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Math 5120, Homework 5 1. (EK 10.17) Finish finding the solution for spreading colonies of microorganisms from class. Namely, we have shown that the traveling wave solution G(z), N (z) satisfies v N cv vcN0 dG =− G− + , dz D D D dN kN G =− . dz v a) Find steady states of this system of ODEs, sketch nullclines, and compute the stability properties of the steady states. b) What solution of the ODE system corrsponds to the traveling wave in the PDE? Determine whether there are constraints on the speed v of the wave (i.e. whether the appropriate solution always exists and remains in the positive GN quadrant. 2. (EK 10.1) Suggest a (set of) PDEs to describe the following processes: (a) a predator-prey system in which both species move randomly in a one-dimensional setting b) a predator-prey system in which predator moves towards higher prey densities and the prey moves towards lower predator densities c) A pair of reacting and diffusing chemicals such that species 1 activates the formation of both sunstances and species 2 inhibits the formaion of both substances. 3. (EK 10.9) Consider equation for bacterial chemotaxis introduced in class: ∂c ∂ ∂B ∂ ∂B χB + µ . = ∂t ∂x ∂x ∂x ∂x Explain how the equation would be modified to incorporate the following: a) The chemotactic sensitivity increases linearly with the chemical concentration b) The random motion decreses as the cell density increases c) The cell population increases logistically with the carrying capacity proportional to the concentration of the chemical 4. (dV 4.5.4) A dingo population which lives in the eastern parts of Australia is prevented form invasion to the west by a fence that runs north-south. Imagine that the fence breaks somewhere at time t = 0. A farm is located on the west side of the fence, exactly 100 miles west of the hole in the fence. The farmers would like to know how long it will take the dingoes to reach their farm. We model the spread of dingoes as: ut = Duxx + ku(1 − u ), K with k = 1 (1/month), and K = 1 (in units of u). a) The region between the fence and farm is flat and the diffusion constant D1 = 100 (miles2 /month). When does the dingo population reach the farm? (consider a traveling wave and calculate wave speed) b) How will the time change if the diffusion constant is D2 = 50 miles2 /month due to some rock and slope in the region? 1