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Models and model analysis have become an integral component of ecology as tools for understanding complex systems, predicating the future, and understanding the past. Traditional model analysis often involves examining the stability of equilibria. However, few systems in nature ever reach an equilibrium. Some systems exhibit periodic behavior caused by external forces such as seasonality or internal forces such as predator-prey dynamics. Floquet analysis is a tool that may be used to examine the stability of periodic models. Eigenvalues or Floquet multipliers may be calculated based upon the population size at the start and end of a period. If the dominant eigenvalue is greater than one, the system is unstable. Floquet analysis has traditionally been used in fields such as engineering and astrophysics. Only recently has it been applied to ecology, in part due to numerical difficulties in finding Floquet multipliers. These recent ecological applications have been to linear or simple linearized systems. We demonstrate how Floquet analysis may be applied to a nonlinear, stage-structured mosquito population model. The mosquito development rates and mortality rates vary based based upon temperature. We used the Trilinos and Sundance C++ libraries to calculate the Floquet multipliers.
Results/Conclusions
Floquet multipliers indicated that the periodic solution to our model was stable. This was in agreement with other numerical simulations that we have done with the model. Our results demonstrate how Floquet analysis maybe used to understand the model behavior of non-linear models. This technique may be applied to any periodic model, and we recommend that Floquet analysis be used more often in theoretical ecology.
