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Flashcards: Dynamic Models

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

h=37

h=30

h=47

h=57

h=50

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Flashcards: Dynamic Models

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

Your answer:

h=37

Your answer:

h=30

Your answer:

h=57

Your answer:

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.

Flashcards: Dynamic Models

In a meadow there are two types of wild flowers growing, a green floor crawler and yellow daisies. The more desirable flower is the yellow daisy as they can be picked and sold for bouquets. The yellow daisies are also the more slow growing flower of the two. The green floor crawlers grow more rapidly and consume more of the meadow land. Yellow daisies compete with the green flower crawlers by growing taller and shading the crawlers new growth. The yellow daisies are also more resistant to certain types of bugs. Can these two types of flowers coexist on one portion of the meadow, or will one over run the other, driving it to extinction?

The Yellow Daisies will go extinct.

Both flowers will thrive.

The Green Floor Crawler will go extinct.

The steady state analysis cannot answer the question.

Both flower will coexist.

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Flashcards: Dynamic Models

Your answer:

The Yellow Daisies will go extinct.

Your answer:

Both flowers will thrive.

Your answer:

The Green Floor Crawler will go extinct.

Your answer:

Both flower will coexist.

Correct answer:

The steady state analysis cannot answer the question.

Explanation:

Since there is competition between the two differing species of flowers, the growth rate function is,

g(P)=rP-aP^2

where is the intrinsic growth rate and a<<r measures the strength of the limitations of resources.

Now identify the question as a dynamic model in steady state.

In other words,

$\\f_1(x_1,...,x_n) \\f_2(x_1,...,x_n)$

From here, identify what is known about the problem in mathematical terms.

$\\D=\text{Yellow Daisy Population} \\C=\text{Green Floor Crawler Population} \\g_D=\text{growth rate of yellow daisies} \\g_C=\text{growth rate of green floor crawler population} \\c_D=\text{loss due to competition for yellow daisies} \\c_C=\text{loss due to competition for green floor crawler}$

$\\g_D=r_1D-a_1D^2 \\g_C=r_2C-a_2C^2 \\c_D=b_1CD \\c_C=b_2CD \\D\geq0, C\geq0 \\r_1,r_2,a_1,a_2,b_1, b_2 \text{are positive reals}$

To answer the question determine whether $C\rightarrow0$ or $D\rightarrow0$ .

To formulate the model let

$\\x_1=D \\x_2=C \\\begin{Bmatrix} (x_1,x_2):x_1\geq0, x_2\geq0 \end{Bmatrix}$

where the steady state equations are as follows.

$\\\\r_1x_1-a_1x_1^2-b_1x_1x_2=0 \\r_2x_2-a_2x_2^2-b_2x_1x_2=0$

For the particular problem, the points on interests are at those where the two functions intersect.

$\\(0,0) \\\left( 0,\frac{r_2}{a_2}\right) \\\left(\frac{r_1}{a_1},0\right)$

and the intersection of

$\\a_1x_1+b_1x_2=r_1 \\b_2x_1+a_2x_2=r_2$

Using Cramer's rule to solve results in the following.

$\\x_1=\frac{r_1a_2-r_2b_1}{a_1a_2-b_1b_2} \\\\\\x_2=\frac{a1_r2-b_2r_1}{a_1a_2-b_1b_2}$

and when the question of coexistence is ask it is found that,

$\frac{r_2}{a_2}<\frac{r_1}{b_1}$ and $\frac{r_1}{a_1}<\frac{r_2}{b_2}$

Which after further examination learns to the conclusion that the steady state analysis cannot answer the question.

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Flashcards: Dynamic Models

Flashcards: Dynamic Models

Flashcards: Dynamic Models

Flashcards: Dynamic Models

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Flashcards: Dynamic Models

Flashcards: Dynamic Models

Flashcards: Dynamic Models

Flashcards: Dynamic Models

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