Ecosystem word problem

[jarettbrock]

In the studies of ecosystem,s predator-prey models are often used to study the interactions between species. Consider a population of tundra wolves, given by W(t), and caribou, given by C(t) in northern Canada. The interaction has been modeled by the equations
(dC/dt)= aC-bCW (DW/dt)= -cW + dCW

A) What values of (dC/dt) and (dW/dt) correspond to stable populations?
B) How would the statement "The caribou go extinct" be represnted mathematically?
C) Suppose that a=0.05, b=0.001, c=0.05, and d=0.0001. Find all population pair (C,W) that lead to stable populations. According to this model, is it possible for the species to live in harmont or will on or both species become extinct?



We have a big set of problems due and they are all for accuracy and i cant seem to figure this one out, maybe its because ive been trying for like 30 minutes and my mind is fixed on only a couple of approaches. Help is greatly appreciated.
You guys who help on math are SAMARITANS!

 

[Deleted member 4993]

In the studies of ecosystem,s predator-prey models are often used to study the interactions between species. Consider a population of tundra wolves, given by W(t), and caribou, given by C(t) in northern Canada. The interaction has been modeled by the equations
(dC/dt)= aC-bCW (DW/dt)= -cW + dCW

A) What values of (dC/dt) and (dW/dt) correspond to stable populations?

What is the definition (criterion) for stability? What happens to change during stability? What happens to rate of change?

B) How would the statement "The caribou go extinct" be represnted mathematically?
C) Suppose that a=0.05, b=0.001, c=0.05, and d=0.0001. Find all population pair (C,W) that lead to stable populations. According to this model, is it possible for the species to live in harmont or will on or both species become extinct?



We have a big set of problems due and they are all for accuracy and i cant seem to figure this one out, maybe its because ive been trying for like 30 minutes and my mind is fixed on only a couple of approaches. Help is greatly appreciated.
You guys who help on math are SAMARITANS!

Please show us your work, indicating exactly where you are stuck - so that we know where to begin to help you.

 

[jarettbrock]

O, ill start with part (a). Im not exactly sure what is being asked.. Am i supposed to just plug in constants into the equation and see what makes it equal? I dont understand where i am supposed to get numbers from when im only given variables.

 

[Deleted member 4993]

What do you understand by the statement "stable population"?

 

[jarettbrock]

I think it means that the population of the caribou and wolves will be the same?

 

[Deleted member 4993]

From answers.com

stable population:This type of population will grow at a constant rate.

What does your text-book say?

 

[jarettbrock]

our text book doesnt have anything relating to stable population problems. my teacher expects us to be able to go home and understand a large variety of different problems from one or two examples. he is a firm believer in trial and error. but in times like these, i dont even know what to do cause its so different. and he counts our problem sets for accuracy so it pisses me off when this kind of crap happens.

 

[BigGlenntheHeavy]

$\displaystyle A)I'm \ assuming \ stable \ populations\ means \ a \ state \ of \ equilibrium \ exists \ between \ prey \ and$$\displaystyle predator.$

$\displaystyle Hence, \ C \ = \ 0 \ and \ W \ = \ 0, \ then \frac{dC}{dt} \ = \ 0 = \ \frac{dW}{dt}$

$\displaystyle This \ makes \ sense, \ for \ if \ we \ have \ no \ caribou, \ the \ wolves \ starve \ off \ and \ we \ have \ a \ stable$$\displaystyle \ population, \ namely \ zero.$

$\displaystyle We \ will \ also \ have \ a \ stable \ population \ if \ the \ wolves \ stay \ at \ \frac{a}{b} \ and \ the \ caribou \ stay \ at \ \frac{c}{d},$$\displaystyle as \ each \ species \ birth \ rate \ is \ exactly \ equal \ to \ its \ death \ rate, \ and \ these \ populations$$\displaystyle will \ be \ maintained \ indefinitely.$

$\displaystyle B)\frac{dC}{dt} \ = \ 0, \ implies \ that\frac{dW}{dt} \ = \ -cW$

$\displaystyle C) \frac{dC}{dt} \ = \ .05C-.001CW \ and \ \frac{dW}{dt} \ = \ -.05W+.0001CW$

$\displaystyle If (C,W) \ = \ (0,0) \ or \ (500,50), \ we \ will \ have \ stable \ populations.$