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(a) Which system of differential equations describes a situation where the two species compete and which system describes pair of cooperative species? Explain your reasoning, making sure to note what the dif- ferent terms in each DE tell you about the population context. (A) (B) dx = -5x+2xy dt dx = 3x(1-)-10* -ху dt dy = -4y+3xy dy 1 - dt dt 10

(a) Which system of differential equations describes a situation where the two species compete and which system describes pair of cooperative species? Explain your reasoning, making sure to note what the dif- ferent terms in each DE tell you about the population context. (A) (B) dx = -5x+2xy dt dx = 3x(1-)-10* -ху dt dy = -4y+3xy dy 1 - dt dt 10

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 65EQ
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5. In previous problems dealing with two species, one of the animals was the predator and the other was the prey. In this problem we recall the Bees & Flowers example and revisit systems of differential equations designed to model two species that are either competitive (that is both species are harmed by interaction) or cooperative (that is both species benefit from interaction). (a) Which system of differential equations describes a situation where the two species compete and which system describes pair of cooperative species? Explain your reasoning, making sure to note what the dif- ferent terms in each DE tell you about the population context. (A) (B) dx = -5x+2xy dt dx = 3x|1 dt 1 ху 10 - 3 :-4y+3xy dt dy dy = 2y(1 y y[l - 10)-5*y 1 dt (b) For system (A), plot all nullclines and use this plot to determine all equilibrium solutions. Verify your equilibrium solutions algebraically. (c) Use your results from 5b to sketch in the long-term behavior of solutions for system (A) with initial con- ditions anywhere in the first quadrant of the phase plane. For example, describe the long-term behavior of solutions if the initial condition is in such-and-such region of the first quadrant. Provide a sketch of your analysis in the xy-plane and justify any conjectures that you have about the long-term outcome for the two populations based on the initial conditions.
Transcribed Image Text:5. In previous problems dealing with two species, one of the animals was the predator and the other was the prey. In this problem we recall the Bees & Flowers example and revisit systems of differential equations designed to model two species that are either competitive (that is both species are harmed by interaction) or cooperative (that is both species benefit from interaction). (a) Which system of differential equations describes a situation where the two species compete and which system describes pair of cooperative species? Explain your reasoning, making sure to note what the dif- ferent terms in each DE tell you about the population context. (A) (B) dx = -5x+2xy dt dx = 3x|1 dt 1 ху 10 - 3 :-4y+3xy dt dy dy = 2y(1 y y[l - 10)-5*y 1 dt (b) For system (A), plot all nullclines and use this plot to determine all equilibrium solutions. Verify your equilibrium solutions algebraically. (c) Use your results from 5b to sketch in the long-term behavior of solutions for system (A) with initial con- ditions anywhere in the first quadrant of the phase plane. For example, describe the long-term behavior of solutions if the initial condition is in such-and-such region of the first quadrant. Provide a sketch of your analysis in the xy-plane and justify any conjectures that you have about the long-term outcome for the two populations based on the initial conditions.
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