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patrickmoloney

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## Homework Statement

Hey guys I'm struggling to find much information of modelling single species population dynamics that relates to this question. A question like this is going to be coming up in my final exam and I need to be able to solve it. I'm struggling to even know where to start. I'm completely new to modelling populations. help would be much appreciated.

A population, initially consisting of [itex]M_0[/itex] mice, has per-capita birth rate of [itex]8 \frac{1}{week}[/itex] and a per-capita death rate of [itex]2\frac{1}{week}[/itex]. Also, 20 mouse traps are set each fortnight and they are always filled.

(a)Write down the word equation for the mice population [itex]M(t)[/itex]

(b) Write the differential rate equation for the number of mice.

(c) Solve the differential rate equation to obtain the formula for the mice population [itex]M(t)[/itex] at any time [itex]t[/itex] in terms of the initial population [itex]M_0[/itex]

(d) Find the equilibrium solution [itex]M_e[/itex]

(e)Find the long-term solution with the dependence on [itex]M_0[/itex]. What happens when [itex]M_0 = M_e[/itex]

(f)Find the time interval on which [itex]M(t)\ge 0[/itex] with dependence on [itex]M_0[/itex]

No idea what's being asked here. I can't seem to find any information. It doesn't help that there is no decent books about this online that can be found easily.

thanks for all the help.

## Homework Equations

## The Attempt at a Solution

(a) [tex]\Bigg(

\text{Rate of change}

\text{ of number of mice}

\Bigg) = \Big(\text{Rate of Births}\Big)-\Big(\text{Normal rate of Reaths}\Big) - \Big(\text{Rate of deaths by mousetraps}\Big)

[/tex]

(b) [tex]\dfrac{dM}{dt} = 16M - 4M - 20[/tex] [tex]M(0) = M_0[/tex]

(c) The differential equation I've got is[tex]\frac{dM}{dt}= 12M -20[/tex]. Using separation of variables we can find [itex]M(t)[/itex]

[tex]M(t) = \dfrac{e^{12t+12c}+20}{12}[/tex] The question is asking for it in terms of initial population [itex]M_0[/itex]. We know that [itex]M(0)= M_0[/itex]. Therefore

[tex]M_0 = \dfrac{e^{12c}+20}{12}[/tex]

We know that [itex]e^{c_1}= C[/itex]. Then we can find [itex]M(t)[/itex] in terms of [itex]M_0[/itex]

Which is [tex]M(t) = \dfrac{(12M_0 -20)e^{12t}+20}{12}[/tex]

Is this what they mean by "in terms of the intial population [itex]M_0[/itex]"?

(d) I think this has to do when the rate of change population stops changing. When [itex]\frac{dM}{dt}=0[/itex] I don't know how to find the solutions though.

(e) I'm not sure how to find the long term behaviour, what does the question want me to find?

(f) i also don't know what they are asking here since it's probably related to part (e).

Thanks for all the help. I'm really struggling with this problem. been at it for about 4 hours now and there is nothing online that is even like this question.

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