
-
Previous Article
A dynamic model of CT scans for quantifying doubling time of ground glass opacities using histogram analysis
- MBE Home
- This Issue
-
Next Article
Quantifying the impact of early-stage contact tracing on controlling Ebola diffusion
Dynamics of delayed mosquitoes populations models with two different strategies of releasing sterile mosquitoes
1. | School of Mathematics and Statistics, Xinyang Normal University, Xinyang 46400, China |
2. | Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899, USA |
3. | Department of Mathematics, Columbus State University, Columbus, Georgia 31907, USA |
To prevent the transmissions of mosquito-borne diseases (e.g., malaria, dengue fever), recent works have considered the problem of using the sterile insect technique to reduce or eradicate the wild mosquito population. It is important to consider how reproductive advantage of the wild mosquito population offsets the success of population replacement. In this work, we explore the interactive dynamics of the wild and sterile mosquitoes by incorporating the delay in terms of the growth stage of the wild mosquitoes. We analyze (both analytically and numerically) the role of time delay in two different ways of releasing sterile mosquitoes. Our results demonstrate that in the case of constant release rate, the delay does not affect the dynamics of the system and every solution of the system approaches to an equilibrium point; while in the case of the release rate proportional to the wild mosquito populations, the delay has a large effect on the dynamics of the system, namely, for some parameter ranges, when the delay is small, every solution of the system approaches to an equilibrium point; but as the delay increases, the solutions of the system exhibit oscillatory behavior via Hopf bifurcations. Numerical examples and bifurcation diagrams are also given to demonstrate rich dynamical features of the model in the latter release case.
References:
[1] |
R. Abdul-Ghani, H. F. Farag, A. F. Allam and A. A. Azazy,
Measuring resistant-genotype transmission of malaria parasites: challenges and prospects, Parasitol Res., 113 (2014), 1481-1487.
doi: 10.1007/s00436-014-3789-9. |
[2] |
P. L. Alonso, G. Brown, M. Arevalo-Herrera, et al, A research agenda to underpin malaria eradication, PLoS Med., 8 (2011), e1000406.
doi: 10.1371/journal.pmed.1000406. |
[3] |
L. Alphey, M. Benedict, R. Bellini, G. G. Clark, D. A. Dame, M. W. Service and S. L. Dobson,
Sterile-insect methods for control of mosquito-borne diseases: An analysis, Vector Borne Zoonotic Dis., 10 (2010), 295-311.
doi: 10.1089/vbz.2009.0014. |
[4] |
J. Arino, L. Wang and G. S. Wolkowicz,
An alternative formulation for a delayed logistic equation, J. Theor. Biol., 241 (2006), 109-119.
doi: 10.1016/j.jtbi.2005.11.007. |
[5] |
M. Q. Benedict and A. S. Robinson,
The first releases of transgenic mosquitoes: An argument for the sterile insect technique, Trends Parasitol, 19 (2003), 349-355.
doi: 10.1016/S1471-4922(03)00144-2. |
[6] |
E. Beretta and Y. Kuang,
Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165.
doi: 10.1137/S0036141000376086. |
[7] |
J. G. Breman,
The ears of the hippopotamus: Manifestations, determinants, and estimates of the malaria burden, Am. J. Trop. Med. Hyg., 64 (2001), 1-11.
doi: 10.4269/ajtmh.2001.64.1. |
[8] |
W. G. Brogdon and J. C. McAllister, Insecticide resistance and vector control, J. Agromedicine, 6 (1999), 41-58. Google Scholar |
[9] |
L. Cai, S. Ai and J. Li,
Dynamics of mosquitoes populations with different strategies for releasing sterile mosquitoes, SIAM, J. Appl. Math., 74 (2014), 1786-1809.
doi: 10.1137/13094102X. |
[10] |
K. Cooke, P. van den Driessche and X. Zou,
Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol., 39 (1999), 332-352.
doi: 10.1007/s002850050194. |
[11] |
H. Diaz, A. A. Ramirez, A. Olarte and C. Clavijo,
A model for the control of malaria using genetically modified vectors, J. Theor. Biol., 276 (2011), 57-66.
doi: 10.1016/j.jtbi.2011.01.053. |
[12] |
J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1960. |
[13] |
Y. Dumont and J. M. Tchuenche,
Mathematical studies on the sterile insect technique for the Chikungunya disease and Aedes albopictus, J. Math. Biol., 65 (2012), 809-854.
