Dynamic properties of Lotka-Volterra systems corresponding to the colonization model

The colonization model, also known as the Levins model, has been developed to understand the mechanisms that drive species coexistence under interspecific competition. Previous simulation studies have shown that the dynamic properties of the model significantly depend on the encounter mode between propagules and colonization sites. Perfect mass action encounters result in convergence towards equilibrium, while perfect ratio-dependent encounters lead to multiple continuously transient trajectories that depend on the initial condition. In the present study, I investigate the properties of the dynamics by transforming the colonization model into a Lotka-Volterra model. I show that the eigenvalues of the Jacobian matrix indicate stability of the equilibrium under perfect mass action encounters, while the Lyapunov function shows the existence of an infinite number of continuously transient trajectories under perfect ratio-dependent encounters. These results highlight new properties of Lotka-Volterra systems and the colonization model, and provide new insights into the mechanisms and dynamic processes involved in the coexistence of multiple species.