Permanence of a nonlinear mutualism model with time varying delay
-
1574
Downloads
-
3110
Views
Authors
Runxin Wu
- Mathematics and Physics Institute, Fujian University of Technology, Fuzhou, Fujian, 350014, P. R. China.
Abstract
Sufficient conditions are obtained for the
permanence of the following nonlinear mutualism model with time varying delay
\[
\frac{dN_1(t)}{dt}= r_1(t)N_1(t)\left[\frac{K_1(t)+\alpha_1(t)N_2^{\beta_1}(t-\tau_2(t))}{
1+ N_2^{\beta_1}(t-\tau_2(t))}-N_1^{\delta_1}(t-\sigma_1(t))\right],
\]
\[
\frac{dN_2(t)}{dt}= r_2(t)N_2(t)\left[\frac{K_2(t)+\alpha_2(t)N_1^{\beta_2}(t-\tau_1(t))}{
1+N_1^{\beta_2}(t-\tau_1(t)}-N_2^{\delta_2}(t-\sigma_2(t))\right],
\]
where \(r_i, K_i, \alpha_i\), \(\tau_i\), and \(\sigma_i, i=1,2\) are continuous
functions bounded above and below by positive constants,
\( \alpha_i>K_i, i=1,2 ,\) and \(\beta_i, \delta_i, i=1, 2\) are all positive constants.
Share and Cite
ISRP Style
Runxin Wu, Permanence of a nonlinear mutualism model with time varying delay, Journal of Mathematics and Computer Science, 19 (2019), no. 2, 129--135
AMA Style
Wu Runxin, Permanence of a nonlinear mutualism model with time varying delay. J Math Comput SCI-JM. (2019); 19(2):129--135
Chicago/Turabian Style
Wu, Runxin. "Permanence of a nonlinear mutualism model with time varying delay." Journal of Mathematics and Computer Science, 19, no. 2 (2019): 129--135
Keywords
- Mutualism
- nonlinear
- delay
- permanence
MSC
References
-
[1]
D. H. Boucher, The biology of mutualism: ecology and evolution, Croom Helm, London (1985)
-
[2]
F. D. Chen, On a nonlinear non--autonomous predator--prey model with diffusion and distributed delay, J. Comput. Appl. Math., 180 (2005), 33--49
-
[3]
B. G. Chen, The influence of commensalism on a Lotka--Volterra commensal symbiosis model with Michaelis--Menten type harvesting, Adv. Difference Equ., 2019 (2019), 14 pages
-
[4]
L. J. Chen, L. J. Chen, Z. Li, Permanence of a delayed discrete mutualism model with feedback controls, Math. Comput. Modelling, 50 (2009), 1083--1089
-
[5]
J. H. Chen, R. X. Wu, A commensal symbiosis model with non--monotonic functional response, Comm. Math. Biol. Neur., 2017 (2017), 8 pages
-
[6]
F. D. Chen, H. L. Wu, X. D. Xie, Global attractivity of a discrete cooperative system incorporating harvesting, Adv. Difference Equ., 2016 (2016), 12 pages
-
[7]
L. J. Chen, X. D. Xie, Permanence of an $n$-species cooperation system with continuous time delays and feedback controls, Nonlinear Anal. Real World Appl., 12 (2001), 34--38
-
[8]
L. J. Chen, X. D. Xie, L. J. Chen, Feedback control variables have no influence on the permanence of a discrete N-species cooperation system, Discrete Dyn. Nat. Soc., 2009 (2009), 10 pages
-
[9]
F. D. Chen, X. D. Xie, X. F. Chen, Dynamic behaviors of a stage--structured cooperation model, Comm. Math. Biol. Neur., 2015 (2015), 19 pages
-
[10]
F. D. Chen, X. D. Xie, Z. S. Miao, L. Q. Pu, Extinction in two species nonautonomous nonlinear competitive system, Appl. Math. Comput., 274 (2016), 119--124
-
[11]
F. D. Chen, Y. L. Xue, Q. F. Lin, X. D. Xie, Dynamic behaviors of a Lotka--Volterra commensal symbiosis model with density dependent birth rate, Adv. Difference Equ., 2018 (2018), 14 pages
-
[12]
F. D. Chen, J. H. Yang, L. J. Chen, X. D. Xie, On a mutualism model with feedback controls, Appl. Math. Comput., 214 (2009), 581--587
-
[13]
A. M. Dean, A simple model of mutualism, Amer. Natural., 121 (1983), 409--417
-
[14]
K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, Dordrecht (1992)
-
[15]
