Pre-images of invariant sets of a discrete-time two-species competition model

This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, redistribution , reselling , loan, sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. In this paper, we explore the structure of pre-images of invariant sets of a discrete-time two-species competition model with singularity at the origin. We first show that this competition model is persistent with respect to the total population of two species, i.e. all initial conditions in R 2 þ n{ð0; 0Þ} are attracted to a compact set which is bounded away from the origin. Then we study the properties of pre-images of this system and give the explicit structure of all pre-images of invariant sets for this system under certain parameter range. These results are analogous to the one-dimensional discrete system. Our study is the first step to explore the structure of the basins of attractions of interior attractors of a general discrete-time two-species model (e.g. the locally asymptotically stable interior period-2 orbit). Finally, we discuss how our results give useful insights on the future study for coexistence of the species and list some open problems related to our system. 1. Introduction It is common that ecosystems have multiple attractors, in which the final densities of species, or even the persistence of species, depend on their initial densities [8,12]. The basins of attractions of attractors can measure the resilience of ecosystems and the possible initial conditions allowing the coexistence of species. Thus, understanding the structure of the basins of attractions of their attractors can provide important information on the strategies for the sustainable management of such ecosystems [12]. Discrete-time ecological models are known to exhibit multiple attractors including periodic and chaotic attractors. However, except for a few exceptions (e.g. Elaydi and Sacker [4]), the qualitative studies of the structure of the basins of attraction of these periodic and chaotic attractors are rare. Since pre-images of invariant …