March 2018, 23(2): 809-836. doi: 10.3934/dcdsb.2018044

Boundedness of positive solutions of a system of nonlinear delay differential equations

Department of Mathematics, University of Pannonia, Veszprém, H-8200, Hungary

Received  December 2016 Revised  August 2017 Published  December 2017

In this manuscript the system of nonlinear delay differential equations $\dot{x}_i(t) =\sum\limits_{j =1}^{n}\sum\limits_{\ell =1}^{n_0}α_{ij\ell} (t) h_{ij}(x_j(t-τ_{ij\ell}(t)))$$-β_i(t)f_i(x_i(t))+ρ_i(t)$, $t≥0$, $1≤i ≤n$ is considered. Sufficient conditions are established for the uniform permanence of the positive solutions of the system. In several particular cases, explicit formulas are given for the estimates of the upper and lower limit of the solutions. In a special case, the asymptotic equivalence of the solutions is investigated.

Citation: István Győri, Ferenc Hartung, Nahed A. Mohamady. Boundedness of positive solutions of a system of nonlinear delay differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 809-836. doi: 10.3934/dcdsb.2018044
References:
[1]

J. Baštinec, L. Berezansky, J. Diblík and Z. Šmarda, On a delay population model with quadratic nonlinearity, Adv. Difference Equ. 2012 (2012), 9pp. doi: 10.1186/1687-1847-2012-230.

[2]

J. BaštinecL. BerezanskyJ. Diblik and Z. Šmarda, On a delay population model with a quadratic nonlinearity without positive steady state, Appl. Math. Comput., 227 (2014), 622-629. doi: 10.1016/j.amc.2013.11.061.

[3]

J. BélairS. A. Campbell and P. van den Driessche, Frustration, stability, and delay-induced oscillations in a neural network nodel, SIAM J. Appl. Math., 56 (1996), 245-255. doi: 10.1137/S0036139994274526.

[4]

L. BerezanskyE. Braverman and L. Idels, Nicholson's blowflies differential equations revisited: main results and open problems, Appl. Math. Model., 34 (2010), 1405-1417. doi: 10.1016/j.apm.2009.08.027.

[5]

L. Berezansky and E. Braverman, On stability of cooperative and hereditary systems with a distributed delay, Nonlinearity, 28 (2015), 1745-1760. doi: 10.1088/0951-7715/28/6/1745.

[6]

L. Berezansky and E. Braverman, Boundedness and persistence of delay differential equations with mixed nonlinearity, Appl. Math. Comput., 279 (2016), 154-169. doi: 10.1016/j.amc.2016.01.015.

[7]

L. BerezanskyL. Idels and L. Troib, Global dynamics of Nicholson-type delay systems with applications, Nonlinear Anal. Real World Appl., 12 (2011), 436-445. doi: 10.1016/j.nonrwa.2010.06.028.

[8]

G. I. Bischi, Compartmental analysis of economic systems with heterogeneous agents: An introduction, in Beyond the Representative Agent, ed. A. Kirman, M. Gallegati (Elgar Pub. Co., 1998), 181-214.

[9]

R. F. Brown, Compartmental system analysis: State of the art, IEEE Trans. Biomed. Eng., BME-27 (1980), 1-11. doi: 10.1109/TBME.1980.326685.

[10]

M. Budincevic, A comparison theorem of differential equations, Novi Sad J. Math., 40 (2010), 55-56.

[11]

A. ChenL. Huang and J. Cao, Existence and stability of almost periodic solution for BAM neural networks with delays, Appl. Math. Comput., 137 (2003), 177-193. doi: 10.1016/S0096-3003(02)00095-4.

[12]

P. DasA. B. Roy and A. Das, Stability and oscillations of a negative feedback delay model for the control of testosterone secretion, BioSystems, 32 (1994), 61-69. doi: 10.1016/0303-2647(94)90019-1.

[13]

P. van den Driessche and X. Zou, Global attractivity in delayed Hopfield neural network models, SIAM J. on Appl. Math., 58 (1998), 1878-1890. doi: 10.1137/S0036139997321219.

[14]

T. Faria, A note on permanence of nonautonomous cooperative scalar population models with delays, Appl. Math. Comput., 240 (2014), 82-90. doi: 10.1016/j.amc.2014.04.040.

[15]

T. Faria, Persistence and permanence for a class of functional differential equations with infinite delay, J. Dyn. Diff. Equat., 28 (2016), 1163-1186. doi: 10.1007/s10884-015-9462-x.