X. Y. Guan, F. D. Chen, Dynamical analysis of a two species amensalism model with Beddington--DeAngelis functional response and Allee effect on the second species, Nonlinear Anal. Real World Appl., 48 (2019), 71--93
-
[16]
R. Y. Han, X. D. Xie, F. D. Chen, Permanence and global attractivity of a discrete pollination mutualism in plant--pollinator system with feedback controls, Adv. Difference Equ., 2016 (2016), 17 pages
-
[17]
M. X. He, F. D. Chen, Extinction and stability of an impulsive system with pure delays, Appl. Math. Lett., 91 (2019), 128--136
-
[18]
M. X. He, Z. Li, F. D. Chen, Permanence, extinction and global attractivity of the periodic Gilpin-Ayala competition system with impulses, Nonlinear Anal. Real World Appl., 11 (2010), 1537--1551
-
[19]
C. Q. Lei, Dynamic behaviors of a non--selective harvesting May cooperative system incorporating partial closure for the populations, Comm. Math. Biol. Neur., 2018 (2018), 23 pages
-
[20]
C. Q. Lei, Dynamic behaviors of a stage--structured commensalism system, Adv. Difference Equ., 2018 (2018), 20 pages
-
[21]
Y. K. Li, On a periodic mutualism model, ANZIAM J., 42 (2001), 569--580
-
[22]
Z. Li, F. D. Chen, Extinction and almost periodic solutions of a discrete Gilpin--Ayala type population model, J. Difference Equ. Appl., 19 (2013), 719--737
-
[23]
T. X. Li, Y. V. Rogovchenko, Oscillation criteria for second--order superlinear Emden--Fowler neutral differential equations, Monatsh. Math., 184 (2017), 489--500
-
[24]
Y. K. Li, G. T. Xu, Positive periodic solutions for an integrodifferential model of mutualism, Appl. Math. Lett., 14 (2001), 525--530
-
[25]
Q. X. Lin, X. D. Xie, F. D. Chen, Q. F. Lin, Dynamical analysis of a logistic model with impulsive Holling type--II harvesting, Adv. Difference Equ., 2018 (2018), 22 pages
-
[26]
Z. S. Miao, X. D. Xie, L. Q. Pu, Dynamic behaviors of a periodic Lotka--Volterra commensal symbiosis model with impulsive, Comm. Math. Biol. Neur., 2015 (2015), 15 pages
-
[27]
L. Q. Pu, X. D. Xie, F. D. Chen, Z. S. Miao, Extinction in two--species nonlinear discrete competitive system, Discrete Dyn. Nat. Soc., 2016 (2016), 10 pages
-
[28]
C. L. Wolin, L. R. Lawlor, Models of facultative mutualism: density effects, Amer. Natural., 144 (1984), 843--862
-
[29]
R. X. Wu, Dynamic behaviors of a nonlinear amensalism model, Adv. Difference Equ., 2018 (2018), 13 pages
-
[30]
R. X. Wu, L. Li, Q. F. Lin, A Holling type commensal symbiosis model involving Allee effect, Comm. Math. Biol. Neur., 2018 (2018), 13 pages
-
[31]
X. D. Xie, F. D. Chen, Y. L. Xue, Note on the stability property of a cooperative system incorporating harvesting, Discrete Dyn. Nat. Soc., 2014 (2014), 5 pages
-
[32]
X. D. Xie, F. D. Chen, K. Yang, Y. L. Xue, Global attractivity of an integro--differential model of mutualism, Abstract Appl. Anal., 2014 (2014), 6 pages
-
[33]
X. D. Xie, Y. L. Xue, R. X. Wu, Global attractivity in a discrete mutualism model with infinite deviating arguments, Discrete Dyn. Nat. Soc., 2017 (2017), 8 pages
-
[34]
Y. L. Xue, X. D. Xie, F. D. Chen, R. Y. Han, Almost periodic solution of a discrete commensalism system, Discrete Dyn. Nat. Soc., 2015 (2015), 11 pages
-
[35]
K. Yang, Z. S. Miao, F. D. Chen, X. D. Xie, Influence of single feedback control variable on an autonomous Holling--II type cooperative system, J. Math. Anal. Appl., 435 (2016), 874--888
-
[36]
K. Yang, X. D. Xie, F. D. Chen, Global stability of a discrete mutualism model, Abstr. Appl. Anal., 2014 (2014), 7 pages
-
[37]
L. Y. Yang, X. D. Xie, F. D. Chen, Dynamic behaviors of a discrete periodic predator--prey--mutualist system, Discrete Dyn. Nat. Soc., 2015 (2015), 11 pages
