The Ricker equation is Xt + 1 =Xo = 0.5, c = 31.00.5-0.5-1.0Xt0.7 Xt0.60.50.40.30.20.127●cxtext. Plot enough terms of the Ricker equation to see how the terms behave.ext●6Does the sequence appear to be convergent?YesNo●0008●100.60.40.21.2 Ext1.00.80.60.40.222468......6co810t10 If the sequence is convergent, estimate the limit and then, assuming the limit exists, calculate its exact value. (If the quantity diverges, enter DIVERGES.)estimateexact valueIf the sequence is divergent, describe the behavior of the terms. (If the sequence is convergent, just state so.)

Answered: The Ricker equation is Xt + 1 = Cxtext.…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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The Ricker equation is Xt + 1 =
Xo = 0.5, c = 3
1.0
0.5
-0.5
-1.0
Xt
0.7 Xt
0.6
0.5
0.4
0.3
0.2
0.1
2
7
●
cxtext. Plot enough terms of the Ricker equation to see how the terms behave.
ext
●
6
Does the sequence appear to be convergent?
Yes
No
●
000
8
●
10
0.6
0.4
0.2
1.2 Ext
1.0
0.8
0.6
0.4
0.2
2
2
4
6
8
......
6
co
8
10
t
10

If the sequence is convergent, estimate the limit and then, assuming the limit exists, calculate its exact value. (If the quantity diverges, enter DIVERGES.)
estimate
exact value
If the sequence is divergent, describe the behavior of the terms. (If the sequence is convergent, just state so.)

Expert Solution

Step 1

Consider the given Ricker equation,

$x_{t + 1} = c x_{t} e^{- x_{t}}$ , where $x_{0} = 0.5, c = 3$

So, we get,

$x_{t + 1} = 3 x_{t} e^{- x_{t}}$

Now,

$\begin{array}{rcl} x_{1} & = & 3 x_{0} e^{- x_{0}} = 3 (0.5) e^{- 0.5} = 0.909 \\ x_{2} & = & 3 x_{1} e^{- x_{1}} = 3 (0.909) e^{- 0.909} = 1.0980 \\ x_{3} & = & 3 x_{2} e^{- x_{2}} = 3 (1.098) e^{- 1.098} = 1.0986 \\ x_{4} & = & 3 x_{3} e^{- x_{3}} = 3 (1.0986) e^{- 1.0986} = 1.09801 \end{array}$