#1 For each of the DDS given below, do the following: Determine if the DDS is linear or not; if it is linear, write the explicit solution. Computex, (using either explicit solution or recursive equation). Find the equilibrium value(s) for the DDS (linear or not). If the DDS is a linear, use the stability criterion to determine whether each equilibrium is stable or not. a) Xn+1=X²-2, x₁ = 1

Solve the problem. = = Let V₁ - [8]-V₂-[-₂-[E] V2 V3 Determine if the set {V₁, V₂, V3} is linearly independent.

The Lotka-Volterra model is often used to characterize predator-prey interactions. For example, if R is the population of rabbits (which reproduce autocatlytically), G is the amount of grass available for rabbit food (assumed to be constant), L is the population of Lynxes that feeds on the rabbits, and D represents dead lynxes, the following equations represent the dynamic behavior of the populations of rabbits and lynxes: R+G→ 2R (1) L+R→ 2L (2) (3) Each step is irreversible since, for example, rabbits cannot turn back into grass. a) Write down the differential equations that describe how the populations of rabbits (R) and lynxes (L) change with time. b) Assuming G and all of the rate constants are unity, solve the equations for the evolution of the animal populations with time. Let the initial values of R and L be 20 and 1, respectively. Plot your results and discuss how the two populations are related.

The demand for watches is given by p = 7000 − 2q dollars, and the supply of watches is given by p = 0.01q + 2q + 1000 dollars,            where q is the number of watches demanded and supplied when the price per watch is p dollars. Based on the information provided, calculate the equilibrium quantity and the equilibrium price for watches.