Vector/Pest control is essential to reduce the risk of vector-borne diseases or losses in crop fields. Among biological control tools, the sterile insect technique (SIT), is the most promising one. SIT control generally consists of massive releases of sterile insects in the targeted area in order to reach elimination or to lower the pest population under a certain threshold. The models presented here are minimalistic with respect to the number of parameters and variables. The first model deals with the dynamics of the vector population while the second model, the SIT model, tackles the interaction between treated males and wild female vectors.For the vector population model, the elimination equilibrium 0 is globally asymptotically stable when the basic offspring number, R, is lower or equal to one, whereas 0 becomes unstable and one stable positive equilibrium exists, with well determined basins of attraction, when R > 1. For the SIT model, we obtain a threshold number of treated male vectors above which the control of wild female vectors is effective: the massive release control. When the amount of treated male vectors is lower than the aforementioned threshold number, the SIT model experiences a bistable situation involving the elimination equilibrium and a positive equilibrium. However, practically, massive releases of sterile males are only possible for a short period of time. That is why, using the bistability property, we develop a new strategy to maintain the wild population under a certain threshold, for a permanent and sustainable low level of SIT control. We illustrate our theoretical results with numerical simulations, in the case of SIT mosquito control.

We present and analyze a model aiming at recovering as dynamical outcomes of tree-grass interactions the wide range of vegetation physiognomies observable in the savanna biome along rainfall gradients at regional/continental scales. The model is based on two ordinary differential equations (ODE), for woody and grass biomass. It is parameterized from literature and retains mathematical tractability, since we restricted it to the main processes, notably tree-grass asymmetric interactions (either facilitative or competitive) and the grass-fire feedback. We used a fully qualitative analysis to derive all possible long term dynamics and express them in a bifurcation diagram in relation to mean annual rainfall and fire frequency. We delineated domains of monostability (forest, grassland, savanna), of bistability (e.g. forest-grassland or forest-savanna) and even tristability. Notably, we highlighted regions in which two savanna equilibria may be jointly stable (possibly in addition to forest or grassland). We verified that common knowledge about decreasing woody biomass with increasing fire frequency is recovered for all levels of rainfall, contrary to previous attempts using analogous ODE frameworks. Thus, this framework appears able to render more realistic and diversified outcomes than often thought of. Our model can help figure out the ongoing dynamics of savanna vegetation in large territories for which local data are sparse or absent. To explore the bifurcation diagram with different combinations of the model parameters, we have developed a user-friendly R-Shiny application freely available at : https://gitlab.com/cirad-apps/tree-grass.

Many systems in life sciences have been modeled by reaction-diffusion equations. However, under some circumstances, these biological systems may experience instantaneous and periodic perturbations (e.g. harvest, birth, release, fire events, etc) such that an appropriate formalism like impulsive reaction-diffusion equations is necessary to analyze them. While several works tackled the issue of traveling waves for monotone reaction-diffusion equations and the computation of spreading speeds, very little has been done in the case of monotone impulsive reaction-diffusion equations. Based on vector-valued recursion equations theory, we aim to present in this paper results that address two main issues of monotone impulsive reaction-diffusion equations. Our first result deals with the existence of traveling waves for monotone systems of impulsive reaction-diffusion equations. Our second result tackles the computation of spreading speeds for monotone systems of impulsive reaction-diffusion equations. We apply our methodology to a planar system of impulsive reaction-diffusion equations that models tree-grass interactions in fire-prone savannas. Numerical simulations, including numerical approximations of spreading speeds, are finally provided in order to illustrate our theoretical results and support the discussion.

Many systems in life sciences have been modeled by reaction-diffusion equations. However, under some circumstances, these biological systems may experience instantaneous and periodic perturbations (e.g. harvest, birth, release, fire events, etc) such that an appropriate formalism is necessary, using, for instance, impulsive reaction-diffusion equations. While several works tackled the issue of traveling waves for monotone reaction-diffusion equations and the computation of spreading speeds, very little has been done in the case of monotone impulsive reaction-diffusion equations. Based on vector-valued recursion equations theory, we aim to present in this paper results that address two main issues of monotone impulsive reactiondiffusion equations. First, they deal with the existence of traveling waves for monotone systems of impulsive reaction-diffusion equations. Second, they allow the computation of spreading speeds for monotone systems of impulsive reaction-diffusion equations. We apply our methodology to a planar system of impulsive reaction-diffusion equations that models tree-grass interactions in fire-prone savannas. Numerical simulations, including numerical approximations of spreading speeds, are finally provided in order to illustrate our theoretical results and support the discussion.