Project # 3 Assignment: Two-Species Lotka-Volterra Systems & the Pendulum Re-visited

Transcription

1 Project # 3 Assignment: Two-Species Lotka-Volterra Systems & the Penulum Re-visite In 196 Volterra came up with a moel to escribe the evolution of preator an prey fish populations in the Ariatic Sea. Let N(t) enote the prey population an P(t) the preator population at time t 0. He assume that: (1). In the absence of preators (P 0) the per capita prey growth rate of the prey population N was constant, but fell linearly as a function of preator population P when preation was present (P > 0);. In the absence of prey (N 0) the per capita growth rate of the preator was constant (an negative), an increase linearly with the prey population N when prey was present (N > 0). Thus the moel introuce by Volterra reas: 1 N t N a bp 1 P t P cn where a, b, c, > 0 are constants. It turns out that this moel can be treate by separation of variables. We fin that: cn N t N a bp P t P 0 Page 1 of 9

2 cn N N a bp P 0 P cn N N a bp P P K cn ln N aln P bp K Since P>0 an N>0, we can replace the absolute values of the variables with the variables. cn lnn alnp bp K Differentiate both sies with repect to "t". t cn lnn alnp bp 0 Define, "H" as follows. c HNP N lnn alnp bp Page of 9

3 Then H is constant along a solution (N(t), P(t)) (for t where the solution exists. We consier the solutions for various initial populations (N0, P0) (N(0), P(0)). Suppose that (N0, P0) is in the 1st quarant. Then H(N0, P0) is finite an all trajectories (N(t), P(t)) evolve so that H(N(t), P(t)) H(N(0), P(0)) H(N0, P0) constant. It is easy to see that H is a strictly concave function. Moreover, "H" has a unique maximum: N H c 0 an N P H a b 0, which yiels the point: P c a b. Since H is strictly concave with a unique maximum in the positive quarant, every trajectory with N0 > 0, P0 > 0 must be a close curve. Thus the interior orbits are a set of close curves each passing through the initial conition: (N(0), P(0)). Now, let's stuy a specific example that has nothing to o with preator-prey behavior per se but is mathematically structurally equivalent. We turn to a familiar example, the penulum. Just like preator-prey relationships it is a "zero sum game" provie that the penulum is unampe. That means that k0 in Project #, which yiels this ifferential equation with abitrary initial conitions. t 4sin 0 0 a t 0 b Page 3 of 9

4 Here is couple, first orer equivalent system. t x 1 x x 1 0 a t x 4sin x 1 x 0 b Let's fin the relationship between x 1 an x by Separation-of-Variables. 1 x t x sinx 1 t x 1 A the equations above an you get this result. 1 x t x sinx 1 t x 0 Separate variables. 4sin x 1 x 1 x x 0 Page 4 of 9

5 Integrate both sies. 4cos x 1 x C Impose the initial conitions. 4cos x 1 0 x 0 C C b 4cosa 4cos x 1 x b 4cosa Now suppose we have the unampe penulum poise almost at the top an we let it swing back towars the equilibrium position so that the initial conitions are a an b 0. Then, the relationship between x 1 an x is as follows. 4cos x 1 x 0 4cos 4 x 1 cos x 1 8 8cosx cos x 1 16 cos x 1 x Page 5 of 9

6 Use the initial conitions an t 0 0 in a new script file that calls your previously create function file for Project #. Here is what your new script file must o: (1.) Solve the system of equations for both the ampe penulum with k1 an the unampe penulum k0. (.) Determine the ifference in the squares of t for the unampe an ampe responses in the excursion from the top to the first passing thru equilibrium. This ifference in squares is proportional to the energy issipate by the ampe penulum. Display that result as "EnergyDissipate" in the comman winow (3.) Generate the plots shown in figures 1 an below using the "subplot" comman an the "figure" comman. Figure 1 shoul be self-explanatory. Figure is the graph of this equation: x 16 cos x 1. (4.) Calculate the maximum error between the ODE 45 solutions for the unampe penulum an the equation graphe in Figure. Display that error as "Confirmation" in the comman winow. Page 6 of 9

7 Here are the results: (a.) In the Comman Winow EnergyDissipate Confirmation e-004 Page 7 of 9 (b.) For Figure 1

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9 (c.) For Figure Page 9 of 9

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