In previous problems dealing with two species, one of the animals was the predat
ID: 3123120 • Letter: I
Question
In previous problems dealing with two species, one of the animals was the predator and the other was the prey. In this problem we study systems of rate of change equations designed to inform us about the future populations for 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 rate of change equations describes a situation where the two species compete and which system describes cooperative species? Explain your reasoning. (A) dx/dt = -5x + 2xy dy/dt = -4y + 3 xy (B) dx/dt = 3x(1- x/3) - 1/10 xy dy/dt = 2 y(1- y/10) - 1/5 xy b) For system (A), plot all null clines and use this plot to determine all equilibrium solutions. Verify your equilibrium solutions algebraically. c) Use your results from part b) to sketch in the long-term behavior of solutions with initial conditions 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 x-y plane and write a paragraph summarizing your conclusions and any conjectures that you have about the long-term outcome for the two populations depending on the initial conditions.Explanation / Answer
The x-nullcline is a set of points in the phase plane so that dx/dt=0
Geometrically, these are the points where the vectors are either straight up or straight down. Algebraically, we find the x-nullcline by solving f(x,y)=0
The y-nullcline is a set of points in the phase plane so that dy/dt=0. Geometrically, these are the points where the vectors are horizontal, going either to the left or to the right. Algebraically, we find the y-nullcline by solving g(x,y)=0.