MATH-142 | PROJECT 1 | FALL 2016 | NAME: 1 Instructions: To plot phase trajectories (asked in the exercises below) use any computer software capable of plotting vector fields or isoclines (for instance MatLab or Mathematica). Problem 1 (Predator-Prey). Consider the system of Lotka-Volterra that monitors the interactions of sharks and fish: \u0001 dF = F a bF cS dt (1) \u0001 dS = S F k . dt Let = k = b = c = 1 and a = 2. Use the method of isoclines to plot the phase trajectories and the nullclines of the system (1) in the region F, S > 0. Give an ecological interpretation of the trajectories. Problem 2 (Competition). If N1 , N2 are two species that live in 'separate' environments or simply do not interact with each other, the logistic equations \u0010 Ni \u0011 dNi = i Ni 1 , dt Ki i = 1, 2 would be appropriate for modeling the dynamics of these populations. Here the constant Ki is called the carrying capacity of the population Ni (recall that Ni Ki as t ). Suppose now that the two-species N1 and N2 live in the same environment and compete for the same food source. In that case, the growth rate of the population N1 is affected negatively by N2 and vice versa. The simplest system for competition then would be \u0010 \u0011 dN1 N1 = 1 N1 1 a21 N2 dt K1 (2) \u0010 \u0011 dN2 N2 = 2 N2 1 a12 N1 . dt K2 Objective: Consider the system (2) with 1 = 2 = 2 and K1 = K2 = 1. Use the method of isoclines to plot the phase trajectories and the nullclines for this system in the following cases: (i) a12 = a21 = 0.75. (ii) a12 = a21 = 1.25. (iii) a12 = 1.25, a21 = 0.75. For each case give an ecological interpretation of the trajectories. Problem 3 (Mutualism). Suppose that two-species N1 and N2 live in the same environment and benefit from the presence of each other. In that case, the growth rate of the population N1 is affected positively by N2 and vice versa. The appropriate system for modeling of such interactions would be \u0010 \u0011 N1 dN1 = 1 N1 1 + a21 N2 dt K1 (3) \u0010 \u0011 dN2 N2 = 2 N2 1 + a12 N1 . dt K2 Note, that in the absence of mutualistic interactions (a12 = a21 = 0) each species grows to its respective carrying capacity Ki . 2 NAME: MATH-142 | MIDTERM-I | FALL 2016 Objective: Consider the system (3) with 1 = 2 = 2 and K1 = K2 = 1. Using the method of isoclines plot the phase trajectories and the nullclines for this system in the following cases: (i) a12 = 0.4, a21 = 0.3. (ii) a12 = 2, a21 = 1. For each case give an ecological interpretation of the trajectories