North Carolina State University Department of Mathematics An Introduction to Applied Mathematics MA 325 Assignment #1 DUE Date: April 3, 2017 1. A family of salmon fish living off the Alaskan Coast obeys the Malthusian law of population growth dp(t)/dt = 0.003p(t), where t is measured in minutes. It is also assumed that at time t = 0 there are one million salmon. (a) Using SimBiology, simulate the salmon population, p(t). (b) Now, assuming that at time t = 0 a group of sharks establishes residence in these waters and begins attacking the salmon. The rate at which salmon are killed by the sharks is 0.001p2 (t), where p(t) is the population of salmon at time t. Moreover, since an undesirable element (the sharks) has moved into their neighborhood, 0.002 salmon per minute leave the Alaska waters. i. Modify the Malthusian law of population growth to take these two factors into account. Using SimBiology to simulate the salmon population, p(t). What happens as t ? ii. Show that the above model is not realistic. Hint: Show, according to this model, the salmon population decreases from one million to about one thousand very quickly (in about one minute). 2. The population of New York City would satisfy the logistic law dp 1 1 = p p2 , dt 25 25 106 where t is measured in years, if we neglected the high emigration and homicide rates. Assume that population of New York City was 8, 000, 000 in 1970. (a) Using SimBiology, find the population, p(t). What happens as t ? (b) Modify the logistic equation to take into account the fact that 9, 000 people per year move from the city, and 1, 000 people per year are murdered. Using SimBiology, find the population, p(t). What happens as t ? 3. Consider a predator-prey model of foxes and rabbits. If we let R(t) and F (t) denote the number of rabbits and foxes, respectively, then the Lotka-Volterra model is: dR dt dF dt = aRbRF = c R F d F, where a = 0.04, b = 0.0005, c = 0.00005, and d = 0.2. (a) Using SimBiology with various initial conditions (including the cases where the number of rabbits is larger than and less than foxes, initially), describe the dynamics of rabbits and foxes from your simulations. (b) Now, assuming that in the absence of foxes, rabbits grow according to a logistic law, then we have the system dR dt dF dt = aRbRF eRR = cRF dF Assuming e = 0.002, describe the dynamics of rabbits and foxes (for various initial conditions) (Hint: logistic law). What values of parameter e resulting in periodic solutions? In the report that you turn in, please include the followings: differential equations that you exported from SimBiology for each problem. the plots for each simulation your write-ups to the questions