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# The predators survive by eating the prey and the prey

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Unformatted text preview: tors survive by eating the prey and the prey exist on an independent source of food. A classical situation involves foxes and rabbits. Let P (t) and p(t), respectively, denote the populations of predators and prey at time t. Given the initial populations P0 and p0 of predators and prey, their subsequent populations satisfy the Lotka-Volterra equations dp = p(a ; P ) t>0 p(0) = p0 dt dP = P (;c + p) t>0 P (0) = P0 dt where a, c, , and are positive constants corresponding to the prey's natural growth rate, the predator's natural death rate, the prey's death rate upon coming into contact with predators, and the predator's growth rate upon coming into contact with prey. This example involves an IVP for a system of two rst-order nonlinear ODEs. The ODEs are nonlinear because: The system is rst order because: 3. Column Buckling. The lateral displacement y(t) at position t of a clamped-hinged bar of length l that is subjected to a load P (Figure 1.1.2) may be approximated by the Euler-Bernoulli equations d4 y + d2y = 0 0<t<l dt4 dt2 with the boundary conditions 2 y (0) y(l) = d dt(l) = 0: y(0) = dydt = 0 2 4 The parameter = P=EI , where EI is the exural rigidity of the bar...
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