Consider the following system of equations:
Consider the following system of equations:
(a) Suppose a = 1. Sketch the nullclines of these equations.
(b) Determine what happens if a is increased to large values (a >> 1). How does this influence the number of steady states?
(c) Determine what happens when a is decreased (0 < a < 1). How does this influence the number of steady states?
(d) Find a condition necessary for the intersection of the nullclines to occur exactly at the minimum of the x-nullcline.
Homework Answers
3. Consider the system of differential equations = z+9-1 (a) Sketch the r-nullcline, where soluti...
just (A) (B) (C)
3. Consider the system of differential equations = z+9-1 (a) Sketch the r-nullcline, where solutions must travel vertically. Identify the regions (b) On a separate set of axes, sketch the y-nullcline, where solutions must travel horizon in the plane where solutions will move toward the right, and where solutions move toward the right tally. Identify the regions in the plane where solutions will move upward, and where solutions move downward. (c) On a third set of...
Consider the system of equations dxdt=x(3−x−4y) dydt=y(1−3x), taking (x,y)>0. (1 point) Consider the system of equations...
Consider the system of equations dxdt=x(3−x−4y) dydt=y(1−3x),
taking (x,y)>0.
(1 point) Consider the system of equations de = 2(3 – 2 – 49) = y(1 - 33), taking (2,y) > 0. (a) Write an equation for the (non-zero) vertical (-)nullcline of this system: (Enter your equation, e.g., y=x.) And for the (non-zero) horizontal (y-)nullcline: (Enter your equation, e.g. y=x.) (Note that there are also nullclines lying along the axes.) (b) What are the equilibrium points for the system? Equilibria =...
Consider a system of differential equations describing the progress of a disease in a population,...
Consider a system of differential equations describing the progress of a disease in a population, given byF, ) for a vector-valued function F. In our particular case, this IS. where z(t) is the number of susceptible individuals at time t and y(t) is the number of infected individuals at time t. The number of individuals is counted in units of 1,000 individuals a) Find the nullclines (simplest form) of this system of differential equations. The x-nullcline is y 2/3 The...
(3) - F(2,4) to Consider a system of differential equations describing the progress of a disease...
(3) - F(2,4) to Consider a system of differential equations describing the progress of a disease in a population, given by for a vector-valued function F. In our particular case, this is: t' = 3 – 3zy - 12 y' – 3ay – 2y where I (t) is the number of susceptible individuals at time t and y(t) is the number of infected individuals at time t. The number of individuals is counted in units of 1,000 individuals. and =...
Consider the system of linear differential equations z,(t)-17/11 z(t) + 9/11 y(t) y,(t)-18/11 z(t...
Consider the system of linear differential equations z,(t)-17/11 z(t) + 9/11 y(t) y,(t)-18/11 z(t) + 38/11 y(t) a) Find the equation of the x-nullcline. Write your answer as an equation in z and y Answer b) Find the equation of the y-nullcline. Write your answer as an equation of z and y Answer. c) The nullclines divide the plane into four regions as illustrated below. 忽聡 2 -2 2 -2 For each of the regions, determine the direction of the...
Exercise 3, Section 9.5. Modified Lotka- Volterra Predator-Prey model Consider two species (rabbits and foxes) such that the population R (rabbits) and F (foxrs) obey the system of equations dR d...
Exercise 3, Section 9.5. Modified Lotka- Volterra Predator-Prey model Consider two species (rabbits and foxes) such that the population R (rabbits) and F (foxrs) obey the system of equations dR dt dF dt R2-R)-12RF . What happens to the population of rabbits if the number of foxes is arro? (Use the phase line analysis from Chapter 2) What happens to the population of foxes if the number of rabbits is zero? 3. Using the method of nullclines, draw an approximate...
Assignment 8 Remaining Time: 131:53:22 Question 1 Consider a system of differential equations describing the progress...
Assignment 8 Remaining Time: 131:53:22 Question 1 Consider a system of differential equations describing the progress of a disease in a population, given by F(x, y) for 1 point How Did I Do? a vector-valued function F. In our particular case, this is: d' = 3 – 3xy - 12 y' = 3xy – 34 where x(t) is the number of susceptible individuals at time t and y(t) is the number of infected individuals at time t. The number of...
2. (28 marks) This questions is about the following system of equations x = (2-x)(y-1) (a) Find all equilibrium solutio...
2. (28 marks) This questions is about the following system of equations x = (2-x)(y-1) (a) Find all equilibrium solutions and determine their type (e.g., spiral source, saddle) Hint: you should find three equilibria. b) For each of the equilibria you found in part (a), draw a phase portrait showing the behaviour of solutions near that equilibrium. -2 (c) Find the nullclines for the system and sketch them on the answer sheet provided. Show the direction of the vector field...
