Extending Theorems for Strongly Anti-Competitive Maps to Apply to Weakly Anti-Competitive Maps

Chris Lynd

doi:10.5644/SJM.21.02.05

Authors

Chris Lynd Commonwealth University of Pennsylvania, Department of Mathematics, Computer Science, USA

DOI:

https://doi.org/10.5644/SJM.21.02.05

Keywords:

difference equations, competitive, anti-competitive, invariant manifold, fixed-point, bifurcation

Abstract

In the paper Global Dynamics of Anti-Competitive Systems in the Plane [4], the authors proved two major theorems that can be used to determine the global dynamics of anti-competitive systems of difference equations. These theorems require three hypotheses to be satisfied: (1) the corresponding map must be strongly anti-competitive, (2) the determinant of the Jacobian matrix of the map, evaluated at an interior fixed-point, does not equal zero, and (3) the only point mapped onto a fixed-point is the fixed-point itself.
In this paper, we prove theorems that obtain the same results as in [4], but do not require any of these hypotheses; furthermore, the new theorems use weaker hypotheses that extend the scope of the theorems to apply to many more cases. Finally, we demonstrate how to use the modified theorems to determine the global dynamics of a weakly anti-competitive system where hypothesis (1) is false in every region of parameter space.

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References

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