Population Dynamics with Two Competing Species
Consider two types of fish in a pond that do not prey on one another but compete for the available food. The governing equations are:[more]
Restrict the discussion to the first quadrant, since and are species populations.
There are up to four steady states in the first quadrant:
(or and ).
, , and are always in the first quadrant. pertains when both species are present in the first quadrant if and .
Vary the growth rate coefficients , , the self-inhibition parameters , , and the interaction parameters , . This Demonstration gives the corresponding stream plot.
The steady states and nullclines and (blue and brown) are shown in the plot.
From the snapshots, is an unstable node, and are either saddle points or stable nodes, and is either a saddle point or a stable node.
The linearized analysis (see the stability tab) confirm these conclusions. Indeed, for , both eigenvalues are positive. For , , and , there are either two negative eigenvalues or two real eigenvalues of opposite sign.
Coexistence (i.e., steady-state ) is possible only if the self-inhibition term dominates the interaction term ().[less]
 A. Varma and M. Morbidelli, Mathematical Methods in Chemical Engineering, New York: Oxford University Press, 1997.