[16]

T. Faria and G. Röst, Persistence, permanence and global stability for an n-dimensional Nicholson system, J. Dyn. Diff. Equat., 26 (2014), 723-744. doi: 10.1007/s10884-014-9381-2.

[17]

K. Gopalsamy and X. He, Stability in asymmetric Hopfield nets with transmission delays, Phys. D., 76 (1994), 344-358. doi: 10.1016/0167-2789(94)90043-4.

[18]

W. S. C. GurneyS. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. doi: 10.1038/287017a0.

[19]

I. Győri, Connections between compartmental systems with pipes and integro-differential equations, Math. Model., 7 (1986), 1215-1238. doi: 10.1016/0270-0255(86)90077-1.

[20]

I. Győri and J. Eller, Compartmental systems with pipes, Math. Biosci., 53 (1981), 223-247. doi: 10.1016/0025-5564(81)90019-5.

[21]

I. GyőriF. Hartung and N. A. Mohamady, On a nonlinear delay population model, Appl. Math. Comput., 270 (2015), 909-925. doi: 10.1016/j.amc.2015.08.090.

[22]

I. GyőriF. Hartung and N. A. Mohamady, Existence and uniqueness of positive solutions of a system of nonlinear algebraic equations, Period. Math. Hung., 75 (2017), 114-127. doi: 10.1007/s10998-016-0179-3.

[23]

J. J. Hopfield, Neurons with graded response have collective computational properties like those of two-stage neurons, Proc. Nat. Acad. Sci., U. S. A. 81 (1984), 3088-3092.

[24]

J. A. Jacquez and C. P. Simon, Qualitative theory of compartmental systems with lags, Math. Biosci., 180 (2002), 329-362. doi: 10.1016/S0025-5564(02)00131-1.

[25]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, MA, 1993.

[26]

B. Liu, Global stability of a class of Nicholson's blowflies model with patch structure and multiple time-varying delays, Nonlinear Anal. Real World Appl., 11 (2010), 2557-2562. doi: 10.1016/j.nonrwa.2009.08.011.

show all references

References:
[1]

J. Baštinec, L. Berezansky, J. Diblík and Z. Šmarda, On a delay population model with quadratic nonlinearity, Adv. Difference Equ. 2012 (2012), 9pp. doi: 10.1186/1687-1847-2012-230.

[2]

J. BaštinecL. BerezanskyJ. Diblik and Z. Šmarda, On a delay population model with a quadratic nonlinearity without positive steady state, Appl. Math. Comput., 227 (2014), 622-629. doi: 10.1016/j.amc.2013.11.061.

[3]

J. BélairS. A. Campbell and P. van den Driessche, Frustration, stability, and delay-induced oscillations in a neural network nodel, SIAM J. Appl. Math., 56 (1996), 245-255. doi: 10.1137/S0036139994274526.

[4]

L. BerezanskyE. Braverman and L. Idels, Nicholson's blowflies differential equations revisited: main results and open problems, Appl. Math. Model., 34 (2010), 1405-1417. doi: 10.1016/j.apm.2009.08.027.

[5]

L. Berezansky and E. Braverman, On stability of cooperative and hereditary systems with a distributed delay, Nonlinearity, 28 (2015), 1745-1760. doi: 10.1088/0951-7715/28/6/1745.

[6]

L. Berezansky and E. Braverman, Boundedness and persistence of delay differential equations with mixed nonlinearity, Appl. Math. Comput., 279 (2016), 154-169. doi: 10.1016/j.amc.2016.01.015.

[7]

L. BerezanskyL. Idels and L. Troib, Global dynamics of Nicholson-type delay systems with applications, Nonlinear Anal. Real World Appl., 12 (2011), 436-445. doi: 10.1016/j.nonrwa.2010.06.028.

[8]

G. I. Bischi, Compartmental analysis of economic systems with heterogeneous agents: An introduction, in Beyond the Representative Agent, ed. A. Kirman, M. Gallegati (Elgar Pub. Co., 1998), 181-214.

[9]

R. F. Brown, Compartmental system analysis: State of the art, IEEE Trans. Biomed. Eng., BME-27 (1980), 1-11. doi: 10.1109/TBME.1980.326685.

[10]

M. Budincevic, A comparison theorem of differential equations, Novi Sad J. Math., 40 (2010), 55-56.

[11]

A. ChenL. Huang and J. Cao, Existence and stability of almost periodic solution for BAM neural networks with delays, Appl. Math. Comput., 137 (2003), 177-193. doi: 10.1016/S0096-3003(02)00095-4.

