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Global dynamics of a novel delayed logistic equation arising from cell biology. (English) Zbl 1448.92170

The paper studies a new type of delay logistic equation including well-posedness, existence, and local stability of the equilibrium.
The asymptotic behavior of solutions around the global attractor also investigated.

MSC:

92D25 Population dynamics (general)
92C37 Cell biology
34K20 Stability theory of functional-differential equations

References:

[1] Arino, J.; Wang, L.; Wolkowicz, Gs, An alternative formulation for a delayed logistic equation, J. Theor. Biol., 241, 1, 109-119 (2006) · Zbl 1447.92326 · doi:10.1016/j.jtbi.2005.11.007
[2] Bacaër, N., A Short History of Mathematical Population Dynamics (2011), Berlin: Springer, Berlin · Zbl 1321.92028
[3] Baker, Re; Simpson, Mj, Correcting mean-field approximations for birth-death-movement processes, Phys. Rev. E, 82, 4, e041905 (2010) · doi:10.1103/PhysRevE.82.041905
[4] Bánhelyi, B.; Csendes, T.; Krisztin, T.; Neumaier, A., Global attractivity of the zero solution for Wright’s equation, SIAM J. Appl. Dyn. Syst., 13, 1, 537-563 (2014) · Zbl 1301.34094 · doi:10.1137/120904226
[5] Chow, S-N; Mallet-Paret, J., Integral averaging and bifurcation, J. Differ. Equ., 26, 112-0159 (1977) · Zbl 0367.34033 · doi:10.1016/0022-0396(77)90101-2
[6] Diekmann, O.; Van Gils, Sa; Lunel, Sm; Walther, Ho, Delay Equations: Functional, Complex, and Nonlinear Analysis (2012), Berlin: Springer, Berlin
[7] Erneux, T., Applied Delay Differential Equations (2009), Berlin: Springer, Berlin · Zbl 1201.34002
[8] Faria, T., Asymptotic stability for delayed logistic type equations, Math. Comput. Model., 43, 3-4, 433-445 (2006) · Zbl 1145.34043 · doi:10.1016/j.mcm.2005.11.006
[9] Faria, T.; Huang, W.; Wu, J., Travelling waves for delayed reaction-diffusion equations with global response, Proc. R. Soc. Lond. A Math. Phys. Eng. Sci., 462, 2065, 229-261 (2006) · Zbl 1149.35368 · doi:10.1098/rspa.2005.1554
[10] Faria, T.; Huang, W.; Wu, J., Smoothness of center manifolds for maps and formal adjoints for semilinear FDEs in general Banach spaces, SIAM J. Math. Anal., 34, 1, 203-1730 (2002) · Zbl 1085.34064 · doi:10.1137/S0036141001384971
[11] Faria, T.; Trofimchuk, S., Nonmonotone travelling waves in a single species reaction-diffusion equation with delay, J. Differ. Equ., 228, 1, 357-376 (2006) · Zbl 1217.35102 · doi:10.1016/j.jde.2006.05.006
[12] Farin, A.; Suzuki, So; Weiker, M.; Goldman, Je; Bruce, Jn; Canoll, P., Transplanted glioma cells migrate and proliferate on host brain vasculature: a dynamic analysis, Glia, 53, 8, 799-808 (2006) · doi:10.1002/glia.20334
[13] Fowler, Ac, An asymptotic analysis of the delayed logistic equation when the delay is large, IMA J. Appl. Math., 28, 1, 41-49 (1982) · Zbl 0488.34075 · doi:10.1093/imamat/28.1.41
[14] Geritz, Sa; Kisdi, É., Mathematical ecology: why mechanistic models?, J. Math. Biol., 65, 6, 1411-1415 (2012) · Zbl 1294.92030 · doi:10.1007/s00285-011-0496-3
[15] Giese, A.; Bjerkvig, R.; Berens, Me; Westphal, M., Cost of migration: invasion of malignant gliomas and implications for treatment, J. Clin. Oncol., 21, 1624-1636 (2003) · doi:10.1200/JCO.2003.05.063
[16] Gopalsamy, K.; Zhang, Bg, On a neutral delayed logistic equation, Dyn. Stab. Syst., 2, 3-4, 183-195 (1988) · Zbl 0665.34066
[17] Grotta-Ragazzo, C.; Malta, Cp; Pakdaman, K., Metastable periodic patterns in singularly perturbed delayed equations, J. Dyn. Differ. Equ., 22, 2, 203-252 (2010) · Zbl 1220.34100 · doi:10.1007/s10884-010-9158-1
[18] Gurney, Wsc; Blythe, Sp; Nisbet, Rm, Nicholson’s blowflies revisited, Nature, 287, 5777, 17-21 (1980) · doi:10.1038/287017a0
[19] Győri, I.; Nakata, Y.; Röst, G., Unbounded and blow-up solutions for a delayed logistic equation with positive feedback, Commun. Pure Appl. Anal., 17, 6, 2845-2854 (2018) · Zbl 1397.34114 · doi:10.3934/cpaa.2018134
