Predator-prey system

[justme.h]

Hello,
I have to solve the following task with predator-prey method.

The dynamics of self-regulating "predator-prey" populations in the population is described by the model:
dN1/dt = (a - bN2 - αN1); dN2/dt = (-c + mN1)N2 (1),
where α is coefficient of internal victim struggle,
and a>0,b>0,α>0,c>0,m>0
With this change N1=k1x, N2=k2y,t=k3z (2) the system (1) can be reduced to: dx/dz=x(E - Ax - y); dy/dz=y(-1 + x) (3)
And here is the questions:
1)Find the coefficients ki of element (2), where i =1,3 (irrationals)
2)Find the relationship/connection between params A and E from (3) and the params from (1)
3)Find the equilibrium (specific points) of system (3)
4)Examine the stability of the equilibrium position of (3)
5)Build a Phase Portrait of (3)

Thank you in advance!

 

[HallsofIvy]

If \(\displaystyle N_1= k_1x\) then \(\displaystyle dN_1/dt= k_1(dx/dt)\) and with \(\displaystyle t= k_3z\), \(\displaystyle dx/dt= dx/dz (dz/dt)= (1/k_3)dx/dz\). So \(\displaystyle dN_1/dt= (k_1/k_3)(dx/dz)\)
If \(\displaystyle N_2= k_2y\) then \(\displaystyle dN_2/dt= k_2(dx/dt)\) so \(\displaystyle dN_2/dt= (k_2/k_3)(dy/dz)\).

Put those into \(\displaystyle dN_1/dt= (a+ bN_1+ \alpha N_2)\) becomes \(\displaystyle (k_1/k_3)(dx/dz)= a+ bk_1x+ \alpha k_2y\) .

 

[justme.h]

If \(\displaystyle N_1= k_1x\) then \(\displaystyle dN_1/dt= k_1(dx/dt)\) and with \(\displaystyle t= k_3z\), \(\displaystyle dx/dt= dx/dz (dz/dt)= (1/k_3)dx/dz\). So \(\displaystyle dN_1/dt= (k_1/k_3)(dx/dz)\)
If \(\displaystyle N_2= k_2y\) then \(\displaystyle dN_2/dt= k_2(dx/dt)\) so \(\displaystyle dN_2/dt= (k_2/k_3)(dy/dz)\).

Put those into \(\displaystyle dN_1/dt= (a+ bN_1+ \alpha N_2)\) becomes \(\displaystyle (k_1/k_3)(dx/dz)= a+ bk_1x+ \alpha k_2y\) .

Hello, thank you for your answer. Your explanation for which point 1-5 is?

 

[HallsofIvy]

I have no idea what you mean by "point 1- 5".

 

[LCKurtz]

There were 5 tasks the problem listed at the end of the original post.

 

[justme.h]

I have no idea what you mean by "point 1- 5".

I have 5 tasks (points) to do.

1)Find the coefficients ki of element (2), where i =1,3 (irrationals)
2)Find the relationship/connection between params A and E from (3) and the params from (1)
3)Find the equilibrium (specific points) of system (3)
4)Examine the stability of the equilibrium position of (3)
5)Build a Phase Portrait of (3)

And i did not understand which point of the task you are answering.