doi: 10.1007/s00285-011-0477-6. |
[14] |
V. A. Dyck, J. Hendrichs and A. S. Robinson, Sterile insect technique -principles and practice in area-wide integrated pest management, Springer, The Netherlands, 2005. Google Scholar |
[15] |
C. Dye,
Models for the population dynamics of the yellow fever mosquito, Aedes aegypti, J. Anim. Ecol., 53 (1984), 247-268.
doi: 10.2307/4355. |
[16] |
L. Esteva and H. M. Yang,
Assessing the effects of temperature and dengue virus load on dengue transmission, J. Biol. Syst., 23 (2015), 527-554.
doi: 10.1142/S0218339015500278. |
[17] |
L. Esteva and H. M. Yang,
Mathematical model to assess the control of Aedes aegypti mosquitoes by the sterile insect technique, Math. Biosci., 198 (2005), 132-147.
doi: 10.1016/j.mbs.2005.06.004. |
[18] |
J. E. Gentile, S. Rund and G. R Madey,
Modelling sterile insect technique to control the population of Anopheles gambiae, Malaria J., 14 (2015), 92-103.
doi: 10.1186/s12936-015-0587-5. |
[19] |
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equation, Springer, New York, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[20] |
J. Ito, A. Ghosh, L. A. Moreira, E. A. Wilmmer and M. Jacobs-Lorena, Transgenic anopheline mosquitoes impaired in transmission of a malria parasite, Nature, 417 (2002), 452-455. Google Scholar |
[21] |
M. Jankovic and S. Petrovskii,
Are time delays always destabilizing? Revisiting the role of time delays and the Allee effect, Theor Ecol., 7 (2014), 335-349.
doi: 10.1007/s12080-014-0222-z. |
[22] |
E. F. Knipling, Possibilities of insect control or eradication through the use of sexually sterile males, J. Econ. Entomol., 48 (1955), 459-462. Google Scholar |
[23] |
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993. |
[24] |
S. S. Lee, R. E. Baker, E. A. Gaffney and S. M. White,
Modelling Aedes aegypti mosquito control via transgenic and sterile insect techniques: Endemics and emerging outbreaks, J. Theor. Biol., 331 (2013), 78-90.
doi: 10.1016/j.jtbi.2013.04.014. |
[25] |
M. A. Lewis and P. van den Driessche,
Waves of extinction from sterile insect release, Math. Biosci., 116 (1993), 221-247.
doi: 10.1016/0025-5564(93)90067-K. |
[26] |
J. Li,
New revised simple models for interactive wild and sterile mosquito populations and their dynamics, J. Biol. Dyna., 11 (2017), 316-333.
doi: 10.1080/17513758.2016.1216613. |
[27] |
J. Li, L. Cai and Y. Li,
Stage-structured wild and sterile mosquito population models and their dynamics, J. Biol.Dyna., 11 (2017), 79-101.
doi: 10.1080/17513758.2016.1159740. |
[28] |
J. Lu and J. Li,
Dynamics of stage-structured discrete mosquito population, J. Appl. Anal. Comput., 1 (2011), 53-67.
|
[29] |
G. J. Lycett and F. C. Kafatos, Anti-malaria mosquitoes?, Nautre, 417 (2002), 387-388. Google Scholar |
[30] |
C. W. Morin and A. C. Comrie,
Regional and seasonal response of a West Nile virus vector to climate change, PNAS, 110 (2013), 15620-15625.
doi: 10.1073/pnas.1307135110. |
[31] |
W. W. Murdoch, C. J. Briggs and R. M. Nisbet, Consumer-resource dynamics, Princeton University Press, New Jersey, USA, 2003. Google Scholar |
[32] |
H. K. Phuc, M. H. Andreasen, et al, Late-acting dominant lethal genetic systems and mosquito control, BMC. Biol., 5 (2007), 11–16.
doi: 10.1186/1741-7007-5-11. |
[33] |
E. P. Pliego, J. Vel$\acute{a}$zquez-Castro and A. F. Collar,
Seasonality on the life cycle of Aedes aegypti mosquito and its statistical relation with dengue outbreaks, Appl. Math. Model., 50 (2017), 484-496.
