Persistence and stability for a class of forced positive nonlinear delay-differential systems

Franco, Daniel; Guiver, Chris; Logemann, Hartmut

Abstract

Persistence and stability properties are considered for a class of forced positive non-linear delay-differential systems which arise in mathematical ecology and other applied contexts.The inclusion of forcing incorporates the effects of control actions (such as harvesting or breeding programmes in an ecological setting), disturbances induced by seasonal or environmental variation, or migration. We provide necessary and sufficient conditions under which the states of these models are semi-globally persistent, uniformly with respect to the initial conditions and forcing terms. Under mild assumptions, the model under consideration naturally admits two steady states (equilibria) when unforced: the origin and a unique non-zero steady state.We present sufficient conditions for the non-zero steady state to be stable in a sense which is reminiscent of input-to-state stability, a stability notion for forced systems developed in control theory. In the absence of forcing, our input-to-sate stability concept is identical to semi-global exponential stability.

Citation

Franco, D., Guiver, C., & Logemann, H. (2021). Persistence and stability for a class of forced positive nonlinear delay-differential systems. Acta applicandae mathematicae, 174(1), Article 1 (2021). https://doi.org/10.1007/s10440-021-00414-5

• Journal Article Type: Article
• Acceptance Date: May 18, 2021
• Online Publication Date: Jun 21, 2021
• Publication Date: 2021-08
• Deposit Date: May 19, 2021
• Publicly Available Date: Jun 21, 2021
• Journal: Acta Applicandae Mathematicae
• Print ISSN: 0167-8019
• Electronic ISSN: 1572-9036
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 174
• Issue: 1
• Article Number: 1 (2021)
• DOI: https://doi.org/10.1007/s10440-021-00414-5
• Keywords: Delay-differential systems, density-dependent population models, environmentalforcing, forced systems, input-to-state stability, persistence, positive systems, semi-global expo-nential stability
• Public URL: http://researchrepository.napier.ac.uk/Output/2773685