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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 different terms in each DE tell you about the population context. (A) (B) dx 3 -5х+ 2ху dt dx 1 = 3x(1-)- = 2y(1-) dt xy 10 dy dy :-4y+3xy dt 1 ху dt

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 different terms in each DE tell you about the population context. (A) (B) dx 3 -5х+ 2ху dt dx 1 = 3x(1-)- = 2y(1-) dt xy 10 dy dy :-4y+3xy dt 1 ху dt

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please help with the attached problem. It references examples from class, but these are not necessary to solve the problem. Everything necessary is included in the attachment, thanks!

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 different terms in each DE tell you about the population context. (A) (В) -3x(1-)- 10* - 2y (1- %)- dx 3x 1- dt dx -5х+2ху ху dt dy :-4y+3xy dt dy 1 dt 10. (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 with initial conditions where in the first quadrant of the phase plane. For example, describe the long-term behavior of solu- tions if the initial condition is in such-and-such region of the first quadrant. Provide a sketch of your analysis in the x-y plane and justify any conjectures that you have about the long-term outcome for the two populations based on the initial conditions. any-
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 different terms in each DE tell you about the population context. (A) (В) -3x(1-)- 10* - 2y (1- %)- dx 3x 1- dt dx -5х+2ху ху dt dy :-4y+3xy dt dy 1 dt 10. (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 with initial conditions where in the first quadrant of the phase plane. For example, describe the long-term behavior of solu- tions if the initial condition is in such-and-such region of the first quadrant. Provide a sketch of your analysis in the x-y plane and justify any conjectures that you have about the long-term outcome for the two populations based on the initial conditions. any-
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