Many animal species use smell to navigate through their environments. A lobster, for example, can use its antennae to detect very small concentrations of chemicals in the water around it (that is, it "smells" the water). The lobster uses this ability to determine which direction a smell is coming from and hence find food in the murky deep. In this investigation, we will assume that our lobster can only move on a flat two-dimensional plane. We will assume that our lobster always moves in the direction that increases the smell the fastest. Suppose the system of differential equations, x' = x - x3 and y' = -y (-2 sxs 2,-2 sy s 2) represents the path the lobster moves on, thus telling us the direction to the food. a) Find the critical points for this system of equations. (2 points) b) Find the linearized system for each of the critical points you found in part a. Find the eigenvalues and eigenvectors of the linear system, classify each critical point as to type and determine its stability. Sketch the trajectories in the neighborhood of each critical point using colored pencils. (2 points) c) Plot enough trajectories of the given system to show clearly the behavior of the solutions. Plot these trajectories in regular pencil making sure not to conceal you colored trajectories from part b. (1 point) d) Write a clear explanation of these solutions. Your explanation should include telling what the lobster does if he starts on the y-axis and why he does it. (4 points) Many animal species use smell to navigate through their environments. A lobster, for example, can use its antennae to detect very small concentrations of chemicals in the water around it (that is, it "smells" the water). The lobster uses this ability to determine which direction a smell is coming from and hence find food in the murky deep. In this investigation, we will assume that our lobster can only move on a flat two-dimensional plane. We will assume that our lobster always moves in the direction that increases the smell the fastest. Suppose the system of differential equations, x' = x - x3 and y' = -y (-2 sxs 2,-2 sy s 2) represents the path the lobster moves on, thus telling us the direction to the food. a) Find the critical points for this system of equations. (2 points) b) Find the linearized system for each of the critical points you found in part a. Find the eigenvalues and eigenvectors of the linear system, classify each critical point as to type and determine its stability. Sketch the trajectories in the neighborhood of each critical point using colored pencils. (2 points) c) Plot enough trajectories of the given system to show clearly the behavior of the solutions. Plot these trajectories in regular pencil making sure not to conceal you colored trajectories from part b. (1 point) d) Write a clear explanation of these solutions. Your explanation should include telling what the lobster does if he starts on the y-axis and why he does it. (4 points)