doi: 10.1016/j.apm.2017.06.003. |
[34] |
M. Rafikov, L. Bevilacqua and A. P. Wyse,
Optimal control strategy of malaria vector using genetically modified mosquitoes, J. Theor. Biol., 258 (2009), 418-425.
doi: 10.1016/j.jtbi.2008.08.006. |
[35] |
S. J. Schreiber,
Allee effect, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209.
doi: 10.1016/S0040-5809(03)00072-8. |
[36] |
J. Smith, M. Amador and R. Barrera,
Seasonal and habitat effects on dengue and West Nile Virus Vectors in San Juan, Puerto Rico, J. Am. Mosq. Control. Assoc., 25 (2009), 38-46.
doi: 10.2987/08-5782.1. |
[37] |
H. Townson, SIT for African malaria vectors: Epilogue, Malar. J., 8 (2009), S10.
doi: 10.1186/1475-2875-8-S2-S10. |
[38] |
WHO, 10 facts on malaria, http://www.who.int/features/factfiles/malaria/en/. Google Scholar |
[39] |
J. Wu, H. R. Thieme, Y. Lou and G. Fan,
Stability and persistence in ODE models for populations with many stages, Math. Biosc. Eng., 12 (2015), 661-686.
doi: 10.3934/mbe.2015.12.661. |
[40] |
B. Zheng, M. Tang and J. Yu,
Modeling Wolbachia spread in mosquitoes through delay differential equations, SIAM J. Appl. Math., 74 (2014), 743-770.
doi: 10.1137/13093354X. |
[41] |
B. Zheng, M. Tang, J. Yu and J. Qiu,
Wolbachia spreading dynamics in mosquitoes with imperfect maternal transmission, J. Math. Biol., 76 (2018), 235-263.
doi: 10.1007/s00285-017-1142-5. |
show all references
References:
[1] |
R. Abdul-Ghani, H. F. Farag, A. F. Allam and A. A. Azazy,
Measuring resistant-genotype transmission of malaria parasites: challenges and prospects, Parasitol Res., 113 (2014), 1481-1487.
doi: 10.1007/s00436-014-3789-9. |
[2] |
P. L. Alonso, G. Brown, M. Arevalo-Herrera, et al, A research agenda to underpin malaria eradication, PLoS Med., 8 (2011), e1000406.
doi: 10.1371/journal.pmed.1000406. |
[3] |
L. Alphey, M. Benedict, R. Bellini, G. G. Clark, D. A. Dame, M. W. Service and S. L. Dobson,
Sterile-insect methods for control of mosquito-borne diseases: An analysis, Vector Borne Zoonotic Dis., 10 (2010), 295-311.
doi: 10.1089/vbz.2009.0014. |
[4] |
J. Arino, L. Wang and G. S. Wolkowicz,
An alternative formulation for a delayed logistic equation, J. Theor. Biol., 241 (2006), 109-119.
doi: 10.1016/j.jtbi.2005.11.007. |
[5] |
M. Q. Benedict and A. S. Robinson,
The first releases of transgenic mosquitoes: An argument for the sterile insect technique, Trends Parasitol, 19 (2003), 349-355.
doi: 10.1016/S1471-4922(03)00144-2. |
[6] |
E. Beretta and Y. Kuang,
Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165.
doi: 10.1137/S0036141000376086. |
[7] |
J. G. Breman,
The ears of the hippopotamus: Manifestations, determinants, and estimates of the malaria burden, Am. J. Trop. Med. Hyg., 64 (2001), 1-11.
doi: 10.4269/ajtmh.2001.64.1. |
[8] |
W. G. Brogdon and J. C. McAllister, Insecticide resistance and vector control, J. Agromedicine, 6 (1999), 41-58. Google Scholar |
[9] |
L. Cai, S. Ai and J. Li,
Dynamics of mosquitoes populations with different strategies for releasing sterile mosquitoes, SIAM, J. Appl. Math., 74 (2014), 1786-1809.
doi: 10.1137/13094102X. |
[10] |
K. Cooke, P. van den Driessche and X. Zou,
Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol., 39 (1999), 332-352.
doi: 10.1007/s002850050194. |
[11] |
H. Diaz, A. A. Ramirez, A. Olarte and C. Clavijo,
A model for the control of malaria using genetically modified vectors, J. Theor. Biol., 276 (2011), 57-66.
