A discrete-time dynamical system is represented by a state variable $x_t$ that describes the system's state at time $t$, and a dynamic rule $x_{t+1} = f(x_t)$ that governs how the state evolves over time. Here's how the given events can be formulated:

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(a) Number of deer in a forest increases by 15 percent per year

Let $x_t$ represent the number of deer in year $t$.

The dynamic rule can be expressed as: $x_{t+1} = x_t + 0.15x_t$ or equivalently: $x_{t+1} = 1.15x_t$

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(b) A population of bacteria doubles every hour. Every hour, before reproduction, 854 bacteria are removed

Let $x_t$ represent the number of bacteria at time $t$ (before removal and reproduction).

The dynamic rule is: $x_{t+1} = 2(x_t - 854)$ This accounts for removing 854 bacteria first, then doubling the remaining population.

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(c) A patient's body absorbs 3% of medication per hour. Every hour, the patient is given 5 units of medication to top up the amount absorbed

Let $x_t$ represent the amount of medication in the patient’s body at hour $t$.

The dynamic rule can be expressed as: $x_{t+1} = 0.97x_t + 5$ This accounts for the fact that 3% of the medication is absorbed, leaving 97% in the body, and an additional 5 units are added.

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Questions for Expansion:

1. How can these discrete-time systems be converted into continuous-time models?
2. What happens to the deer population if an additional fixed number of deer are introduced annually?
3. How does the bacteria population change if the removal rate fluctuates hourly?
4. What is the steady-state amount of medication in the patient’s body if the system reaches equilibrium?
5. How can stochastic variations (e.g., random changes in growth rate) be incorporated into these models?

Tip: For stability analysis, calculate equilibrium points by setting $x_{t+1} = x_t$ and solve for $x$.