Dynamics of delayed models in ecology, epidemiology, and cytology: Existence of non-positive solutions

Abstract

Delay differential equations are widely used to model processes which are influenced by past states. Time delays can alter fundamental properties and introduce new dynamics in a delayed system over its corresponding non-delayed model. Positivity of solutions is an essential property in ecological and epidemiological models. However positive solution is often unchecked or it is incorrectly established. Our extensive literature review of delayed models in ecology, epidemiology, and cancer research reveals that many studies either: (i) do not discuss the positivity of solutions, (ii) assume the positivity of solutions trivially true, or (iii) provide flawed proofs. Hence, the positivity of solutions for delayed models is still poorly understood. We uncover, for the first time, that a class of delayed systems can exhibit (i) non-positive transient solutions converging to a positive equilibrium, (ii) non-positive transient solutions leading to a positive limit cycle, and (iii) non-positive limit cycles. To address these issues, we propose a refined approach for proving positivity in certain delayed models. We believe that, the ambiguities and challenges we have addressed here, will benefit future researchers while developing and exploring delayed models. © The Author(s), under exclusive licence to Springer Nature B.V. 2025.