High School

Lizards are infected by mites. If \( x_t \) is the number of mites at time \( t \), then the number of mites at time \( t+1 \) is given by the formula:

\[ x_{t+1} = 0.5x_t + 1 \]

(a) What is the updating function for this dynamical system?

(b) If the initial number of mites is 10, find the number of mites at times \( t=1 \) and \( t=2 \).

(c) Find the equilibrium (fixed) point of this dynamical system.

(d) If the initial number of mites is 100, what is the number of mites in the population at time \( t=10 \)? At time \( t=250 \)?

Answer :

Final Answer

(a) The updating function for this dynamical system is x_(t+1) = 0.5x_t.

(b) If the initial number of mites is 10, the number of mites at times t=1 and t=2 are 5 and 2.5, respectively.

(c) The equilibrium (fixed) point of this dynamical system is x = 0.

(d) If the initial number of mites is 100, the number of mites at time t=10 is 0, and at time t=250, it remains at 0.

Explanation

(a) The updating function represents how the number of mites changes from one time step to the next. In this case, it's a simple exponential decay function where x_(t+1) is half of x_t. This means that each time step, half of the mites present in the previous step will remain.

(b) Starting with an initial population of 10 mites, at t=1, you have 0.5 * 10 = 5 mites. At t=2, you have 0.5 * 5 = 2.5 mites. This reflects the exponential decrease in mite population over time.

(c) The equilibrium point is where the population of mites no longer changes; it remains constant. In this case, when x_t is 0, x_(t+1) is also 0. Therefore, the equilibrium point is at x = 0.

(d) If you start with 100 mites, the population will continue to decrease exponentially. At t=10, it reaches 0 mites, and it remains at 0 for subsequent time steps, including t=250.

In summary, this dynamical system models the decline of mite populations over time due to the updating function x_(t+1) = 0.5x_t. Starting with 10 or 100 mites, the population ultimately reaches an equilibrium of 0.

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Final answer:

(a) The updating function for this system is xt+1 = 0.5xt + 1. (b) If the initial number of mites is 10, the number of mites at times t=1 and t=2 are 6 and 4, respectively. (c) The equilibrium (fixed) point of the system is x = 2. (d) If the initial number of mites is 100, the number of mites at time t=10 is approximately 2.002 and at time t=250 is also approximately 2.002.

Explanation:

(a) The updating function for this dynamical system is given by:

xt+1 = 0.5xt + 1

(b) If the initial number of mites is 10, we can find the number of mites at times t=1 and t=2 by plugging in the values:

For t = 1:
x1 = 0.5(10) + 1 = 6

For t = 2:

x2 = 0.5(6) + 1 = 4

(c) To find the equilibrium (fixed) point of the dynamical system, we set xt+1 = xt:

x = 0.5x + 1

0.5x - x = -1

-0.5x = -1

x = 2

Therefore, the equilibrium (fixed) point is x = 2.

(d) If the initial number of mites is 100, we can find the number of mites at time t=10 and t=250 by repeatedly applying the updating function:

For t = 10:
x10 = 0.5(0.5(0.5(0.5(0.5(0.5(0.5(0.5(0.5(0.5(100+1)+1)+1)+1)+1)+1)+1)+1)+1)+1)+1) ≈ 2.002

For t = 250:
x250 ≈ 2.002

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