# 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
##### References:
 [1] J. A. D. Appleby, I. Győri and D. W. Reynolds, History-dependent decay rates for a logistic equation with infinite delay, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 23-44. doi: 10.1017/S0308210510000016. [2] B. Bánhelyi, T. Csendes, T. Krisztin and A. Neumaier, Global attractivity of the zero solution for Wright's equation, SIAM Journal on Applied Dynamical Systems, 13 (2014), 537-563. doi: 10.1137/120904226. [3] O. Diekmann, S. A. van Gils, S. M. V. Lunel and H. O. Walther, Delay Equations Functional, Complex and Nonlinear Analysis, Springer Verlag, 1991. doi: 10.1007/978-1-4612-4206-2. [4] T. Faria and E. Liz, Boundedness and asymptotic stability for delayed equations of logistic type, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 1057-1073. doi: 10.1017/S030821050000281X. [5] E. Liz and G. Röst, Dichotomy results for delay differential equations with negative Schwarzian, Nonlinear Analysis: Real World Applications, 11 (2010), 1422-1430. doi: 10.1016/j.nonrwa.2009.02.030. [6] K. Gopalsamy, Stability and Oscillation in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, 1992. doi: 10.1007/978-94-015-7920-9. [7] I. Győri, A new approach to the global asymptotic stability problem in a delay Lotka-Volterra differential equation, Mathematical and Computer Modelling, 31 (2000), 9-28. doi: 10.1016/S0895-7177(00)00043-1. [8] I. Győri and F. Hartung, Fundamental solution and asymptotic stability of linear delay differential equations, Dynamics of Continuous, Discrete and Impulsive Systems, 13 (2006), 261-288. [9] I. Győri and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford, 1991. [10] X. He, Global stability in nonautonomous Lotka-Volterra systems of "pure-delay type", Differential and Integral Equations, 11 (1998), 293-310. [11] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, San Diego, 1993. [12] S. M. Lenhart and C. C. Travis, Global stability of a biological model with time delay, Proc. Amer. Math. Sot., 96 (1986), 75-78. doi: 10.2307/2045656. [13] H. Li and R. Yuan, An affirmative answer to the extended Gopalsamy and Liu's conjecture on the global asymptotic stability in a population model, Nonlinear Anal. Real World Appl., 11 (2010), 3295-3308. doi: 10.1016/j.nonrwa.2009.11.022. [14] S. Ruan, Delay differential equations in single species dynamics, In Delay Differential Equations and Applications, Springer, (2006), 477-517. doi: 10.1007/1-4020-3647-7_11. [15] G. Stépán, Retarded Dynamical Systems: Stability and Characteristic Function, Wiley, New York, 1989. [16] Z. Teng, Permanence and stability in non-autonomous logistic systems with infinite delay, Dyn. Syst., 17 (2002), 187-202. doi: 10.1080/14689360110102312. [17] J. B. van den Berg and J. Jaquette, A proof of Wright's conjecture arXiv: 1704.00029v1 [mathDS] 31 Mar 2017. doi: 10.1016/j.jde.2018.02.018.

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##### References:
 [1] J. A. D. Appleby, I. Győri and D. W. Reynolds, History-dependent decay rates for a logistic equation with infinite delay, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 23-44. doi: 10.1017/S0308210510000016. [2] B. Bánhelyi, T. Csendes, T. Krisztin and A. Neumaier, Global attractivity of the zero solution for Wright's equation, SIAM Journal on Applied Dynamical Systems, 13 (2014), 537-563. doi: 10.1137/120904226. [3] O. Diekmann, S. A. van Gils, S. M. V. Lunel and H. O. Walther, Delay Equations Functional, Complex and Nonlinear Analysis, Springer Verlag, 1991. doi: 10.1007/978-1-4612-4206-2. [4] T. Faria and E. Liz, Boundedness and asymptotic stability for delayed equations of logistic type, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 1057-1073. doi: 10.1017/S030821050000281X. [5] E. Liz and G. Röst, Dichotomy results for delay differential equations with negative Schwarzian, Nonlinear Analysis: Real World Applications, 11 (2010), 1422-1430. doi: 10.1016/j.nonrwa.2009.02.030. [6] K. Gopalsamy, Stability and Oscillation in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, 1992. doi: 10.1007/978-94-015-7920-9. [7] I. Győri, A new approach to the global asymptotic stability problem in a delay Lotka-Volterra differential equation, Mathematical and Computer Modelling, 31 (2000), 9-28. doi: 10.1016/S0895-7177(00)00043-1. [8] I. Győri and F. Hartung, Fundamental solution and asymptotic stability of linear delay differential equations, Dynamics of Continuous, Discrete and Impulsive Systems, 13 (2006), 261-288. [9] I. Győri and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford, 1991. [10] X. He, Global stability in nonautonomous Lotka-Volterra systems of "pure-delay type", Differential and Integral Equations, 11 (1998), 293-310. [11] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, San Diego, 1993. [12] S. M. Lenhart and C. C. Travis, Global stability of a biological model with time delay, Proc. Amer. Math. Sot., 96 (1986), 75-78. doi: 10.2307/2045656. [13] H. Li and R. Yuan, An affirmative answer to the extended Gopalsamy and Liu's conjecture on the global asymptotic stability in a population model, Nonlinear Anal. Real World Appl., 11 (2010), 3295-3308. doi: 10.1016/j.nonrwa.2009.11.022. [14] S. Ruan, Delay differential equations in single species dynamics, In Delay Differential Equations and Applications, Springer, (2006), 477-517. doi: 10.1007/1-4020-3647-7_11. [15] G. Stépán, Retarded Dynamical Systems: Stability and Characteristic Function, Wiley, New York, 1989. [16] Z. Teng, Permanence and stability in non-autonomous logistic systems with infinite delay, Dyn. Syst., 17 (2002), 187-202. doi: 10.1080/14689360110102312. [17] J. B. van den Berg and J. Jaquette, A proof of Wright's conjecture arXiv: 1704.00029v1 [mathDS] 31 Mar 2017. doi: 10.1016/j.jde.2018.02.018.
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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