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 ( x ^ , y ^ ) the Jacobian matrix of equation ( 5 ) is as follows: 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 ^ ) ) The determinant of the above matrix is calculated as follows: Δ = | 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 ^ ) | = ( r 1 K 1 ( K 1 − 2 x ^ − α 12 y ^ ) ) ( r 2 K 2 ( K 2 − 2 y ^ − α 21 x ^ ) ) − r 1 r 2 K 1 K 2 ( α 21 y ^ ) ( α 12 x ^ ) It is given that K 2 α 21 < K 1 . Solve the above inequality as follows: K 2 α 21 < K 1 K 1 K 2 α 21 > 1 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 )

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Chapter 11.4, Problem 15E

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To show: The point X^=(x^,y^) is a saddle point in Lotka-Voltera model if K1α12<K2 and K2α21<K1.

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Chapter 11.4, Problem 14E

Chapter 11.4, Problem 16E

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Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E Ch. 11.1 - Prob. 6E Ch. 11.1 - Prob. 7E Ch. 11.1 - Prob. 8E Ch. 11.1 - Prob. 9E Ch. 11.1 - Prob. 10E

Ch. 11.1 - Prob. 11E Ch. 11.1 - Prob. 12E Ch. 11.1 - Prob. 13E Ch. 11.1 - Prob. 14E Ch. 11.1 - Prob. 15E Ch. 11.1 - Prob. 16E Ch. 11.1 - Prob. 23E Ch. 11.1 - Prob. 24E Ch. 11.1 - Prob. 25E Ch. 11.1 - Prob. 26E Ch. 11.1 - Prob. 27E Ch. 11.1 - Prob. 28E Ch. 11.1 - Prob. 29E Ch. 11.1 - Prob. 30E Ch. 11.2 - Prob. 17E Ch. 11.2 - Prob. 18E Ch. 11.2 - Prob. 19E Ch. 11.2 - Prob. 20E Ch. 11.2 - Prob. 21E Ch. 11.2 - Prob. 22E Ch. 11.2 - Prob. 23E Ch. 11.2 - Prob. 24E Ch. 11.2 - Prob. 25E Ch. 11.2 - Prob. 26E Ch. 11.3 - Prob. 1E Ch. 11.3 - Prob. 2E Ch. 11.3 - Prob. 3E Ch. 11.3 - Prob. 4E Ch. 11.3 - Prob. 5E Ch. 11.3 - Prob. 6E Ch. 11.3 - Prob. 7E Ch. 11.3 - Prob. 8E Ch. 11.3 - Prob. 9E Ch. 11.3 - Prob. 10E Ch. 11.3 - Prob. 11E Ch. 11.3 - Prob. 12E Ch. 11.3 - Prob. 13E Ch. 11.3 - Prob. 14E Ch. 11.3 - Prob. 15E Ch. 11.3 - Prob. 16E Ch. 11.3 - Prob. 17E Ch. 11.3 - Prob. 18E Ch. 11.3 - Prob. 19E Ch. 11.3 - Prob. 20E Ch. 11.3 - Prob. 21E Ch. 11.3 - Prob. 22E Ch. 11.3 - Prob. 23E Ch. 11.3 - Prob. 24E Ch. 11.3 - Prob. 25E Ch. 11.3 - Prob. 26E Ch. 11.3 - Prob. 27E Ch. 11.3 - Prob. 28E Ch. 11.3 - Prob. 29E Ch. 11.3 - Prob. 30E Ch. 11.3 - Prob. 31E Ch. 11.3 - Prob. 32E Ch. 11.3 - Prob. 33E Ch. 11.3 - Prob. 34E Ch. 11.3 - Prob. 35E Ch. 11.3 - Prob. 36E Ch. 11.3 - Prob. 37E Ch. 11.3 - Prob. 38E Ch. 11.3 - Prob. 39E Ch. 11.4 - Prob. 1E Ch. 11.4 - Prob. 2E Ch. 11.4 - Prob. 3E Ch. 11.4 - Prob. 4E Ch. 11.4 - Prob. 5E Ch. 11.4 - Prob. 6E Ch. 11.4 - Prob. 7E Ch. 11.4 - Prob. 8E Ch. 11.4 - Prob. 9E Ch. 11.4 - Prob. 11E Ch. 11.4 - Prob. 12E Ch. 11.4 - Prob. 13E Ch. 11.4 - Prob. 14E Ch. 11.4 - Prob. 15E Ch. 11.4 - Prob. 16E Ch. 11.4 - Prob. 17E Ch. 11.4 - Prob. 18E Ch. 11.4 - Prob. 19E Ch. 11.4 - Prob. 20E Ch. 11.4 - Prob. 21E Ch. 11.4 - Prob. 22E Ch. 11 - Prob. 1CR Ch. 11 - Prob. 2CR Ch. 11 - Prob. 3CR Ch. 11 - Prob. 4CR Ch. 11 - Prob. 5CR Ch. 11 - Prob. 6CR Ch. 11 - Prob. 7CR Ch. 11 - Prob. 8CR Ch. 11 - Prob. 11CR Ch. 11 - Prob. 12CR Ch. 11 - Prob. 13CR Ch. 11 - Prob. 15CR Ch. 11 - Prob. 16CR Ch. 11 - Prob. 17CR

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The point X ^ = ( x ^ , y ^ ) is a saddle point in Lotka-Voltera model if K 1 α 12 < K 2 and K 2 α 21 < K 1 .

Solution Summary: The author explains how the point stackrelX=left is a saddle point in Lotka-Voltera model.

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To show: The point X^=(x^,y^) is a saddle point in Lotka-Voltera model if K1α12<K2 and K2α21<K1.

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Chapter 11 Solutions

The point X ^ = ( x ^ , y ^ ) is a saddle point in Lotka-Voltera model if K 1 α 12 &lt; K 2 and K 2 α 21 &lt; K 1 .

Solution Summary: The author explains how the point stackrelX=left is a saddle point in Lotka-Voltera model.

Concept explainers

Videos

To show: The point X^=(x^,y^) is a saddle point in Lotka-Voltera model if K1α12<K2 and K2α21<K1.

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Chapter 11 Solutions

The point X ^ = ( x ^ , y ^ ) is a saddle point in Lotka-Voltera model if K 1 α 12 < K 2 and K 2 α 21 < K 1 .