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Math Modeling : Chaos and Fractals

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Example Question #1 : Chaos And Fractals

There are two similar species of monkey that inhabit the same forest. The intrinsic growth rate is 4% for species A and 7% for species B. The capacity of the forest is estimated at 125,000 specie A monkeys and 450,000 species B monkeys. Poaching these monkeys over the last 100 years has caused for a decline in each species. Species A has a population of 4,000 and species B, a population of 80,000. Use a discrete model to project the population growth with a time step of several years. How large a time step can be used to maintain a continuous time model with an equivalent qualitative behavior?

Assume $\alpha =10^{-8}$ , x_b=100

Possible Answers:

h=37

h=30

h=57

h=47

h=50

Correct answer:

h=47

Explanation:

Identify the known information and the dynamical system.

$\\A=\text{number of species A monkey} \\B=\text{number of species B monkey} \\g_A=\text{growth rate of species A per year} \\g_B=\text{growth rate of species B per year} \\c_A=\text{competition on species A} \\c_B=\text{competition on species B} \\\\g_A=0.04A(1-A/125,000) \\g_B=0.07B(1-B/450,000) \\c_A=c_B=\alpha AB \\A\geq 0, b\geq 0 \\\alpha \epsilon \mathbb{R}>0$

$\\\frac{dx_1}{dt}=f_1(x_1,...,x_n) \\\vdots \\\\\frac{dx_n}{dt}=f_n(x_1,...,x_n)$

The dynamical system equation is:

$\frac{dx}{dt}=F(x)$

From here solve by using the Euler Method for a discrete time dynamical system.

$\frac{\Delta x}{\Delta t}=F(x)$

Now formulate the model

$\\\frac{dx_1}{dt}=f_1(x_1,x_2)=0.04x_1\left(1-\frac{x_1}{125,000} \right )-\alpha x_1x_2 \\\\\frac{dx_2}{dt}=f_1(x_1,x_2)=0.07x_1\left(1-\frac{x_1}{450,000} \right )-\alpha x_1x_2$

$\\x_1=\text{species A monkey} \\x_2=\text{species B monkey}$

Now transform these equations into a set of difference equations.

$\\\Delta x_1=f_1(x_1,x_2)\Delta t \\\Delta x_2=f_2(x_1,x_2)\Delta t$

$\\\Delta x_1= \text{change in the species A monkey population} \\\Delta x_2= \text{change in the species B monkey population}$

Assume

$\\\alpha =10^{-8} \\x_1(0)=4,000 \\x_2(0)=80,000\\N=50$

where represents the number of iterations the computer will use when implementing the Euler Method.

The time step $h=\Delta t$ will need to be changed and the model reformulated to calculate how large the time step can be before it doesn't resemble a continuous time model.

Using computer implementation of the Euler Method on the population of species B monkeys the following graph is produced.

Screen shot 2016 05 05 at 6.06.50 am

It is seen that h=47 is the largest time step before the function begins to have chaos and not represent the behavior of a continuous system.

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Math Modeling : Chaos and Fractals

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Example Question #1 : Chaos And Fractals

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