doi: 10.1016/j.jtbi.2011.01.053. |
[12] |
J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1960. |
[13] |
Y. Dumont and J. M. Tchuenche,
Mathematical studies on the sterile insect technique for the Chikungunya disease and Aedes albopictus, J. Math. Biol., 65 (2012), 809-854.
doi: 10.1007/s00285-011-0477-6. |
[14] |
V. A. Dyck, J. Hendrichs and A. S. Robinson, Sterile insect technique -principles and practice in area-wide integrated pest management, Springer, The Netherlands, 2005. Google Scholar |
[15] |
C. Dye,
Models for the population dynamics of the yellow fever mosquito, Aedes aegypti, J. Anim. Ecol., 53 (1984), 247-268.
doi: 10.2307/4355. |
[16] |
L. Esteva and H. M. Yang,
Assessing the effects of temperature and dengue virus load on dengue transmission, J. Biol. Syst., 23 (2015), 527-554.
doi: 10.1142/S0218339015500278. |
[17] |
L. Esteva and H. M. Yang,
Mathematical model to assess the control of Aedes aegypti mosquitoes by the sterile insect technique, Math. Biosci., 198 (2005), 132-147.
doi: 10.1016/j.mbs.2005.06.004. |
[18] |
J. E. Gentile, S. Rund and G. R Madey,
Modelling sterile insect technique to control the population of Anopheles gambiae, Malaria J., 14 (2015), 92-103.
doi: 10.1186/s12936-015-0587-5. |
[19] |
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equation, Springer, New York, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[20] |
J. Ito, A. Ghosh, L. A. Moreira, E. A. Wilmmer and M. Jacobs-Lorena, Transgenic anopheline mosquitoes impaired in transmission of a malria parasite, Nature, 417 (2002), 452-455. Google Scholar |
[21] |
M. Jankovic and S. Petrovskii,
Are time delays always destabilizing? Revisiting the role of time delays and the Allee effect, Theor Ecol., 7 (2014), 335-349.
doi: 10.1007/s12080-014-0222-z. |
[22] |
E. F. Knipling, Possibilities of insect control or eradication through the use of sexually sterile males, J. Econ. Entomol., 48 (1955), 459-462. Google Scholar |
[23] |
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993. |
[24] |
S. S. Lee, R. E. Baker, E. A. Gaffney and S. M. White,
Modelling Aedes aegypti mosquito control via transgenic and sterile insect techniques: Endemics and emerging outbreaks, J. Theor. Biol., 331 (2013), 78-90.
doi: 10.1016/j.jtbi.2013.04.014. |
[25] |
M. A. Lewis and P. van den Driessche,
Waves of extinction from sterile insect release, Math. Biosci., 116 (1993), 221-247.
doi: 10.1016/0025-5564(93)90067-K. |
[26] |
J. Li,
New revised simple models for interactive wild and sterile mosquito populations and their dynamics, J. Biol. Dyna., 11 (2017), 316-333.
doi: 10.1080/17513758.2016.1216613. |
[27] |
J. Li, L. Cai and Y. Li,
Stage-structured wild and sterile mosquito population models and their dynamics, J. Biol.Dyna., 11 (2017), 79-101.
doi: 10.1080/17513758.2016.1159740. |
[28] |
J. Lu and J. Li,
Dynamics of stage-structured discrete mosquito population, J. Appl. Anal. Comput., 1 (2011), 53-67.
|
[29] |
G. J. Lycett and F. C. Kafatos, Anti-malaria mosquitoes?, Nautre, 417 (2002), 387-388. Google Scholar |
[30] |
C. W. Morin and A. C. Comrie,
Regional and seasonal response of a West Nile virus vector to climate change, PNAS, 110 (2013), 15620-15625.
doi: 10.1073/pnas.1307135110. |
[31] |
W. W. Murdoch, C. J. Briggs and R. M. Nisbet, Consumer-resource dynamics, Princeton University Press, New Jersey, USA, 2003. Google Scholar |
[32] |
H. K. Phuc, M. H. Andreasen, et al, Late-acting dominant lethal genetic systems and mosquito control, BMC. Biol., 5 (2007), 11–16.
doi: 10.1186/1741-7007-5-11. |
[33] |
E. P. Pliego, J. Vel$\acute{a}$zquez-Castro and A. F. Collar,
Seasonality on the life cycle of Aedes aegypti mosquito and its statistical relation with dengue outbreaks, Appl. Math. Model., 50 (2017), 484-496.
