 − 4 Δ = ( r 1 + r 2 ) 2 − 4 ( r 1 r 2 ) = r 1 2 + r 2 2 + 2 r 1 r 2 − 4 r 1 r 2 = r 1 2 + r 2 2 − 2 r 1 r 2 = ( r 1 − r 2 ) 2 When r 1 ≠ r 2 the value of Δ = r 1 r 2 > 0 , τ = r 1 + r 2 > 0 , τ 2 − 4 Δ = ( r 1 + r 2 ) 2 − 4 ( r 1 r 2 ) > 0 . The condition for a critical point to be an unstable node is that the value of τ 2 − 4 Δ > 0 , τ > 0 and Δ > 0 . For the critical point ( 0 , 0 ) the above conditions are satisfied. Therefore, the critical point ( 0 , 0 ) is an unstable node. At the critical point ( K 1 , 0 ) the Jacobian matrix of equation ( 5 ) is as follows: g ′ ( k 1 , 0 ) = ( − r 1 − r 1 α 12 0 r 2 ( 1 − K 1 K 2 α 21 ) ) The determinant of the above matrix is calculated as follows: Δ = | − r 1 − r 1 α 12 0 r 2 ( 1 − K 1 K 2 α 21 ) | = − r 1 [ r 2 ( 1 − K 1 K 2 α 21 ) ] − 0 = − r 1 r 2 ( 1 − K 1 K 2 α 21 ) Consider the trace point τ and τ is equal to the sum of the diagonal elements of the matrix. τ = − r 1 + r 2 ( 1 − K 1 K 2 α 21 ) … … ( 6 ) Assume the new coefficient as follows: c = 1 − K 1 K 2 α 21 … … ( 7 ) Substitute 1 − K 1 K 2 α 21 = c in equation ( 6 )

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 10.4, Problem 14E

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To show: The points (0,0),(K1,0) and (0,K2) are unstable in Lotka-Voltera if X^=(x^,y^) is a stable node.

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Chapter 10.4, Problem 13E

Chapter 10.4, Problem 15E

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Differential Equations with Boundary-Value Problems (MindTap Course List)

Ch. 10.1 - In Problems 16 write the given nonlinear...Ch. 10.1 - Prob. 2E Ch. 10.1 - In Problems 16 write the given nonlinear...Ch. 10.1 - Prob. 4E Ch. 10.1 - In Problems 16 write the given nonlinear...Ch. 10.1 - In Problems 16 write the given nonlinear...Ch. 10.1 - Prob. 7E Ch. 10.1 - Prob. 8E Ch. 10.1 - Prob. 9E Ch. 10.1 - Prob. 10E