[12]

P. DasA. B. Roy and A. Das, Stability and oscillations of a negative feedback delay model for the control of testosterone secretion, BioSystems, 32 (1994), 61-69. doi: 10.1016/0303-2647(94)90019-1.

[13]

P. van den Driessche and X. Zou, Global attractivity in delayed Hopfield neural network models, SIAM J. on Appl. Math., 58 (1998), 1878-1890. doi: 10.1137/S0036139997321219.

[14]

T. Faria, A note on permanence of nonautonomous cooperative scalar population models with delays, Appl. Math. Comput., 240 (2014), 82-90. doi: 10.1016/j.amc.2014.04.040.

[15]

T. Faria, Persistence and permanence for a class of functional differential equations with infinite delay, J. Dyn. Diff. Equat., 28 (2016), 1163-1186. doi: 10.1007/s10884-015-9462-x.

[16]

T. Faria and G. Röst, Persistence, permanence and global stability for an n-dimensional Nicholson system, J. Dyn. Diff. Equat., 26 (2014), 723-744. doi: 10.1007/s10884-014-9381-2.

[17]

K. Gopalsamy and X. He, Stability in asymmetric Hopfield nets with transmission delays, Phys. D., 76 (1994), 344-358. doi: 10.1016/0167-2789(94)90043-4.

[18]

W. S. C. GurneyS. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. doi: 10.1038/287017a0.

[19]

I. Győri, Connections between compartmental systems with pipes and integro-differential equations, Math. Model., 7 (1986), 1215-1238. doi: 10.1016/0270-0255(86)90077-1.

[20]

I. Győri and J. Eller, Compartmental systems with pipes, Math. Biosci., 53 (1981), 223-247. doi: 10.1016/0025-5564(81)90019-5.

[21]

I. GyőriF. Hartung and N. A. Mohamady, On a nonlinear delay population model, Appl. Math. Comput., 270 (2015), 909-925. doi: 10.1016/j.amc.2015.08.090.

[22]

I. GyőriF. Hartung and N. A. Mohamady, Existence and uniqueness of positive solutions of a system of nonlinear algebraic equations, Period. Math. Hung., 75 (2017), 114-127. doi: 10.1007/s10998-016-0179-3.

[23]

J. J. Hopfield, Neurons with graded response have collective computational properties like those of two-stage neurons, Proc. Nat. Acad. Sci., U. S. A. 81 (1984), 3088-3092.

[24]

J. A. Jacquez and C. P. Simon, Qualitative theory of compartmental systems with lags, Math. Biosci., 180 (2002), 329-362. doi: 10.1016/S0025-5564(02)00131-1.

[25]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, MA, 1993.

[26]

B. Liu, Global stability of a class of Nicholson's blowflies model with patch structure and multiple time-varying delays, Nonlinear Anal. Real World Appl., 11 (2010), 2557-2562. doi: 10.1016/j.nonrwa.2009.08.011.

