American Institute of Mathematical Sciences

November  2018, 17(6): 2845-2854. doi: 10.3934/cpaa.2018134

Unbounded and blow-up solutions for a delay logistic equation with positive feedback

 1 University of Pannonia, Veszprém, Hungary 2 Shimane University, Matsue, Japan 3 University of Oxford, Oxford, United Kingdom 4 University of Szeged, Szeged, Hungary

* Corresponding author

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.

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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References:
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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