[R-sig-eco] (Off-topic) Doubt About Isoclines for 2 competing population (Logistic Growth)

[Zoltan Botta-Dukat]

Dear Augusto,

A product is zero only if one of the multipliers is zero.
r2=0 and/or N2=0 are mathematically correct, but biologically 
meaningless solution of the equation.
If r2>0 and N2>0 than the only solution is

(1 - alpha21*N1 - alpha22*N2)=0

Best wishes

Zoltan


2013.08.13. 22:09 keltezéssel, Augusto Ribas írta:
> Hello dear list members, i was reading the book Primer of Ecology with R
> from M Henry H Stevens and got a simple doubt, its not about R but hope
> someone could spare one moment with it, its a little off-topic but maybe
> could of general interest.
>
> On the chapter 5, more specifically 5.2.1 he start to teach about
> iscoclines for 2 populations.
> He set the change of one species to zero and solve for the other. My doubt
> is about this part:
>
> We have logistic growth for one species and one competitor:
>
> Dn2/dt = r2*N2 * (1 - alpha21*N1 - alpha22*N2)
>
> Then he set Dn2/dt=0, zero change and solve for N2, and we got:
>
> 0 = r2*N2 * (1 - alpha21*N1 - alpha22*N2)
>
> The next step is:
>
> 0 = 1 - alpha21*N1 - alpha22*N2
>
> And we end with:
>
> N2 = 1/alpha22 - N1 * alpha21 / alpha22
>
> And this end in a line equation that is the iscocline for this 2 species
> system, the isocline line for species 2.
> I got this part, my doubt is what happens between this two steps:
>
> 0 = r2*N2 * (1 - alpha21*N1 - alpha22*N2)
>
> To
>
> 0 = 1 - alpha21*N1 - alpha22*N2
>
>
> How the r2*N2 that multiply everything in the parenthesis vanishes? does it
> is simply put on the other side and when divided by zero it becomes zero?
> Or there is something more about this?
> I dont know if i got this right.
>


-- 
Botta-Dukát Zoltán
--------------------------------
Ökológiai és Botanikai Intézet
Magyar Tudományos Akadémia
Ökológiai Kutatóközpont
--------------------------------
2163. Vácrátót, Alkotmány u. 2-4.
tel: +36 28 360122/157
fax: +36 28 360110
botta-dukat.zoltan at okologia.mta.hu
www.okologia.mta.hu


Zoltán BOTTA-Dukát
--------------------------------
Institute of Ecology and Botany
Hungarian Academy of Sciences
Centre for Ecological Research
--------------------------------
H-2163 Vácrátót, Alkomány u. 2-4.
HUNGARY
Phone: +36 28 360122/157
Fax..: +36 28 360110
botta-dukat.zoltan at okologia.mta.hu
www.okologia.mta.hu