𝑥̇ = (10 − 𝑥)(40 − 𝑥)(50 − 𝑥) + 𝜃
𝑥(0) = 60
1. Consider the above model as having both a fast and a slow dynamic. The slow dynamic is modeled by considering the possibility that the parameter 𝜃 varies "slowly" over time. (Hint: Assume for convenience that if the system is not in equilibrium, 𝜃 will pause. Once the system reaches any equilibrium, 𝜃 will resume and will begin to change values again.)
For this example, consider the case where 𝜃 starts at zero and then monotonically decreases to -1000, and then turns around, and monotonically increases back to zero (pausing whenever the system is not in equilibrium).

(a) Solve for all the equilibrium points when 𝜃 = 0, draw a graph with these equilibria (label the axes), and label each equilibrium point with "S" (stable), "M" (marginally stable), or U "unstable".
(b) Prove that you have correctly labeled all the equilibrium poi. correctly ("S", "M", or "U") by analyzing the corresponding Jacobian matrix. Show your work
(c) Approximately where will the first equilibrium point of x(t) be?
(d) Will the system experience a "market crash"? If so, what will x(t) be right before the crash?
(e) What is the final value of x(t) when 𝜃 returns to the value zero for the second time at the end?

2. Suppose Megacom and Acorn are the only two companies in the cellphone market. We have come up with the following generic model to describe the dynamics of this market, using M to denote the number of Megacom cellphones in the world, and A as the number of Acorn cellphones in the world. Both M and A are functions of time, t.
𝑀̇ = (𝜏 − 𝛼𝐴 − 𝜇𝑀)𝜇𝑀
𝐴̇ = (𝜏 − 𝛼𝐴 − 𝜇𝑀)𝛼𝐴
𝐴(0) = 1, 𝑀(0) = 1

𝜏 is a parameter that represents the total number of potential computer buyers. Assume it is very large. 𝛼 and 𝜇 are parameters in the range of [0,1], that represent how “clunky” Acorn and Megacom’s cellphones are (respectively); a “low-clunky” cellphone only does a few functions but does them well, and “high-clunky” cellphone does lots of functions. Assume that Megacom cellphones are more clunky than Acorn cellphones 𝜇 > 𝛼
(a) Show that the initial growth of the two cellphones will be approximately exponential in shape. Hint: 𝜏 is very large)
(b) Explain why A(t) and M(t) will eventually hit equilibrium and stop moving. Draw a graph of (M, A) labeling all the equilibrium points (Hint: there are infinitely many).
Also calculate the exact values (in terms of all the parameters) of any equilibrium that intersects with an axis
(c) Prove that slight disturbance of the status at equilibrium points will not lead to explosive growth or decay (Hint: a phase diagram with a detailed explanation should be convincing).
(d) Once the companies reach equilibrium, Acorn considers changing 𝛼 to a different value (still in [0,1]. However, assume that Megacom either cannot or will not change its “clunkiness” parameter 𝜇. Show which direction 𝛼 should be changed so that A(t) will grow.
(e) Assume that Acorn does indeed change 𝛼 as recommended in the previous part Th change will also affect the state of Megacom. In the short period of time after this change, will the customer base of Megacom increase or decrease? Whose customer base is changing faster?

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