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a) Interaction between two interaction between the species is described by the following system. (A) fixed points. Assume 0 and y 2 0. (only 1st qundrant) species. Let r and y represent the popul...
The following system can be interpreted as a competition system describing the interaction of two species...
The following system can be interpreted as a competition system describing the interaction of two species with populations x(t) and y(t) x' 40x – 22 – ry y' = 30y - y2 – 0.5xy This system has four critical points (0,0), (0, 30), (40,0), and (20, 20). (a) At critical point (20, 20), find the linearization of the system and its eigenvalues. Deter- mine the type and stability of the critical point (20, 20). Base on your work in part...
1. The populations of two competing species x(t) and y(t) are governed by the non-linear system...
1. The populations of two competing species x(t) and y(t) are governed by the non-linear system of differential equations dx dt 10x – x2 – 2xy, dy dt 5Y – 3y2 + xy. (a) Determine all of the critical points for the population model. (b) Determine the linearised system for each critical point in part (a) and discuss whether it can be used to approximate the behaviour of the non-linear system. (c) For the critical point at the origin: (i)...
1. Let x and y represent two animal populations which satisfy 2(9- 2 - 3y) y(-6...
1. Let x and y represent two animal populations which satisfy 2(9- 2 - 3y) y(-6 + 2x) (a) (5 points) What is the relationship between 2 and y? How does a grow in the absence of y? How does y grow in the absence of x? (b) (5 points) Sketch the nullclines and direction arrows of the system. (c) (4 points) Find the eigenvalues of the interior critical point. (d) (7 points) Sketch the general solution. Be detailed. (e)...
1. Let x and y represent two animal populations which satisfy (9-1-3y) y(-6 + 2x) (a)...
1. Let x and y represent two animal populations which satisfy (9-1-3y) y(-6 + 2x) (a) (5 points) What is the relationship between x and y? How does a grow in the absence of y? How does y grow in the absence of x? (b) (5 points) Sketch the nullclines and direction arrows of the system. (c) (4 points) Find the eigenvalues of the interior critical point. (d) (7 points) Sketch the general solution. Be detailed. (e) (4 points) Sketch...
Requesting the solution to the problem below from Ordinary Differential Equations and Dynamical Systems, Gerald Teschl....
Requesting the solution to the problem below from Ordinary
Differential Equations and Dynamical Systems, Gerald Teschl.
Thanks.
Additional materials:
Problem 7.2 (Volterra principle). Show that for any orbit of the Volterra- Lotka system (7.3), the time average over one period 1 1 T | (0)2 = 1, T | g(t)dt =1 is independent of the orbit. (Hint: Integrate log(r(t)) over one period.) 7.1. Examples from ecology In this section we want to consider a model from ecology. It describes two...
I. Let 2r2y"-ry' + (r' + 1) y = 0. ) Verify that a 0 is a regular singular point. (2 points) ) Find two linearly independent power series solutions about a0. Provide at least3 aon-ze...
I. Let 2r2y"-ry' + (r' + 1) y = 0. ) Verify that a 0 is a regular singular point. (2 points) ) Find two linearly independent power series solutions about a0. Provide at least3 aon-zero terms of each solution. (8 points) 0 0 is reowlar 터 -2 kti I'm Stuck-HEKG-HER- ME FINISH
I. Let 2r2y"-ry' + (r' + 1) y = 0. ) Verify that a 0 is a regular singular point. (2 points) ) Find two linearly independent...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra be the spacing between the inner and outer conductors. (a) Let the radii of the two conductors be only slightly different, so that d << ra. Show that the result derived in Example 24.4 (Section 24.1) for the capacitance of a cylindrical capacitor then reduces to Eq. (24.2), the equation for the capacitance of a parallel-plate capacitor, with A being the surface area of...
The following system can be interpreted as a competition system describing the interaction of two species with populations x(t) and y(t) x' 40x – 22 – ry y' = 30y - y2 – 0.5xy This system has four critical points (0,0), (0, 30), (40,0), and (20, 20). (a) At critical point (20, 20), find the linearization of the system and its eigenvalues. Deter- mine the type and stability of the critical point (20, 20). Base on your work in part...
1. The populations of two competing species x(t) and y(t) are governed by the non-linear system of differential equations dx dt 10x – x2 – 2xy, dy dt 5Y – 3y2 + xy. (a) Determine all of the critical points for the population model. (b) Determine the linearised system for each critical point in part (a) and discuss whether it can be used to approximate the behaviour of the non-linear system. (c) For the critical point at the origin: (i)...
1. Let x and y represent two animal populations which satisfy 2(9- 2 - 3y) y(-6 + 2x) (a) (5 points) What is the relationship between 2 and y? How does a grow in the absence of y? How does y grow in the absence of x? (b) (5 points) Sketch the nullclines and direction arrows of the system. (c) (4 points) Find the eigenvalues of the interior critical point. (d) (7 points) Sketch the general solution. Be detailed. (e)...
1. Let x and y represent two animal populations which satisfy (9-1-3y) y(-6 + 2x) (a) (5 points) What is the relationship between x and y? How does a grow in the absence of y? How does y grow in the absence of x? (b) (5 points) Sketch the nullclines and direction arrows of the system. (c) (4 points) Find the eigenvalues of the interior critical point. (d) (7 points) Sketch the general solution. Be detailed. (e) (4 points) Sketch...
Requesting the solution to the problem below from Ordinary
Differential Equations and Dynamical Systems, Gerald Teschl.
Thanks.
Additional materials:
Problem 7.2 (Volterra principle). Show that for any orbit of the Volterra- Lotka system (7.3), the time average over one period 1 1 T | (0)2 = 1, T | g(t)dt =1 is independent of the orbit. (Hint: Integrate log(r(t)) over one period.) 7.1. Examples from ecology In this section we want to consider a model from ecology. It describes two...
I. Let 2r2y"-ry' + (r' + 1) y = 0. ) Verify that a 0 is a regular singular point. (2 points) ) Find two linearly independent power series solutions about a0. Provide at least3 aon-zero terms of each solution. (8 points) 0 0 is reowlar 터 -2 kti I'm Stuck-HEKG-HER- ME FINISH
I. Let 2r2y"-ry' + (r' + 1) y = 0. ) Verify that a 0 is a regular singular point. (2 points) ) Find two linearly independent...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra be the spacing between the inner and outer conductors. (a) Let the radii of the two conductors be only slightly different, so that d << ra. Show that the result derived in Example 24.4 (Section 24.1) for the capacitance of a cylindrical capacitor then reduces to Eq. (24.2), the equation for the capacitance of a parallel-plate capacitor, with A being the surface area of...
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