In reviewing my biophysics' course on population kinetics I am stuck in finding which equation was actually used to derive from. It uses an example to "explain" the analytical method, in order to apply it on two exercises. So, my problem and question is analytically related, to which no one at other sites for biology, physics or even math have been able to help me.
The example is about a model of hares ($N_{H}$) and lynxes ($N_{L}$) (predator prey) and involving the impact of human hunting on either two.
The course claims to use several equations to derive the impact of little distortions ($\Delta N_{H}$ and $\Delta N_{L}$) on an equilibrium situation.
$\frac{d(\Delta N_{H})}{dt}=-\frac{k_{HL}k_{L,dth}}{k_{LH}}\Delta N_{L}$
$\frac{d(\Delta N_{L})}{dt}=\frac{k_{LH}k_{H}}{k_{HL}}\Delta N_{H}$
It is these resulting equations I'm mindboggled about regarding how she ever got to them. The k's are rates. I'll explain what they are a little more down this post.
The course claims to have used the following theoretical derivations:
$\frac{d(\Delta N_{H})}{dt}=(\frac{\partial f_{1}}{\partial N_{H}})_{dth} \Delta N_{H} + (\frac{\partial f_{1}}{\partial N_{L}})_{dth} \Delta N_{L}$
$\frac{d(\Delta N_{L})}{dt}=(\frac{\partial f_{2}}{\partial N_{H}})_{dth} \Delta N_{H} + (\frac{\partial f_{2}}{\partial N_{L}})_{dth} \Delta N_{L}$
with $f_{1}$ and $f_{2}$ being
$\frac{dN_{H}}{dt}=f_{1}(N_{H},N_{L})= k_{H}N_{H}-k_{HL}N_{L}N_{H}$
$\frac{dN_{L}}{dt}=f_{2}(N_{H},N_{L})= k_{LH}N_{H}N_{L}-k_{L,dth}N_{L}$
The first term in both equations result in the number of hares, respectively lynxes born every year, and the second term term the number of animals dying.
At equilibrium the last two equations become $0$ and thus
$N_{H,dth}=\frac{k_{L,dth}}{k_{LH}}$
$N_{L,dth}=\frac{k_{H}}{k_{HL}}$
When I partially derive $f_{1}$ and $f_{2}$ and plug in the equilibrium values I get
$\frac{d(\Delta N_{H})}{dt}=(k_{H}-k_{HL} N_{L})\Delta N_{H} - (k_{HL}N_{H}) \Delta N_{L}$
$=(k_{H}-\frac{k_{HL}k_{H}}{k_{HL}})\Delta N_{H} - \frac{k_{HL}k_{L,dth}}{k_{LH}} \Delta N_{L}$
$=(k_{H}-k_{H})\Delta N_{H} - \frac{k_{HL}k_{L,dth}}{k_{LH}} \Delta N_{L}$
When I compare this with the result I'm supposed to get, I'm supposed to drop the whole first term, which is possible since the first term $=0$
For the second function it is
$\frac{d(\Delta N_{L})}{dt}=(k_{LH} N_{L})\Delta N_{H} - k_{L,dth} \Delta N_{L}$
$=(\frac{k_{LH} k_{H}}{k_{HL}})\Delta N_{H} - k_{L,dth} \Delta N_{L}$
Again comparing my result I have with what I'm supposed to get, I'm supposed to drop a term, the second one this time. But this time I don't understand why.
Please, can someone explain this to me?
ETA the meaning of the k's and N's. $N_H$:number of hares $N_L$:number of lynxes $k_{H}$:rate of births of hares a year per hare $k_{HL}$: rate of hares taken per lynx per year $k_{LR}$: rate of births of lynxes per year per lynx $k_{L,dth}$: rate of lynxes dying per year
I emailed my professor about it, and it turns out I made a mistake in my partial derivation of f2 $\frac{dN_{L}}{dt}=k_{LH}N_{H}N_{L}−k_{L,dth}N_{L}$
It ought to have been
$\frac{d(ΔN_{L})}{dt}=(k_{LH}N_{L})ΔN_{H}+(k_{LH}N_{H}−k_{L,dth})ΔN_{L}$
Substituting the equilibrium equations for $N_{L}$ and $N_{H}$ it becomes
$=(\frac{k_{LH}k_{H}}{k_{HL}})ΔN_{H}+(k_{LH}\frac{k_{L,dth}}{k_{LH}}-k_{L,dth})ΔN_{L}$
$=(\frac{k_{LH}k_{H}}{k_{HL}})ΔN_{H}+ 0 ΔN_{L}$
$=(\frac{k_{LH}k_{H}}{k_{HL}})ΔN_{H}$
I forgot to partially derive the first term of f2 for $N_{L}$