doi: 10.1016/j.apm.2017.06.003. |
[34] |
M. Rafikov, L. Bevilacqua and A. P. Wyse,
Optimal control strategy of malaria vector using genetically modified mosquitoes, J. Theor. Biol., 258 (2009), 418-425.
doi: 10.1016/j.jtbi.2008.08.006. |
[35] |
S. J. Schreiber,
Allee effect, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209.
doi: 10.1016/S0040-5809(03)00072-8. |
[36] |
J. Smith, M. Amador and R. Barrera,
Seasonal and habitat effects on dengue and West Nile Virus Vectors in San Juan, Puerto Rico, J. Am. Mosq. Control. Assoc., 25 (2009), 38-46.
doi: 10.2987/08-5782.1. |
[37] |
H. Townson, SIT for African malaria vectors: Epilogue, Malar. J., 8 (2009), S10.
doi: 10.1186/1475-2875-8-S2-S10. |
[38] |
WHO, 10 facts on malaria, http://www.who.int/features/factfiles/malaria/en/. Google Scholar |
[39] |
J. Wu, H. R. Thieme, Y. Lou and G. Fan,
Stability and persistence in ODE models for populations with many stages, Math. Biosc. Eng., 12 (2015), 661-686.
doi: 10.3934/mbe.2015.12.661. |
[40] |
B. Zheng, M. Tang and J. Yu,
Modeling Wolbachia spread in mosquitoes through delay differential equations, SIAM J. Appl. Math., 74 (2014), 743-770.
doi: 10.1137/13093354X. |
[41] |
B. Zheng, M. Tang, J. Yu and J. Qiu,
Wolbachia spreading dynamics in mosquitoes with imperfect maternal transmission, J. Math. Biol., 76 (2018), 235-263.
doi: 10.1007/s00285-017-1142-5. |










[1] |
Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042 |
[2] |
Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084 |
[3] |
Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344 |
[4] |
Thazin Aye, Guanyu Shang, Ying Su. On a stage-structured population model in discrete periodic habitat: III. unimodal growth and delay effect. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021005 |
[5] |
Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020342 |
[6] |
Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021013 |
[7] |
Yiwen Tao, Jingli Ren. The stability and bifurcation of homogeneous diffusive predator–prey systems with spatio–temporal delays. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021038 |
[8] |
Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074 |
[9] |
Yangjian Sun, Changjian Liu. The Poincaré bifurcation of a SD oscillator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1565-1577. doi: 10.3934/dcdsb.2020173 |
[10] |
Wenxiong Chen, Congming Li, Shijie Qi. A Hopf lemma and regularity for fractional $ p $-Laplacians. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3235-3252. doi: 10.3934/dcds.2020034 |
[11] |
Ripeng Huang, Shaojian Qu, Xiaoguang Yang, Zhimin Liu. Multi-stage distributionally robust optimization with risk aversion. Journal of Industrial & Management Optimization, 2021, 17 (1) : 233-259. doi: 10.3934/jimo.2019109 |
[12] |
Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020461 |
[13] |
Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020464 |
[14] |
Kimie Nakashima. Indefinite nonlinear diffusion problem in population genetics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3837-3855. doi: 10.3934/dcds.2020169 |
[15] |
Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3883-3907. doi: 10.3934/dcds.2020129 |
[16] |
Adrian Viorel, Cristian D. Alecsa, Titus O. Pinţa. Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020407 |
[17] |
Pablo Neme, Jorge Oviedo. A note on the lattice structure for matching markets via linear programming. Journal of Dynamics & Games, 2020 doi: 10.3934/jdg.2021001 |
[18] |
Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020426 |
[19] |
Mengting Fang, Yuanshi Wang, Mingshu Chen, Donald L. DeAngelis. Asymptotic population abundance of a two-patch system with asymmetric diffusion. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3411-3425. doi: 10.3934/dcds.2020031 |
[20] |
Attila Dénes, Gergely Röst. Single species population dynamics in seasonal environment with short reproduction period. Communications on Pure & Applied Analysis, 2021, 20 (2) : 755-762. doi: 10.3934/cpaa.2020288 |
2018 Impact Factor: 1.313
Tools
Metrics
Other articles
by authors
[Back to Top]