2. Find the equilibrium solution(s) for the following systems of Des.(a) (Prey-Predator Model)(b)where a, b, c, d are constant parameters,respectively) population at any time t.x'= ax - bxyy'= cxy - dyand x = = x(t) (y = y(t)) represents the prey (predator,x' = -2x + yy' = 3x - 5y

We have to find the equilibrium solutions for the following systems of DEs.

Answered: 2. Find the equilibrium solution(s) for…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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6. For the competing species population model dx dt dy = 2x (1 - -/-) - ху = 3y (1-1)-2xy dt studied in this section, we showed that the equilibrium point (1, 1) is a saddle. (a) Find the linearized system near each of the other equilibrium points. (b) Classify each equilibrium point (as either a source, a sink, a saddle, ...). (c) Sketch the phase portrait of each linearized system. (d) Give a brief description of the phase portrait near each equilibrium point of the nonlinear system.
Can somebody give me the rules on how to classify the equilibrium points?
3. The populations x(t) and y(t) of two species satisfy the equations x' x(4-3y), y = y (4-3x), (x, y ≥ 0). Find the steady states of the system and characterise them.
B2. Suppose the non-linear system dx/dt = f(x, y), dy/dt = g(x,y) has a fixed point at (x*,*). Derive the linearisation of the system about (x*,*). Under what circumstances are the solution paths of the non-linear system close to those of the linearisation? In a model of pollution, capital K and pollution P satisfy the pair of differential equations dk/dt = K(ska-1-8), dP/dt = K³ −yP where 0 0, y > 0 and ẞ>1. Find the fixed point (K*,P*) satisfying K*> 0 and P* > 0 and show it is locally stable for the non-linear system. Find an explicit expression for K(t) when K(0) = K₁ ≥0, and find its limit as t→∞.
Find a particular solution of the indicated linear system that satisfies the initial conditions x, (0) = 4, x,(0) = - 6. 7 -2 5t x' = , X2 = 12 -7 6 The particular solution is x, (t) = and x2(t) =
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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
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Find a particular solution of the indicated linear system that satisfies the initial conditions x₁ (0)=1, x₂ (0) = 4, and x3 (0) = 6. x' = - 22-26-2 20 24 - 20 - 20 2 x; x₁ = e 2 -2t 6 -5 5 1 x₂ = ²t - 1 1 The particular solution is x₁ (t) = x₂(t)=, and x3 (t) = . 1 - 1 0
Find a particular solution of the indicated linear system that satisfies the initial conditions x, (0) =7, x2(0) = - 8. 9. x' = 14 - 4 5t x; x, = e 1 - 5t = e X2 - 9 The particular solution is x, (t) = and x2(t) =.

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2. Find the equilibrium solution(s) for the following systems of Des. (a) (Prey-Predator Model) (b) = ax - bxy x' = y' = cxy - dy where a, b, c, d are constant parameters, and x = = x(t) (y = y(t)) represents the prey (predato: respectively) population at any time t. x' = −2x+y y' =3x-5y