Equilibrium Points: The model has equilibrium points where both populations remain constant over time. These equilibrium points can be stable or unstable, depending on the values of the model parameters. A stable equilibrium represents a situation where predator and prey popula- tions coexist in a relatively stable manner. In the Lotka-Volterra Model it is rare to find perfect equilibrium solutions, more often the solutions exhibit oscillatory behaviour around the equilibrium points rather than staying at them. To find the equilibrium points for this system we set both dx/dt and dy/dt to zero and solve dx = ax - bxy dt dy dt == -cy + dxy We find we have 2 sets of equilibrium values, (0,0) when both predator and prey populations vanish, and (c/d, a/b) when there are just enough prey to support a constant predator population but there are not too many predators. Answer the following question for the system shown below. dx dt = 2.4x-1.2xy 33 dy dt = −y + xy 11. Find the equilibrium values for the system and use the Slopes App Phase Plane Activityto graph a solution in the plase plane with intial conditions as close as you can get to the equilibrium value. Upload your file to GS showing both the solution and the phase plane portrait.

Equilibrium Points: The model has equilibrium points where both populations remain constant over time. These equilibrium points can be stable or unstable, depending on the values of the model parameters. A stable equilibrium represents a situation where predator and prey popula- tions coexist in a relatively stable manner. In the Lotka-Volterra Model it is rare to find perfect equilibrium solutions, more often the solutions exhibit oscillatory behaviour around the equilibrium points rather than staying at them. To find the equilibrium points for this system we set both dx/dt and dy/dt to zero and solve dx = ax - bxy dt dy dt == -cy + dxy We find we have 2 sets of equilibrium values, (0,0) when both predator and prey populations vanish, and (c/d, a/b) when there are just enough prey to support a constant predator population but there are not too many predators. Answer the following question for the system shown below. dx dt = 2.4x-1.2xy 33 dy dt = −y + xy 11. Find the equilibrium values for the system and use the Slopes App Phase Plane Activityto graph a solution in the plase plane with intial conditions as close as you can get to the equilibrium value. Upload your file to GS showing both the solution and the phase plane portrait.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 23EQ: 23. Consider a simple economy with just two industries: farming and manufacturing. Farming consumes...

See similar textbooks

Similar questions

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.
The demand and supply functions for two interdependent commodities are given by:Qd1 = 205 − 2.5P1 − P2Qs1 = 1.5P1 − 30Qd2 = 147.5 − 0.5P1− 1.5P2Qs2 =P2 − 60Determine the equilibrium price and quantity for this two-commodity model.
Consider one application in which either a first order or second order IVP is formed to find a solution in a model problem.
Draw the phase diagram of the system; list all the equilibrium points; determine the stability of the equilibrium points; and describe the outcome of the system from various initial points. You should consider all four quadrants of the xy-plane. Include the coordinate axes; all the isoclines; all the equilibrium points; the allowed directions of motion (both vertical and horizontal) in all the regions into which the isoclines divide the xx plane; direction of motion along isoclines, where applicable. dx dt || 7-y₁ dy dt =x-7.
Pleas don't provide handwritten solution ...
Ок Denote the owl and wood rat populations at time k by x = 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 = (0.2)0k + (0.5)RK Rk+1=(-0.16)0k + (1.1)Rk Give a formula for xk- xk = 4 ( D +C₂1
Solve the 2 Market commodities of market equilibrium using Cramer's Rule • Find the equilibrium price and quantity for each of the following models: D = S D = 3 - 2P S = -1+5P D1 = S1 D1 = 5 – 2P, + P2 S = -3 + 5P, – 2P2 D2 = S2 D2 = 2 + 2P, – P2 S2 = -43 – 2P1 + 3P2 a. b. %3D %3D %3D
(i) Explain the biological meaning of each parameter in model (5). (ii) Find all nullclines of system (5) and sketch them in the x- y plane, clearly showing the location of the steady states.
For the following two-population system, first describe the type of x- and y-populations involved (exponential or logistic) and the nature of their interaction-competition, cooperation, or predation. Then find and characterize the system's critical points (as to type and stability). Determine what nonzero x- and y-populations can coexist. Finally, construct a phase plane portrait that enables you to describe the long-term behavior of the two populations in terms of their initial populations x(0) and y(0). dx dt dy dt=xy-4y = 5xy-10x CICCES Describe the type of x- and y-populations involved. Select the correct choice below. OA. The populations involved are naturally declining populations in competition. OB. The populations involved are naturally growing populations in cooperation. OC. The populations involved are naturally declining populations in cooperation. OD. The populations involved are naturally growing populations in competition.
Suppose 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.
Find the equilibrium point(s) of each nonlinear system given below. Then determine the stability of each equilibrium point. Simulate these systems and plot their phase plane portraits and determine if any possess periodic orbits.
Denote the owl and wood rat populations at time k by xk Ok Rk 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 xx.) 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 = (0.1)0k + (0.6)RK Rk+1=(-0.15)0k +(1.1)Rk Give a formula for XK- = XK C +0₂ , where k is in months, Ok is the number of owls,