(e) In particular, what do you think will happen to the populations ast o if theinitial population sizes are N, (0) = 2.001 and N2(0) = 0.001?%3D%3D Interaction of two species of squirrels fiercely competing for the same ecological niche onan island is described by Lotka-Volterra-Gause equationsdN1N1(2 – N1 – 2N2) = f(N1, N2),dt(1)dN2N2(3 – N2 – 3N1) = g(N1, N2),dtwhere N1 = N1(t) and N2 = N2(t) are the population densities of the competing species.

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Answered: (e) In particular, what do you think…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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3. Maximum Entropy Estimation: Shannon's entropy is defined as: where H(p)=1 Pi ln pi Possibility of each event: pi = p(xi) = [0, 1] You are given a six-sided dice numbered from 1 to 6. You are given the information that the average result from 10000 times of rolling dice is E[x]=3.5. What is your estimation of the probabilities associated with different sides (what is the probabilities of having 1, respectively)? 2,..., The following nonlinear optimization problem 6 max H (p) subject to Σi±1 P₁ = 1, Σ²±1 xipi = = E[x] i=1 gives the least-biased probability distribution (a) Solve this problem by calling an optimization solver. Include your script and result output. Plot how the probabilities change as E[x] varies between 1 and 6. (b) Derive the optimality conditions using Lagrange Multiplier Method. Save this result, as we will come back to it in the next homework assignment.
16. Suppose the true average growth μ of one type of plant during a 1-year period is identical to that of a second type, but the variance of growth for the first type is σ², whereas for the second type the variance is 40². Let X₁,..., X be m independent growth observations on the first type [so E(X;) = µ, V(X;) = σ²], and let Y₁,..., Y be n independent growth observations on the second type [E(Y) =μ, V(Y) = 4σ²]. a. Show that the estimator û = 8X + (1 - 8) is unbi- ased for μ (for 0 < 8 <1, the estimator is a weighted average of the two individual sample means). b. For fixed m and n, compute V(u), and then find the value of 8 that minimizes V(i). [Hint: Differentiate V(u) with respect to 8.]
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(e) In particular, what do you think will happen to the populations as t - o if the initial population sizes are N(0) = 2.001 and N2(0) 0.001? %3D