On variational and topological methods in nonlinear difference equations
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November 2018, 17(6): 2845-2854. doi: 10.3934/cpaa.2018134
Unbounded and blow-up solutions for a delay logistic equation with positive feedback
István Győri 1, , Yukihiko Nakata 2, and Gergely Röst 3,4,,
University of Pannonia, Veszprém, Hungary
Shimane University, Matsue, Japan
University of Oxford, Oxford, United Kingdom
University of Szeged, Szeged, Hungary
Received September 2017 Revised January 2018 Published June 2018
We study bounded, unbounded and blow-up solutions of a delay logistic equation without assuming the dominance of the instantaneous feedback. It is shown that there can exist an exponential (thus unbounded) solution for the nonlinear problem, and in this case the positive equilibrium is always unstable. We obtain a necessary and sufficient condition for the existence of blow-up solutions, and characterize a wide class of such solutions. There is a parameter set such that the non-trivial equilibrium is locally stable but not globally stable due to the co-existence with blow-up solutions.
Keywords: Delay logistic equation, stability, blow-up, delay differential equation.
Mathematics Subject Classification: Primary: 34K12, 35B44; Secondary: 34K20.
Citation: István Győri, Yukihiko Nakata, Gergely Röst. Unbounded and blow-up solutions for a delay logistic equation with positive feedback. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2845-2854. doi: 10.3934/cpaa.2018134
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Figure 2.1. Stability region for the positive equilibrium in the $(\alpha,r)$-parameter plane. The shaded region is the stability region given by (2.1) and (2.2). The positive equilibrium is globally stable for $\alpha\le-1$ and is unstable above the stability boundary. Exponential solutions exist on the denoted curve. Blow-up solutions exist for $\alpha>0$, hence we can observe a region where the positive equilibrium is locally stable yet blow-up solutions also exist
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István Győri Yukihiko Nakata Gergely Röst