6. Consider the following system 3 a) Take the case a = . Use a trapping region as in Poincare Be...
6. Consider the following system 3 a) Take the case a = . Use a trapping region as in Poincare Bendixson's Theorem to show that there is at least one periodic orbit. [2 marks] b) How is the case a-1 different? Do you think there is a periodic orbit in this case? What happens to the periodic orbit as a is increased from to 1? You may use a computer to investigate this. When does the qualitative change in the...
please explain every step. thanks Consider the following system of linear equations ri (a) For what...
please explain every step. thanks
Consider the following system of linear equations ri (a) For what values of r and s is this system of linear equations inconsistent? (b) For what values of and s does this system of linear equations have infinitely many solutions? (ey For what values of and s does this system of linear equations have a unique solution?
just (A) (B) (C)
3. Consider the system of differential equations = z+9-1 (a) Sketch the r-nullcline, where solutions must travel vertically. Identify the regions (b) On a separate set of axes, sketch the y-nullcline, where solutions must travel horizon in the plane where solutions will move toward the right, and where solutions move toward the right tally. Identify the regions in the plane where solutions will move upward, and where solutions move downward. (c) On a third set of...
Consider the system of equations dxdt=x(3−x−4y) dydt=y(1−3x),
taking (x,y)>0.
(1 point) Consider the system of equations de = 2(3 – 2 – 49) = y(1 - 33), taking (2,y) > 0. (a) Write an equation for the (non-zero) vertical (-)nullcline of this system: (Enter your equation, e.g., y=x.) And for the (non-zero) horizontal (y-)nullcline: (Enter your equation, e.g. y=x.) (Note that there are also nullclines lying along the axes.) (b) What are the equilibrium points for the system? Equilibria =...
Consider a system of differential equations describing the progress of a disease in a population, given byF, ) for a vector-valued function F. In our particular case, this IS. where z(t) is the number of susceptible individuals at time t and y(t) is the number of infected individuals at time t. The number of individuals is counted in units of 1,000 individuals a) Find the nullclines (simplest form) of this system of differential equations. The x-nullcline is y 2/3 The...
(3) - F(2,4) to Consider a system of differential equations describing the progress of a disease in a population, given by for a vector-valued function F. In our particular case, this is: t' = 3 – 3zy - 12 y' – 3ay – 2y where I (t) is the number of susceptible individuals at time t and y(t) is the number of infected individuals at time t. The number of individuals is counted in units of 1,000 individuals. and =...
Consider the system of linear differential equations z,(t)-17/11 z(t) + 9/11 y(t) y,(t)-18/11 z(t) + 38/11 y(t) a) Find the equation of the x-nullcline. Write your answer as an equation in z and y Answer b) Find the equation of the y-nullcline. Write your answer as an equation of z and y Answer. c) The nullclines divide the plane into four regions as illustrated below. 忽聡 2 -2 2 -2 For each of the regions, determine the direction of the...
Exercise 3, Section 9.5. Modified Lotka- Volterra Predator-Prey model Consider two species (rabbits and foxes) such that the population R (rabbits) and F (foxrs) obey the system of equations dR dt dF dt R2-R)-12RF . What happens to the population of rabbits if the number of foxes is arro? (Use the phase line analysis from Chapter 2) What happens to the population of foxes if the number of rabbits is zero? 3. Using the method of nullclines, draw an approximate...
Assignment 8 Remaining Time: 131:53:22 Question 1 Consider a system of differential equations describing the progress of a disease in a population, given by F(x, y) for 1 point How Did I Do? a vector-valued function F. In our particular case, this is: d' = 3 – 3xy - 12 y' = 3xy – 34 where x(t) is the number of susceptible individuals at time t and y(t) is the number of infected individuals at time t. The number of...
2. (28 marks) This questions is about the following system of equations x = (2-x)(y-1) (a) Find all equilibrium solutions and determine their type (e.g., spiral source, saddle) Hint: you should find three equilibria. b) For each of the equilibria you found in part (a), draw a phase portrait showing the behaviour of solutions near that equilibrium. -2 (c) Find the nullclines for the system and sketch them on the answer sheet provided. Show the direction of the vector field...
6. Consider the following system 3 a) Take the case a = . Use a trapping region as in Poincare Bendixson's Theorem to show that there is at least one periodic orbit. [2 marks] b) How is the case a-1 different? Do you think there is a periodic orbit in this case? What happens to the periodic orbit as a is increased from to 1? You may use a computer to investigate this. When does the qualitative change in the...
please explain every step. thanks
Consider the following system of linear equations ri (a) For what values of r and s is this system of linear equations inconsistent? (b) For what values of and s does this system of linear equations have infinitely many solutions? (ey For what values of and s does this system of linear equations have a unique solution?
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