Question

can some on please help me on a "wombat island" question, its a pretty long question, i will show you bit by bit. please help

Answer 1

A model for number of wombats “W” on a island, ‘t’ years after the initial 200 are settled there , takes into account the availability of wombat food. The model is

dW/dt= (m-n-kW)W

Where m is the birth rate and n is the death rate of the wombats. k is a constant related to the amount of food. Suppose the m=0.1 and the n=0.06 and k=0.00005

a) Find an expression for the number of wombats after ‘t’ years?
b) According to the model, find the wombat population after
(i) 10 years
(ii) 100 years

c) At what time is the wombat population increasing most rapidly?

d) Explain the effects of term ‘-kW’ on the growth rate of the wombat population?

Answer 2

come on its my first question , please help someone? >.<

Answer 3

Hello viktorhh, Welcome to Open Study,

Wombats are a function of time W(t) !

\[\frac{\text dW}{\text dt}= (m-n-kW)W \]
\[m=0.1\qquad n=0.06\qquad k=0.00005\]

Answer 4

ye thats right man!

Answer 5

thanks

Answer 6

i think it is probably a good idea to lump the constants \(m\) and \(n\) together
\[m-n=0.1-0.06=0.04=c\]

Answer 7

\[\frac{\text dW}{\text dt}= (c-kW)W\],
now separate the variables
\[\frac{\text dW} {(c-kW)W}={\text dt}\]
and you can integrate
\[\int\frac{\text dW} {(c-kW)W}=\int{\text dt}\]

Answer 8

so, i ll just sub into all the numbers now and get question(a) . thanks

Answer 9

dont put the numbers in yet

Answer 10

how come?

Answer 11

what would happen if you did ?

Answer 12

dont i need to find an expressoin for of number of wombats after ‘t’ years?

Answer 13

yeah before you can determine the number of wombats, you need to turn the differential equation into a equation without derivatives in in
W(t) rather than W'(t)

Answer 14

in it*

Answer 15

ok

Answer 16

you might use partial fractions on the integrand before integrating

Answer 17

i used cas, it come out to be