[20] Győri, I.; Pituk, M., \(L^2\)-perturbation of a linear delay differential equation, J. Math. Anal. Appl., 195, 415-427 (1995) · Zbl 0853.34070 · doi:10.1006/jmaa.1995.1364
[21] Hale, Jk, Asymptotic Behavior of Dissipative Systems. Mathematical Surveys and Monographs (1988), Providence, RI: American Mathematical Society, Providence, RI · Zbl 0642.58013
[22] Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and applications of Hopf bifurcation (Vol. 41). CUP Archive (1981) · Zbl 0474.34002
[23] Holmes, Mh, Introduction to Perturbation Methods (2012), Berlin: Springer, Berlin
[24] Hutchinson, Ge, Circular causal systems in ecology, Ann. N. Y. Acad. Sci., 50, 221-246 (1948) · doi:10.1111/j.1749-6632.1948.tb39854.x
[25] Ivanov, A.; Liz, E.; Trofimchuk, S., Halanay inequality, Yorke 3/2 stability criterion, and differential equations with maxima, Tohoku Math. J. Second Ser., 54, 2, 277-295 (2002) · Zbl 1025.34078 · doi:10.2748/tmj/1113247567
[26] Krisztin, T.; Walther, Ho; Wu, J., Shape, Smoothness, and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback (1999), Providence: American Mathematical Society, Providence · Zbl 1004.34002
[27] Kuang, Y., Delay Differential Equations: with Applications in Population Dynamics (1993), Cambridge: Academic Press, Cambridge · Zbl 0777.34002
[28] Lessard, Jp, Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright’s equation, J. Differ. Equ., 248, 5, 992-1016 (2010) · Zbl 1200.34078 · doi:10.1016/j.jde.2009.11.008
[29] Lin, Cj; Wang, L.; Wolkowicz, Gs, An alternative formulation for a distributed delayed logistic equation, Bull. Math. Biol., 80, 7, 1713-1735 (2018) · Zbl 1396.92071 · doi:10.1007/s11538-018-0432-4
[30] Lindström, T., Monotone dynamics or not? dynamical consequences of various mechanisms for delayed logistic growth, Differ. Equ. Appl., 9, 379-382 (2017) · Zbl 1382.34072
[31] Liz, E., Delayed logistic population models revisited, Publicacions Matemàtiques, EXTRA, 309-331 (2014) · Zbl 1316.34088 · doi:10.5565/PUBLMAT_Extra14_17
[32] May, Rm, Stability and Complexity in Model Ecosystems (1973), Princeton: Princeton University Press, Princeton
[33] Morozov, Ay; Banerjee, M.; Petrovskii, Sv, Long-term transients and complex dynamics of a stage-structured population with time delay and the Allee effect, J. Theor. Biol., 396, 116-124 (2016) · Zbl 1343.92428 · doi:10.1016/j.jtbi.2016.02.016
[34] Noren, Dp; Chou, Wh; Lee, Sh; Qutub, Aa; Warmflash, A.; Wagner, Ds; Popel, Sp; Levchenko, A., Endothelial cells decode VEGF-mediated Ca2+ signaling patterns to produce distinct functional responses, Sci. Signal., 9, 416, ra20 (2016) · doi:10.1126/scisignal.aad3188
[35] Nussbaum, Rd, A global bifurcation theorem with applications to functional differential equations, J. Funct. Anal., 19, 319-338 (1975) · Zbl 0314.47041 · doi:10.1016/0022-1236(75)90061-0
[36] Ruan, S., Delay Differential Equations in Single Species Dynamics. Delay Differential Equations and Applications, 477-517 (2006), Dordrecht: Springer, Dordrecht · Zbl 1130.34059
[37] Smith, Hl, An Introduction to Delay Differential Equations with Applications to the Life Sciences (2011), New York: Springer, New York · Zbl 1227.34001
[38] Van Den Berg, Jb; Jaquette, J., A proof of Wright’s conjecture, J. Differ. Equ., 264, 12, 7412-7462 (2018) · Zbl 1388.34068 · doi:10.1016/j.jde.2018.02.018
[39] Wright, Em, A non-linear difference-differential equation, Journal für die Reine und Angewandte Mathematik, 194, 66-87 (1955) · Zbl 0064.34203
[40] Yan, X.; Shi, J., Stability switches in a logistic population model with mixed instantaneous and delayed density dependence, J. Dyn. Differ. Equ., 29, 1, 113-130 (2017) · Zbl 1369.34095 · doi:10.1007/s10884-015-9432-3
[41] Zou, X., Delay induced traveling wave fronts in reaction-diffusion equations of KPP-Fisher type, J. Comput. Appl. Math., 146, 2, 309-321 (2002) · Zbl 1058.35114 · doi:10.1016/S0377-0427(02)00363-1
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