Question

Biophysics, population kinetics... I have trouble understanding what my prof used to derive the Laplacian regarding population equations. Can someone here help me figure it out? (The equations follow in next post)

Answer 1

The course uses the example of lynxes-hare connection to explain the equations.

First the initial part I get:
\[(1)\frac{ dN _{H} }{dt }= k _{H}N _{H}-k _{HL}N _{L}N _{H}\]
\[(2)\frac{ dN _{L} }{dt }= k _{LH}N _{H}N _{L}-k _{L,dth}N _{L}\]

stationary equilibrium (these I figured out myself)
\[\frac{ dN _{H} }{dt }=0= k _{H}N _{H}-k _{HL}N _{L}N _{H} -> N _{L}=\frac{ k _{H} }{ k _{HL} }\]
\[\frac{ dN _{L} }{dt }= 0=k _{LH}N _{H}N _{L}-k _{L,dth}N _{L}-> N _{H}=\frac{ k _{L,dth} }{k _{LH} }\]

N_h: number of hares
N_lL: number of lynxes
k_h. N_h: number of births for hares
k_hl . N_l. N_h: number of hares killed by lynxes

The course continues to including the hunt on hares and gives the following equations
\[(3) N _{L,dth}=\frac{ (k _{H}-k _{H,dth}) }{ k _{HL} }\]
and
\[N _{H,dth}=\frac{ k_{L,dth} }{ k_{LH} }\]

So it seems to me that the stationary equations I found are the death equations, except that the hare's also includes human hunting, and not just lynxes hunting them.

Now the part that confuses me: we investigate what happens in case of small distortions to the equilibrium

\[\Delta N _{H}= N _{H}-N _{H,dth}\]
\[\Delta N _{L}= N _{L}-N _{L,dth}\]

Then my course says "from (1) (2) and (3) it follows that"

\[(4) \frac{ d (\Delta N _{H})}{ dt}=-\frac{ k _{HL} k _{L,dth}}{ k _{LH} }\Delta N _{L}\]
\[(5) \frac{ d (\Delta N _{L})}{ dt}=\frac{ k _{LH} k _{H}}{ k _{HL} }\Delta N _{H}\]

The course does mention that for the last two she uses...
\[\frac{ dN _{H} }{ dt }=f _{1}(N _{H},N _{L})\]
\[\frac{ dN _{L} }{ dt }=f _{2}(N _{H},N _{L})\]

and so
\[\frac{ d (\Delta N _{H})}{ dt }=\left( \frac{ \delta f _{1} }{ \delta N _{H} } \right)_{dth} \Delta N _{H}+\left( \frac{ \delta f _{1} }{ \delta N _{L} } \right)_{dth} \Delta N _{L}\]
\[\frac{ d (\Delta N _{L})}{ dt }=\left( \frac{ \delta f _{2} }{ \delta N _{H} } \right)_{dth} \Delta N _{H}+\left( \frac{ \delta f _{2} }{ \delta N _{L} } \right)_{dth} \Delta N _{L}\]

But when I'm confused which equations she uses exactly, because when I use (1) and (2) I don't end up with (4) and (5).