FULL QUESTION..{plzz heLp"}

[shivastar]

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?

 

[tkhunny]

As shown yesterday, this is a seperable differential equation. The solution was given as an expression that can be simplified to:

[1/(m-n)]*ln((n-m+k*W)/W) = t+C

This is where we have a problem. The problem statement, as presented, provides no initial values to solve for 'C'. Generally, this should be considered a catastrophic deficiency and there is no more we can do to solve the problem.

Just for today, let's assume C = 0, mostly because I could not find a value for C that helped this doomed population. C = 0 seems simplest, for now. This gives:

[1/(m-n)]*ln((n-m+k*W)/W) = t

Solving for W (really, W(t)), gives:

W(t) = (n-m)/[e^t*(n-m)-k]

Experiment with that. What is the difference when k = 0 and k ≠ 0?

I think there is something wrong with this model unless we are trying to kill wombats. For one thing, with the parameters you have provided, there are no wombats until t > 247. Also, k can't be zero (0).