5. Consider the nonlinear two dimensional Lotka-Volterra (predator-prey) system z'(t) = z(t)[2-2(t)-2y(t)l, y'(t) = y(t)12-y(t)--2(t)] (a)...
5. Consider the nonlinear two dimensional Lotka-Volterra (predator-prey) system z'(t) = z(t)[2-2(t)-2y(t)l, y'(t) = y(t)12-y(t)--2(t)] (a) Find all critical points of this system, and at each determine whether or not the system is locally stable or unstable. (b) We proved in class, using the Bendixson-Dulac theorem, that this system has no periodic solution with trajectory in the first quadrant of the plane. Assuming this, use the Poincare-Bendixson theorem to prove that all trajectories (z(t),y(t)) of the system (2) with initial...
5. Consider the system: dz 4y 1 dy (a) Are these species predator-prey or competing? b) What type of growth does species z exhibit in absence of species y? What type of growth does species y exhibit in absence of species r? (c) Find all critical (equilibrium) points d) Using the Jacobian matrix, classify (if possible) each critical (equilibrium) point as a stable node, a stable spiral point, an unstable node, an unstable spiral point, or a saddle point. (e)...
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...
2. Consider the systems: dz =2x(1-5)-yr dt dy dt a) Which system corresponds to a predator-prey one? Which is the predator and which the prey? Briefly justify your answer. b) Find the equilibrium solutions only for the predator-prey one. c) Sketch its phase plane showing the equilibrium solutions and the behavior on the r- and y-axis (only for the predator-prey one) d) Describe briefly what kind of situation could the other system represent. 2. Consider the systems: dz =2x(1-5)-yr dt...
2. Consider the nonlinear plane autonomous system 3 2 satisfying the initial condition (r(0), y(0)) = (4,0). (a) Change to polar coordinates and find the solution r(t) and (t) of the system (b) As t goes to infinity, (x(t). y(t)) will follow the circle trajectory. Find the radius and period of the circle trajectory. (limit behavior of the solution (a)) 2. Consider the nonlinear plane autonomous system 3 2 satisfying the initial condition (r(0), y(0)) = (4,0). (a) Change to...
3. Consider an autonomous system z'=cx +2y where c is a real constant. 2 (a) Calculate the trace T and the determinant A of the coefficient matrix -2 1 (b) For each following cases of c, classify the stability (stable or unstable) and the type (center, node, saddle, or spiral) of the critical point (0,0). Note that if a critical point is a center, it is stable. (1) c--5 (2) c--3 (3) c1 (4) c 6 3. Consider an autonomous...
Question 410 marks Consider the nonlinear system ェ=(1-y)2(4-12), ỳ=(1-z)y(y2-4) (0<x<2, o<y<2), which has a single fixed point at (1,1) (a) Show that the following expression for K(x, y) is a constant of motion for this system: K(x, y)- 2 ln(ry) + Inl( 2)(y- 2)]-3In(2)(y+2)]. (b) Use the constant of motion to show that the fixed point is a centre of the nonlinear system.
Problem 2 Consider the system of equations 2 1. Show that the z and t are determined as a function of x and y near the point (0, 1,-1). Can we apply the Implicit Function theorem? 2. Compute the partial derivatives of z and t with respect to z, y at (0,1) 3. Without solving the system, what is approximate value of 2(0.001,1.002) (Hint: Use the first order Taylor approximation about the point (1,0) to find the approximation) 4. Compute...
For the Hamiltonian syste m we did in class: 2. 3 Ic (1) Show that it's a Hamiltonian system with a Hamiltonian function (2) Show that for each c > 0, {(z,y) є R2 . H(z,y) c} is a bounded invariant set of the dynamical system (in fact, it's also closed) (3) Find all the equilibria of this system. Show that H-() is made up of one equilibium point and two homoclinic orbits attached to it. (4) Sketch the invariant...
3. (10pts) Consider the follow population curves that are solutions for the worm-robins from a predictor-prey system 1.8 1.6 1.4 1.2 Rabins 05 1 15 2 3 0.8 0.6 0.4 0.2 2 2.4S Problem 3 10 15 25 Time t (a) There are two trajectories drawn on the left. Which one seems to correspond to the solutions? Indicate the initial point and its moving direction. Explain. the rate of population of either warm or robins is zero. system to complete...