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Homework4 - Homework 4 AMATH 383 Autumn 2009 Due Wednesday...

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Homework4 AMATH 383, Autumn 2009 Due: Wednesday, November 18, after class 1. (3+6+5+2 points) PeriodicityoftheVolterraSystem: Consider the Volterra equations x = rx axy y = ky + byx for a predator-prey-model with solutions x ( t ) and y ( t ) . In class we demonstrated that close to the non-trivial stationary pont the solutions are periodic. Here, we want to proof that all positive solutions are periodic except those starting in the stationary point. (a) Show that x , y , t can be scaled such that the equations for the scaled variables ˜ x , ˜ y , ˜ t have the form x = x xy, y = γ ( y + yx ) where we dropped the tildas for simplicity and used the parameter γ = k/r . What are the scales of x , y , and t ? (b) Show that along solution trajectories ( x ( t ) ,y ( t )) of the above systems the quantity A ( x,y )= y ln y + γ ( x ln x ) is constant, that is, d dt A ( x ( t ) ,y ( t ))=0 . This means that the solution curves in the x - y - plane are given by the level lines A ( x,y )= c for different c . (c) Show that A ( x,y ) becomes infinite at the axes x =0 and y =0 , is convex and has a global minimum. At what point is the minimum? (d) Try to imagine the shape of the function A ( x,y ) and argue that the level lines A ( x,y )= c for c > 1+ γ are closed lines. Conclude that the solutions ( x ( t ) ,y ( t )) of the Volterra equations are periodic for any positive initial conditions different from the stationary point. 2. (3+5+6+6+1+2 points) CarnivoresinAustralia: Consider the system of ordinary differential equations dx dt = rx (1 x K 0 ) axy dy dt = ky + bxy describing the interaction of herbivores x ( t ) and carnivores y ( t ) . The environment has a carrying capacity K 0 for herbivores. This problem will walk you through what has been presented in class.
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