Ch. 10.1 - Prob. 11E Ch. 10.1 - Prob. 12E Ch. 10.1 - Prob. 13E Ch. 10.1 - Prob. 14E Ch. 10.1 - Prob. 15E Ch. 10.1 - In Problems 716 find all critical points of the...Ch. 10.1 - In Problems 2326 solve the given nonlinear plane...Ch. 10.1 - In Problems 2326 solve the given nonlinear plane...Ch. 10.1 - In Problems 2326 solve the given nonlinear plane...Ch. 10.1 - In Problems 2326 solve the given nonlinear plane...Ch. 10.1 - Prob. 27E Ch. 10.1 - Prob. 28E Ch. 10.1 - Prob. 29E Ch. 10.1 - Prob. 30E Ch. 10.2 - In Problems 916 classify the critical point (0, 0)...Ch. 10.2 - Prob. 10E Ch. 10.2 - Prob. 11E Ch. 10.2 - Prob. 12E Ch. 10.2 - In Problems 916 classify the critical point (0, 0)...Ch. 10.2 - In Problems 916 classify the critical point (0, 0)...Ch. 10.2 - In Problems 916 classify the critical point (0, 0)...Ch. 10.2 - Prob. 16E Ch. 10.2 - Prob. 17E Ch. 10.2 - Determine a condition on the real constant so...Ch. 10.2 - Prob. 19E Ch. 10.2 - Prob. 20E Ch. 10.2 - Prob. 21E Ch. 10.2 - Prob. 22E Ch. 10.2 - Prob. 23E Ch. 10.2 - In Problems 23-26 a nonhomogeneous linear system...Ch. 10.2 - Prob. 25E Ch. 10.2 - Prob. 26E Ch. 10.3 - Prob. 1E Ch. 10.3 - Prob. 2E Ch. 10.3 - Prob. 3E Ch. 10.3 - Prob. 4E Ch. 10.3 - Prob. 5E Ch. 10.3 - In Problems 310, without solving explicitly,...Ch. 10.3 - In Problems 310, without solving explicitly,...Ch. 10.3 - Prob. 8E Ch. 10.3 - Prob. 9E Ch. 10.3 - In Problems 310, without solving explicitly,...Ch. 10.3 - Prob. 11E Ch. 10.3 - In Problems 1120 classify (if possible) each...Ch. 10.3 - Prob. 13E Ch. 10.3 - Prob. 14E Ch. 10.3 - Prob. 15E Ch. 10.3 - Prob. 16E Ch. 10.3 - Prob. 17E Ch. 10.3 - Prob. 18E Ch. 10.3 - Prob. 19E Ch. 10.3 - Prob. 20E Ch. 10.3 - Prob. 21E Ch. 10.3 - Prob. 22E Ch. 10.3 - Prob. 23E Ch. 10.3 - Prob. 24E Ch. 10.3 - Prob. 25E Ch. 10.3 - Prob. 26E Ch. 10.3 - Prob. 27E Ch. 10.3 - Show that the dynamical system x = x + xy y = 1 y...Ch. 10.3 - Prob. 29E Ch. 10.3 - Prob. 30E Ch. 10.3 - Prob. 31E Ch. 10.3 - Prob. 32E Ch. 10.3 - Prob. 33E Ch. 10.3 - Prob. 34E Ch. 10.3 - Prob. 35E Ch. 10.3 - Prob. 36E Ch. 10.3 - When a nonlinear capacitor is present in an...Ch. 10.3 - Prob. 38E Ch. 10.3 - Prob. 39E Ch. 10.4 - Prob. 1E Ch. 10.4 - Prob. 2E Ch. 10.4 - Prob. 3E Ch. 10.4 - Prob. 4E Ch. 10.4 - Prob. 5E Ch. 10.4 - Prob. 6E Ch. 10.4 - Prob. 7E Ch. 10.4 - Prob. 8E Ch. 10.4 - Prob. 9E Ch. 10.4 - Competition Models A competitive interaction is...Ch. 10.4 - Prob. 12E Ch. 10.4 - Prob. 13E Ch. 10.4 - Prob. 14E Ch. 10.4 - Prob. 15E Ch. 10.4 - Additional Mathematical Models Damped Pendulum If...Ch. 10.4 - Prob. 17E Ch. 10.4 - Prob. 18E Ch. 10.4 - Prob. 19E Ch. 10.4 - Prob. 20E Ch. 10.4 - Prob. 21E Ch. 10.4 - Prob. 22E Ch. 10 - Prob. 1RE Ch. 10 - Prob. 2RE Ch. 10 - Prob. 3RE Ch. 10 - Prob. 4RE Ch. 10 - Prob. 5RE Ch. 10 - Prob. 6RE Ch. 10 - Prob. 7RE Ch. 10 - Prob. 8RE Ch. 10 - Prob. 9RE Ch. 10 - Prob. 10RE Ch. 10 - Prob. 11RE Ch. 10 - Discuss the geometric nature of the solutions to...Ch. 10 - Classify the critical point (0, 0) of the given...Ch. 10 - Prob. 14RE Ch. 10 - Prob. 15RE Ch. 10 - Prob. 16RE Ch. 10 - Prob. 17RE Ch. 10 - Prob. 18RE Ch. 10 - Prob. 19RE Ch. 10 - Prob. 20RE

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The points ( 0 , 0 ) , ( K 1 , 0 ) and ( 0 , K 2 ) are unstable in Lotka-Voltera if X ^ = ( x ^ , y ^ ) is a stable node.

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To show: The points (0,0),(K1,0) and (0,K2) are unstable in Lotka-Voltera if X^=(x^,y^) is a stable node.

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Chapter 10 Solutions