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(Problem 3) Consider the following three-species ecosystems: dF F(a – cS) dt ds S(-k + \F...
(Problem 1) Given the Lotka-Voltera model: -) dF F(a-cs) dt ds S(-k + XF). dt (1) Linearize the model about the equilibrium point (F, S) = (0,0) using Taylor series.
Consider the second version of the Lotka-Volterra model: dF F(a - 6F - cS) dt ds = S(-k + XF). dt (1) Explain the model; i.e. what are the terms in the equation signify? How is this model different from equations (1)? (2) Find the equilibrium point(s). (3) Linearize (2) about the equilibrium point(s). (4) Classify if the equilibrium points are stable or unstable. (5) Pick some values for a, b,c,k, X. Plot the solutions of the model and the...
Do part (1) and (3)
• (Problem 1) Given the Lotka-Voltera model: = F(a-cs) dF dt ds dt = S(-k + \F). (1) Linearize the model about the equilibrium point (F, S) = (0,0) using Taylor series. Hint: see lecture notes. (2) Use the matlab code, lotkavolterra2.m to plot the solutions and the phase portrait. Choose some values for a, b, c, k, 1. (3) From the previous problem, increase the value of k. What does increasing the value of...
= • (Problem 4) Consider the following alternate predator-prey (Leslie) model: dF F(a – bF – cS) dt dS S S(-k +13). dt F Note that the prey model is the same as in the Lotka-Volterra model. However, the predators change in a different manner. Show that if there are many predators for each prey, then the predators cannot cope with the excessive competition for their prey and die off. On the other hand if there are many prey for...
• (Problem 2) Consider the second version of the Lotka-Volterra model: F(a – bF – cS) dF dt ds dt S(-k + \F). (1) Explain the model; i.e. what are the terms in the equation signify? How is this model different from equations (1)? (2) Find the equilibrium point(s). (3) Linearize (2) about the equilibrium point(s). (4) Classify if the equilibrium points are stable or unstable. (5) Pick some values for a, b, c, k, 1. Plot the solutions of...
Problem 2. Consider the predator-prey system: dF dt The graph below shows a solution curve in the RF-phase space. Describe the fate of the prey (R) and the predator (F) population based on that image
Problem 2. Consider the predator-prey system: dF dt The graph below shows a solution curve in the RF-phase space. Describe the fate of the prey (R) and the predator (F) population based on that image
dn (a) Show that L[i" f(t)] = (-1)" (t) for any positive integer n 2 1 dsn a d K(s, t)f(t) dt / ) est = tne-8t and assume that K(s, t)f(t) dt. Hint: (-1)" as ds (b) Use the above formula to compute L[t? cost].
dn (a) Show that L[i" f(t)] = (-1)" (t) for any positive integer n 2 1 dsn a d K(s, t)f(t) dt / ) est = tne-8t and assume that K(s, t)f(t) dt. Hint:...
(25 Problem 3: Steady State Error. Consider Iaput Rs) Output Cs) KGG) with the following transfer function. 5(s+1)-, and H(s)=1. G(S)- (+12s+5) (a) Calculate the error constants (K,, K,, Kg) and the steady state errors for three (20pts) (b) Find the value of K so that the results are valid. Hint: When is the systepí stable? (5 pts) basic types of unit (step, ramp, parabolic) inputs
Consider the unity feedback system is given below Ris) Cs) G(8) with transfer function: -K(s +1) G(S) 52 + 25 + 2 Considering that can have only positive values, the system is unstable when the value of K is ........
Consider the following static (closed-economy) version of the Classical model: Y = F (K, L) C = A + a(Y − T ), with A > 0 and 0 < a < 1, I = B − br, with B, b > 0, where A and B represent respectively the autonomous components of consumption (C) and investment (I). Assume the factor inputs, K (capital) and L (labor), are fixed in supply. Finally, assume that government expenditures (G) and taxes (T)...