4. Consider the Lotka-Volterra model for the interaction between a predator population (wolvesW(t)) and a prey population (moose M(t)),MW = −cW+dMW=aM - bMWwith the four constants all positive.(a) Explain the meaning of the terms.(b) Non-dimensionalize the equations in the form dx/d = x(1 − y) and dy/dî = µy(x − 1).(c) Find the fixed points, linearize, classify their stability and sketch the trajectories forvarious initial conditions (again, using a computer is fine). You should find a center. Canyou trust that this is really a center?(d) To answer this, try to find a conserved quantity for this system (read Strogatz §6.5nonlinear centers form at minima and maxima of this quantity. The most fundamental caseis energyan undamped, unforced oscillator has a nonlinear center). Find the minimaor maxima of this quantity, and show that the center found above lies at the same point.(Hint: to find the conserved quantity, notice that x/x = d(lnx)/dt). 4. Consider the Lotka- Volterra model for the interaction between a predator population (wolvesW(t)) and a prey population (moose M(t)),MW = −cW+dMW=aM - bMWwith the four constants all positive.(a) Explain the meaning of the terms.(b) Non-dimensionalize the equations in the form dx/d = x(1 − y) and dy/dî = µy(x − 1).(c) Find the fixed points, linearize, classify their stability and sketch the trajectories forvarious initial conditions (again, using a computer is fine). You should find a center. Canyou trust that this is really a center?(d) To answer this, try to find a conserved quantity for this system (read Strogatz §6.5nonlinear centers form at minima and maxima of this quantity. The most fundamental caseis energyan undamped, unforced oscillator has a nonlinear center). Find the minimaor maxima of this quantity, and show that the center found above lies at the same point.(Hint: to find the conserved quantity, notice that x/x = d(lnx)/dt).

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