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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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. Google Scholar

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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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

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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