What happens when we take a logistic model for population growth and make the carrying-capacity time? depend on x = x. μ

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What happens when we take a logistic model for population growth and make the carrying-capacity time? depend on x = x. μ. (1 + sin .. ¹ (t))) X We can also view this equation as an autonomous system on the plane: r = r. ·(₁. 1 µ· (1 + sin (0))) — 7² ė 1 -
= gives the following 5. (5 points) Show that the change of variables y differential equation: y = − y(1 − µ · (1 + sint)) + 1. 6. (5 points) Solve the differential equation from the previous part to find y(t) = e¹(μ-1)+µ·(cos(t)−1) es(1-μ) +μ(1-cos (s)) ds + 7. (5 points) Use the result of the previous part to calculate the Poincare Map for x(t): xo e ²π(1-μ) p(xo) 2π xo ²″ es(1-µ)+μ(1−cos (s)) ds + 1 -2π 8. (5 points) Can you compute the integral a = [²″ es(1−µ)+µ(1−cos (s)) ds? Determine whether a can be positive, negative, or zero. Jo