Another assumption, as it guarantees positive alpha, all interactions must be harmful and thus competitive.
Smale showed that for N greater than or equal to 5 this model can exhibit any asymptotic behavior including fixed point or limit-cycle. One significant issue with this model is that no equilibrium point is stable and it assumes there is no environmental complexity. The type I and II functional responses in this model differ as to whether the growth rate of one population will continue to increase linearly with increasing density in the other population.  Another assumption, as it guarantees positive alpha, all interactions must be harmful and thus competitive. For the associated interaction matrix, one can pull in the relevant carrying capacity of each species without changing the equations. For 10 points, name this doubly eponymous set of equations which model predator-prey interactions.
Smale showed that for N greater than or equal to 5 this model can exhibit any asymptotic behavior including fixed point or limit-cycle.
{Lotka}-{Volterra} Equations or {competition} model (prompt on {predator}-{prey equations} until mentioned)