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.
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 :)