In 18, determine the system of differential equations corresponding to each compartment

In Problems 18, determine the system of differential equations corresponding to each compartment model and analyze the stability of the equilibrium (0, 0). The parameters have the same meaning as in Figure 11.33. a = 0.5, b = 0.1, c = 0.05, d = 0.02

In Problems 18, determine the system of differential equations corresponding to each compartment model and analyze the stability of the equilibrium (0, 0). The parameters have the same meaning as in Figure 11.33. a = 0.4, b = 1.2, c = 0.3, d = 0

In Problems 18, determine the system of differential equations corresponding to each compartment model and analyze the stability of the equilibrium (0, 0). The parameters have the same meaning as in Figure 11.33. a = 2.5, b = 0.7, c = 0, d = 0.1

In Problems 18, determine the system of differential equations corresponding to each compartment model and analyze the stability of the equilibrium (0, 0). The parameters have the same meaning as in Figure 11.33. a = 1.7, b = 0.6, c = 0.1, d = 0.3

In Problems 18, determine the system of differential equations corresponding to each compartment model and analyze the stability of the equilibrium (0, 0). The parameters have the same meaning as in Figure 11.33. a = 0, b = 0.1, c = 0, d = 0.3

In Problems 18, determine the system of differential equations corresponding to each compartment model and analyze the stability of the equilibrium (0, 0). The parameters have the same meaning as in Figure 11.33. a = 0.2, b = 0.1, c = 0, d = 0

In Problems 18, determine the system of differential equations corresponding to each compartment model and analyze the stability of the equilibrium (0, 0). The parameters have the same meaning as in Figure 11.33. a = 0.1, b = 1.2, c = 0.5, d = 0.05

In Problems 18, determine the system of differential equations corresponding to each compartment model and analyze the stability of the equilibrium (0, 0). The parameters have the same meaning as in Figure 11.33. a = 0.2, b = 0, c = 0, d = 0.3

In Problems 918, find the corresponding compartment diagram for each system of differential equations. dx1 dt = 0.4x1 + 0.3x2 dx2 dt = 0.1x1 0.5x2

In Problems 918, find the corresponding compartment diagram for each system of differential equations. dx1 dt = 0.4x1 + 3x2 dx2 dt = 0.2x1 3x2

In Problems 918, find the corresponding compartment diagram for each system of differential equations. dx1 dt = 0.2x1 + 0.1x2 dx2 dt = 0.1x2

In Problems 918, find the corresponding compartment diagram for each system of differential equations. dx1 dt = 0.2x1 + 1.1x2 dx2 dt = 0.2x1 1.1x2

In Problems 918, find the corresponding compartment diagram for each system of differential equations. dx1 dt = 2.3x1 + 1.1x2 dx2 dt = 0.2x1 2.3x2

In Problems 918, find the corresponding compartment diagram for each system of differential equations. dx1 dt = 1.6x1 + 0.3x2 dx2 dt = 0.1x1 0.5x2

In Problems 918, find the corresponding compartment diagram for each system of differential equations. dx1 dt = 1.2x1 dx2 dt = 0.3x1 0.2x2

In Problems 918, find the corresponding compartment diagram for each system of differential equations. dx1 dt = 0.2x1 + 0.4x2 dx2 dt = 0.2x1 0.4x2

In Problems 918, find the corresponding compartment diagram for each system of differential equations. dx1 dt = 0.2x1 dx2 dt = 0.3x2

In Problems 918, find the corresponding compartment diagram for each system of differential equations. dx1 dt = x1 dx2 dt = x1 0.5x2

Suppose that a drug is administered to a person in a single dose, and assume that the drug does not accumulate in body tissue, but is excreted through urine. Denote the amount of drug in the body at time t by x1(t) and in the urine at time t by x2(t). If x1(0) = 4 mg and x2(0) = 0, find x1(t) and x2(t) if dx1 dt = 0.3x1(t)

Suppose that a drug is administered to a person in a single dose, and assume that the drug does not accumulate in body tissue, but is excreted through urine. Denote the amount of drug in the body at time t by x1(t) and in the urine at time t by x2(t). If x1(0) = 6mg and x2(0) = 0, find a system of differential equations for x1(t) and x2(t) if it takes 20 minutes for the drug to be at onehalf of its initial amount in the body.

A very simple two-compartment model for gap dynamics in a forest assumes that gaps are created by disturbances (wind, fire, etc.) and that gaps revert to forest as trees grow in the gaps. We denote by x1(t) the area occupied by gaps and by x2(t) the area occupied by adult trees. We assume that the dynamics are given by dx1 dt = 0.2x1 + 0.1x2 (11.44) dx2 dt = 0.2x1 0.1x2 (11.45) (a) Find the corresponding compartment diagram. (b) Show that x1(t) + x2(t) is a constant. Denote the constant by A and give its meaning. [: Show that d dt (x1 + x2) = 0.] (c) Let x1(0)+x2(0) = 20. Use your answer in (b) to explain why this equation implies that x1(t) + x2(t) = 20 for all t > 0. (d) Use your result in (c) to replace x2 in (11.44) by 20 x1, and show that doing so reduces the system (11.44) and (11.45) to dx1 dt = 2 0.3x1 (11.46) with x1(t) + x2(t) = 20 for all t 0. (e) Solve the system (11.44) and (11.45), and determine what fraction of the forest is occupied by adult trees at time t when x1(0) = 2 and x2(0) = 18.What happens as t ?

One simple model for forest succession is a three-compartment model. We assume that gaps in a forest are created by disturbances and are colonized by early successional species, which are then replaced by late successional species. We denote by x1(t) the total area occupied by gaps at time t, by x2(t) the total area occupied by early successional species at time t, and by x3(t) the total area occupied by late successional species at time t. The dynamics are given by dx1 dt = 0.2x2 + x3 2x1 dx2 dt = 2x1 0.7x2 dx3 dt = 0.5x2 x3 (a) Draw the corresponding compartment diagram. (b) Show that x1(t) + x2(t) + x3(t) = A where A is a constant, and give the meaning of A.

Solve d2x dt2 = 4x with x(0) = 0 and dx(0) dt = 6.

Solve d2x dt2 = 9x with x(0) = 0 and dx(0) dt = 12.

Transform the second-order differential equation d2x dt2 = 3x into a system of first-order differential equations.

Transform the second-order differential equation d2x dt2 = 1 2 x into a system of first-order differential equations.

Transform the second-order differential equation d2x dt2 + dx dt = x into a system of first-order differential equations.

Transform the second-order differential equation d2x dt2 2 dx dt = 3x into a system of first-order differential equations.