Diff Eq: Cooperative and Competitive Species

System:

A. dx/dt= -5x+2xy B. dx/dt=6x-x^2-4xy
dy/dt= -4y+3xy dy/dt=5y-2xy-2y^2

Discuss the terms in each system, ie, what does the coefficient to the x term in x' represent, and which system is cooperative/competitive?

Determine all relevant equilibrium points and analyze behavior when x0=0 or y0=0. Determine the curves in the phase plan along which the vector field is either horizontal or vertical; which way does the vector field point along these curves?

Describe possible evolution scenarios.

Question

Diff Eq: Cooperative and Competitive Species

System:

A. dx/dt= -5x+2xy            B.  dx/dt=6x-x^2-4xy
    dy/dt= -4y+3xy                  dy/dt=5y-2xy-2y^2

Discuss the terms in each system, ie, what does the coefficient to the x term in x' represent, and which system is cooperative/competitive?

Determine all relevant equilibrium points and analyze behavior when x0=0 or y0=0.  Determine the curves in the phase plan along which the vector field is either horizontal or vertical; which way does the vector field point along these curves?

Describe possible evolution scenarios.

anonymous · Answer

Does anyone have time to review my thoughts as I work on my response? 

In a cooperative system, the coefficient of x in the equation for dx/dt represents the reliance of the two species upon each other.  The coefficient of y in the equation for dy/dt also represents this reliance.  The greater the coefficient, the less valuable/critical one species is to the other; that is, a dx/dt coefficient of x closer to zero will have less reliance on the opposite species for continued growth.  

EPs: A: (0,0)   (-4/3,-3/2)
       B: (0,0)   (0, 5/2)   (6,0)   (2/3, 5/6)

For all cooperative systems, if either population starts at 0, the system behaves as an exponential decay system.  This is logical, since species rely on each other for growth in a cooperative system.