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 $M_0$ mice, has per-capita birth rate of $8 \frac{1}{week}$ and a per-capita death rate of $2\frac{1}{week}$. Also, 20 mouse traps are set each fortnight and they are always filled.

(a)Write down the word equation for the mice population $M(t)$.

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

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

$$\dfrac{dM}{dt} = 16M - 4M - 10$$ $$M(0) = M_0$$

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

The differential equation I've got is $$\frac{dM}{dt}= 12M -20$$ I don't know what method to use to solve this, I can't figure out if the model is logisitc or some other type.

(d) Find the equilibrium solution $M_e$

I found online that Equilibrium solutions are found by setting the derivative equal to zero and solving the resulting equation. I'll be able to figure that out later once I solve (c)

(e)Find the long-term solution with the dependence on $M_0$. What happens when $M_0 = M_e$

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.

  • $\begingroup$ You have a separable differential equation there. Do you know how to solve that class of equations? $\endgroup$ – Matthew Conroy Mar 19 '17 at 21:02
  • $\begingroup$ What type of model is this? $\endgroup$ – Patrick Moloney Mar 19 '17 at 22:50

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

So from your help I solved the separable differential equation. The final result I got was $$M(t) = \dfrac{e^{12t+12c}+20}{12}$$ The question is asking for it in terms of initial population $M_0$. We know that $M(0)= M_0$. Therefore $$M_0 = \dfrac{e^{12c}+20}{12}$$

Is this what they mean by "in terms of the intial population $M_0$"? Thanks :)


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