Figure 1.  Numerical solution of the System (27).
Figure 2.  Numerical solution of the System (40).
Table 1.  Numerical solution of the System (28)
$k$ $\underline{x}_1^{(k)}$ $\underline{x}_2^{(k)}$ $\underline{x}_3^{(k)}$
$0$ $0$ $0$ $0$
$1$ $0.3761$ $1.7105$ $1.9834$
$2$ $1.8185$ $4.8060$ $3.7077$
$3$ $ 3.6353$ $7.5553$ $5.9214$
$4$ $ 4.0406$ $7.9252$ $6.4602$
$5$ $4.4130$ $8.1962$ $6.9628$
$6$ $4.5364$ $8.2765$ $7.1294$
$7$ $ 4.5767$ $8.3023$ $7.1836$
$8$ $ 4.5958$ $8.3146$ $7.2092$
$9$ $4.5960$ $ 8.3147$ $7.2095$
$10$ $4.5960$ $ 8.3147$ $7.2095$
$k$ $\underline{x}_1^{(k)}$ $\underline{x}_2^{(k)}$ $\underline{x}_3^{(k)}$
$0$ $0$ $0$ $0$
$1$ $0.3761$ $1.7105$ $1.9834$
$2$ $1.8185$ $4.8060$ $3.7077$
$3$ $ 3.6353$ $7.5553$ $5.9214$
$4$ $ 4.0406$ $7.9252$ $6.4602$
$5$ $4.4130$ $8.1962$ $6.9628$
$6$ $4.5364$ $8.2765$ $7.1294$
$7$ $ 4.5767$ $8.3023$ $7.1836$
$8$ $ 4.5958$ $8.3146$ $7.2092$
$9$ $4.5960$ $ 8.3147$ $7.2095$
$10$ $4.5960$ $ 8.3147$ $7.2095$
Table 2.  Numerical solution of the System (30)
$k$ $\overline{x}_1^{(k)}$ $\overline{x}_2^{(k)}$ $\overline{x}_3^{(k)}$
$0$ $0$ $0$ $0$
$1$ $0.6849$ $2.0198$ $ 2.8145$
$2$ $2.9151$ $5.9799$ $ 5.0354$
$3$ $5.5288$ $9.7858$ $7.5194$
$4$ $6.4086$ $10.7362$ $8.3557$
$5$ $6.6740$ $11.0053$ $8.6081$
$6$ $6.7520$ $11.0838$ $8.6822$
$7$ $6.7747$ $11.1067$ $8.7038$
$8$ $ 6.7839$ $11.1159$ $ 8.7125$
$9$ $6.7840$ $11.1161$ $8.7126$
$10$ $6.7840$ $11.1161$ $8.7126$
$k$ $\overline{x}_1^{(k)}$ $\overline{x}_2^{(k)}$ $\overline{x}_3^{(k)}$
$0$ $0$ $0$ $0$
$1$ $0.6849$ $2.0198$ $ 2.8145$
$2$ $2.9151$ $5.9799$ $ 5.0354$
$3$ $5.5288$ $9.7858$ $7.5194$
$4$ $6.4086$ $10.7362$ $8.3557$
$5$ $6.6740$ $11.0053$ $8.6081$
$6$ $6.7520$ $11.0838$ $8.6822$
$7$ $6.7747$ $11.1067$ $8.7038$
$8$ $ 6.7839$ $11.1159$ $ 8.7125$
$9$ $6.7840$ $11.1161$ $8.7126$
$10$ $6.7840$ $11.1161$ $8.7126$
Table 3.  Numerical solution of the System (41)
$k$ $\underline{x}_1^{(k)}$ $\underline{x}_2^{(k)}$
$0$ $0$ $0$
$1$ $3.4641$ $1.0831$
$2$ $4.7663$ $2.5795$
$3$ $5.2031$ $3.7627$
$4$ $5.4246$ $4.8659$
$5$ $5.4659$ $5.1549$
$6$ $5.4721$ $5.2008$
$7$ $5.4751$ $5.2419$
$8$ $5.4777$ $5.2429$
$9$ $ 5.4778$ $5.2430$
$10$ $ 5.4778$ $5.2430$
$k$ $\underline{x}_1^{(k)}$ $\underline{x}_2^{(k)}$
$0$ $0$ $0$
$1$ $3.4641$ $1.0831$
$2$ $4.7663$ $2.5795$
$3$ $5.2031$ $3.7627$
$4$ $5.4246$ $4.8659$
$5$ $5.4659$ $5.1549$
$6$ $5.4721$ $5.2008$
$7$ $5.4751$ $5.2419$
$8$ $5.4777$ $5.2429$
$9$ $ 5.4778$ $5.2430$
$10$ $ 5.4778$ $5.2430$
Table 4.  Numerical solution of the System (43)
$k$ $\overline{x}_1^{(k)}$ $\overline{x}_2^{(k)}$
$0$ $0$ $0$
$1$ $ 4.4721$ $4.6552$
$2$ $6.3445$ $7.3850$
$3$ $7.0199$ $8.5877$
$4$ $7.2586$ $9.0666$
$5$ $7.3436$ $9.2503$
$6$ $7.3744$ $9.3198$
$7$ $7.3918$ $9.3608$
$8$ $7.3920$ $9.3615$
$9$ $7.3921$ $9.3616$
$10$ $7.3921$ $9.3616$
$k$ $\overline{x}_1^{(k)}$ $\overline{x}_2^{(k)}$
$0$ $0$ $0$
$1$ $ 4.4721$ $4.6552$
$2$ $6.3445$ $7.3850$
$3$ $7.0199$ $8.5877$
$4$ $7.2586$ $9.0666$
$5$ $7.3436$ $9.2503$
$6$ $7.3744$ $9.3198$
$7$ $7.3918$ $9.3608$
$8$ $7.3920$ $9.3615$
$9$ $7.3921$ $9.3616$
$10$ $7.3921$ $9.3616$
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