  Ecological resilience refers to the ability of a system to retain its state
when subject to state variables perturbations or parameter changes. While
understanding and quantifying resilience is crucial to anticipate the possible
regime shifts, characterizing the influence of the system parameters on
resilience is the first step towards controlling the system to avoid
undesirable critical transitions. In this paper, we apply tools of qualitative
theory of differential equations to study the resilience of competing
populations as modeled by the classical Lotka-Volterra system. Within the high
interspecific competition regime, such model exhibits bistability, and the
boundary between the basins of attraction corresponding to exclusive survival
of each population is the stable manifold of a saddle-point. Studying such
manifold and its behavior in terms of the model parameters, we characterized
the populations resilience: while increasing competitiveness leads to higher
resilience, it is not always the case with respect to reproduction. Within a
pioneering context where both populations initiate with few individuals,
increasing reproduction leads to an increase in resilience; however, within an
environment previously dominated by one population and then invaded by the
other, an increase in resilience is obtained by decreasing the reproduction
rate. Besides providing interesting insights for the dynamics of competing
population, this work brings near to each other the theoretical concepts of
ecological resilience and the mathematical methods of differential equations
and stimulates the development and application of new mathematical tools for
ecological resilience.
