Mathematical Modeling Question concerning 2-gender interactions

Question

I was revising for my Math Modelling term paper, and was doing this particular question in a textbook, where I got stuck for a whole day. Here's a snippet:

Q1 Reference: Topics in Mathematical Modeling by K.K Tung

Answer 1

Hints:

For part a), for individual subpopulation equilibrium, you need: $\displaystyle \frac{dm}{dt} = \frac{df}{dt} = 0$

That gives you:

$$B_f\frac{m^*f^*}{m^*+f^*} = D_ff^*$$

$$B_m\frac{m^*f^*}{m^*+f^*} = D_mm^*$$

Can you manipulate these to eliminate $f^*$ and $m^*$? Try to get an equation in terms of the birth and death rates alone. That is the required condition.

For the second part, let $\displaystyle \frac mf = k$, i.e. $m = kf$. Rewrite the two differential equations only in terms of $f$ and $k$ (use product rule where needed).

Now set $\displaystyle \frac{dk}{dt} = 0$ to find $k^*$. Remember that for this part, $\displaystyle D_f = D_m = D$.

EDIT: I'd like to have hidden this part under a spoiler tag, but I'm not sure how to do it. If anyone wishes to help me by editing this, please feel free.

To find the equilibrium subpopulations for part a), simplify and rearrange the two equations to:

$$\frac{m^*}{m^*+f^*} = \frac{D_f}{B_f}$$

$$\frac{f^*}{m^*+f^*} = \frac{D_m}{B_m}$$

Now add the two to get:

$$\frac{D_f}{B_f} + \frac{D_m}{B_m} = 1$$

which is the required condition.

For part b), by employing product rule to get $\displaystyle \frac{dm}{dt} = k\frac{df}{dt} + f\frac{dk}{dt}$ and then setting $\displaystyle \frac{dk}{dt} = 0$ when $k = k^*$,

we get:

$$\frac{df}{dt} = B_f\frac{k^*}{k^*+1}f - Df$$

and

$$k^*\frac{df}{dt} = B_m\frac{k^*}{k^*+1}f - k^*Df$$

where $D_m$ and $D_f$ have been collapsed to a single parameter $D$.

Multiply the first equation by $k^*$ and equate to get:

$$k^*B_f\frac{k^*}{k^*+1}f -k^*Df = B_m\frac{k^*}{k^*+1}f - k^*Df$$

which simplifies to give $\displaystyle k^* = \frac{B_m}{B_f}$