Problem 4. (10 points) The populations of a host H(t) and a parasite P(t) are ap

Question

Problem 4. (10 points) The populations of a host H(t) and a parasite P(t) are approximately described by the system where a, b,c and k are positive constants 4.1. (2.5 points) Show there are constants , and such that z aP, y and = that will bring these equations into the simpler form 4.2. (2.5 points) Find the fixed point of this system, and determine the stability character of this fixed point. 4.3. (2.5 points) Sketch some orbits near the fixed points, indicating the direction of the "motion". Clearly identify the axes

Explanation / Answer

4. Population of host, H and parasite P

H' = (a - bP)H

P' = (c - kP/H)P , H, P > 0

a, b, c, k are +ve constants

4.1 let

x = alpha*P

y = beta*H

tau = gamma*t

now

H' = dH/dt = (gamma/beta)dy/d(tau)

P' = dP/dt = (gamma)/alpha)dx/d(tau)

hence

(gamma/beta)dy/d(tau) = (a - b*x/alpha)y/beta

dy/t(tau) = (a/gamma - x(b/alpha*gamma))y

dx/d(tau) = (c/gamma - kx*beta/y*alpha*gamma)x

dy/t(tau) = (a/gamma)(1 - x(b/a*alpha))y

dx/d(tau) = (c/gamma)(1 - k*beta*x/y*alpha*c)x

hence

for

a = gamma

b = a*alpha

c/gamma = k

k*beta = alpha*c

we can compare the equations

hence

b = 1

a = 1/alpha = gamma

and we get

dy/dt = (1 - x)y

dx/dt = kx(1 - x/y) ( whwere t is tau now)

4.3 solving the diffential equations we have

y' = y - xy

x' = kx - kx^2/y

y" = y' - xy' - yx' = y'(1 - x) - yx'

y" = y'(1 - (y - y')/y) - k(y - y') - k(y - y')^2/y^2

y"*y = y'^2 - k(y^2 - yy' + y + y'^2/y - 2y')

this equation can be solved to plot solutions

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