Exercises 25—30 refer to a situation in which models similar to the predator-prey population models arise. Suppose A and B represent two substances that can combine to form a new substance C (chemists would write A + B
30. Suppose A and B are being added to the solution at constant (perhaps unequal) rates, and, in addition to the A + B
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Differential Equations
- 23. Consider a simple economy with just two industries: farming and manufacturing. Farming consumes 1/2 of the food and 1/3 of the manufactured goods. Manufacturing consumes 1/2 of the food and 2/3 of the manufactured goods. Assuming the economy is closed and in equilibrium, find the relative outputs of the farming and manufacturing industries.arrow_forwardConsider the discrete-time dynamical system modeling the concentration of a chemical in a lung. (Note: round all values at the end of the calculations and use 4 decimal places.)ct+1 = (1-p)ct + pβLet V = 2 L, W = 1 L, and β = 6 mmol/LIf c0 = 7 mmol/L, iterate to find the following values:c1 = ____mmol/Lc2 = ____mmol/Lc3 = ____mmol/Lc4 = ____mmol/LFind the equilibrium of this system:c* = ____mmol/Larrow_forwardConsider the discrete-time dynamical system modeling the concentration of a chemical in a lung. (Note: round all values at the end of the calculations and use 4 decimal places.) ct+1 = (1 - p)ct + pβ Let V = 2 L, W = 1 L, and β = 6 mmol/L If c0 = 7 mmol/L, iterate to find the following values: c1 = ____mmol/Lc2 = ____mmol/Lc3 = ____mmol/Lc4 = ____mmol/Larrow_forward
- Consider the following system x = (a 1¹ ) x + (1₁) ₁ U (2) Find the values of a so that the system is stable, or asymptotically stable?arrow_forwardhelp me to solve thisarrow_forwardDraw the phase portraits of the following linear systems and justify the choice of the direction of trajectories. (4.1) * = ( 13 1³ ) *. X, -3 (4.2) = *-(34)x X.arrow_forward
- Graph the following Discrete Dynamical Systems. Explain their long-term behavior. Try to find realistic scenarios that these DDS might explain. 7. a(n + 1) = -1.3 a(n) + 20, a(0) = 9arrow_forward9. discuss the behavior of the dynamical system xk+1= AXk where -0.3] 2 ] (b) A = 1.5 (a) A = 1.2 0.3 0.41 -0.3 1.1arrow_forwardOk Denote the owl and wood rat populations at time k by xk = Rk where k is in months, Ok is the number of owls, and R is the number of rats (in thousands). Suppose OK and RK satisfy the equations below. Determine the evolution of the dynamical system. (Give a formula for XK.) As time passes, what happens to the sizes of the owl and wood rat populations? The system tends toward what is sometimes called an unstable equilibrium. What might happen to the system if some aspect of the model (such as birth rates or the predation rate) were to change slightly? Ok+ 1 = (0.4)0k + (0.9)Rk Rk+1=(-0.2)0k +(1.3) Rkarrow_forward
- 9. discuss the behavior of the dynamical system Xk+1= Axk where -0.31 (@) A = [ (b) A = ["0.3 11 1.5 0.3 (b) A = 0.3 1 3Darrow_forwardUse (1) in Section 8.4 X = eAtc (1) to find the general solution of the given system. 1 X' = 0. X(t) =arrow_forwardSuppose that a patient receives a daily dose of 50mg/L of a certain drug such that 42% of it is eliminated from the body each day. If on a certain Monday, the concentration of the drug measured in their body (shortly after the daily dose) is 55 mg/L, write down the Discrete-Time Dynamical System describing the dynamics of the concentration xt of the drug in the body (in mg/L, t days after that Monday). Then write down the general solution to this DTDS.arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning