In this paper, we mainly consider a eco-epidemiological predator-prey system where delay is time-varying to study the transmission dynamics of Bacterial white spot disease in Litopenaeus Vannamei, which will contribute to the sustainable development of shrimp. First, the permanence and the positiveness of solutions are given. Then, the conditions for the local asymptotic stability of the equilibriums are established. Next, the global asymptotic stability for the system around the positive equilibrium is gained by applying the functional differential equation theory and constructing a proper Lyapunov function. Last, some numerical examples verify the validity and feasibility of previous theoretical results.
Citation: Xue Liu, Xin You Meng. Dynamics of Bacterial white spot disease spreads in Litopenaeus Vannamei with time-varying delay[J]. Mathematical Biosciences and Engineering, 2023, 20(12): 20748-20769. doi: 10.3934/mbe.2023918
[1] | Md Nazmul Hassan, Angela Peace . Mechanistically derived Toxicant-mediated predator-prey model under Stoichiometric constraints. Mathematical Biosciences and Engineering, 2020, 17(1): 349-365. doi: 10.3934/mbe.2020019 |
[2] | Yawen Yan, Hongyue Wang, Xiaoyuan Chang, Jimin Zhang . Asymmetrical resource competition in aquatic producers: Constant cell quota versus variable cell quota. Mathematical Biosciences and Engineering, 2023, 20(2): 3983-4005. doi: 10.3934/mbe.2023186 |
[3] | Huanyi Liu, Hengguo Yu, Chuanjun Dai, Zengling Ma, Qi Wang, Min Zhao . Dynamical analysis of an aquatic amensalism model with non-selective harvesting and Allee effect. Mathematical Biosciences and Engineering, 2021, 18(6): 8857-8882. doi: 10.3934/mbe.2021437 |
[4] | Moitri Sen, Malay Banerjee, Yasuhiro Takeuchi . Influence of Allee effect in prey populations on the dynamics of two-prey-one-predator model. Mathematical Biosciences and Engineering, 2018, 15(4): 883-904. doi: 10.3934/mbe.2018040 |
[5] | Zhenliang Zhu, Yuming Chen, Zhong Li, Fengde Chen . Dynamic behaviors of a Leslie-Gower model with strong Allee effect and fear effect in prey. Mathematical Biosciences and Engineering, 2023, 20(6): 10977-10999. doi: 10.3934/mbe.2023486 |
[6] | Lina Hao, Meng Fan, Xin Wang . Effects of nutrient enrichment on coevolution of a stoichiometric producer-grazer system. Mathematical Biosciences and Engineering, 2014, 11(4): 841-875. doi: 10.3934/mbe.2014.11.841 |
[7] | Yueping Dong, Jianlu Ren, Qihua Huang . Dynamics of a toxin-mediated aquatic population model with delayed toxic responses. Mathematical Biosciences and Engineering, 2020, 17(5): 5907-5924. doi: 10.3934/mbe.2020315 |
[8] | Md Nazmul Hassan, Kelsey Thompson, Gregory Mayer, Angela Peace . Effect of Excess Food Nutrient on Producer-Grazer Model under Stoichiometric and Toxicological Constraints. Mathematical Biosciences and Engineering, 2019, 16(1): 150-167. doi: 10.3934/mbe.2019008 |
[9] | Santanu Bhattacharya, Nandadulal Bairagi . Dynamic optimization of fishing tax and tourism fees for sustainable bioeconomic resource management. Mathematical Biosciences and Engineering, 2025, 22(7): 1751-1789. doi: 10.3934/mbe.2025064 |
[10] | Shuangte Wang, Hengguo Yu . Stability and bifurcation analysis of the Bazykin's predator-prey ecosystem with Holling type Ⅱ functional response. Mathematical Biosciences and Engineering, 2021, 18(6): 7877-7918. doi: 10.3934/mbe.2021391 |
In this paper, we mainly consider a eco-epidemiological predator-prey system where delay is time-varying to study the transmission dynamics of Bacterial white spot disease in Litopenaeus Vannamei, which will contribute to the sustainable development of shrimp. First, the permanence and the positiveness of solutions are given. Then, the conditions for the local asymptotic stability of the equilibriums are established. Next, the global asymptotic stability for the system around the positive equilibrium is gained by applying the functional differential equation theory and constructing a proper Lyapunov function. Last, some numerical examples verify the validity and feasibility of previous theoretical results.
We are pleased to present the edition in Mathematical Biosciences and Engineering of a Special Issue that highlights machine learning in molecular biology. Our aim is to report latest developments both in computational methods and analysis expanding the existed biological knowledge in molecular biological systems. We feature both web-based resources, which provide easy access to users, downloadable tools of particular use for in-house processing, and the inclusion into pipelines being developed in the laboratory.
In this special issue, Zhu et al. [1] developed a new approach to computationally reconstruct the 3D structure of the X-chromosome during XCI, in which the chain of DNA beads representing a chromosome is stored and simulated inside a 3D cubic lattice. They first generated the 3D structures of the X-chromosome before and after XCI by applying simulated annealing and Metropolis-Hastings simulations. Then, Xist localization intensities on the X-chromosome (RAP data) are used to model the traveling speeds or acceleration between all bead pairs during the process of XCI. With their approach, the 3D structures of the X-chromosome at 3 hours, 6 hours, and 24 hours after the start of the Xist expression, which initiates the XCI process, have been reconstructed.
Long noncoding RNAs (lncRNA) play important roles in gene expression regulation in diverse biological contexts. While lncRNA-gene interactions are closely related to the occurrence and development of cancers, the new target genes could be detected from known lncRNA regulated genes. Lu et al. [2] developed a method by using a biclustering approach for elucidating lncRNA-gene interactions, which allows for the identification of particular expression patterns across multiple datasets, indicating networks of lncRNA and gene interactions. Their method was applied and evaluated on the breast cancer RNA-seq datasets along with a set of known lncRNA regulated genes. Their method provides useful information for future studies on lncRNAs.
RNA modification site prediction offers an insight into diverse cellular processing in the regulation of organisms. Deep learning can detect optimal feature patterns to represent input data other than feature engineering from traditional machine learning methods. Sun et al. [3] developed DeepMRMP (Multiple Types RNA Modification Sites Predictor), a predictor for multiple types of RNA modifications method, which is based on the bidirectional Gated Recurrent Unit (BGRU) and transfer learning. Using multiple RNA site modification data and correlation among them, DeepMRMP build predictor for different types of RNA modification sites. DeepMRMP identifies N1-methyladenosine (m1A), pseudouridine (Ψ), 5-methylcytosine (m5C) modification sites through 10-fold cross-validation of the RNA sequences of H. sapiens, M. musculus and S. cerevisiae,
In biomedical research, near infrared spectroscopy (NIRS) is widely applied to analysis of active ingredients in medicinal fungi. Huang et al. [4] introduced an autonomous feature extraction method to model original NIRS vectors using attention based residual network (ABRN). Attention module in ABRN is employed to enhance feature wave bands and to decay noise. Different from traditional NIRS analysis methods, ABRN does not require any preprocessing of artificial feature selections which rely on expert experience. Comparing with other methods on various benchmarks and measurements, ABRN has better performance in autonomously extracting feature wave bands from original NIRS vectors, which can decrease the loss of tiny feature peaks.
Selectively and non-covalently interact with hormone, the soluble carrier hormone binding protein (HBP) plays an important role in the growth of human and other animals. Since experimental methods are still labor intensive and cost ineffective to identify HBP, it's necessary to develop computational methods to accurately and efficiently identify HBP. In Tan et al.'s paper [5], a machine learning-based method named as HBPred2.0 was proposed to identify HBP, in which the samples were encoded by using the optimal tripeptide composition obtained based on the binomial distribution method. The proposed method yielded an overall accuracy of 97.15% in the 5-fold cross-validation test. A user-friendly webserver is also provided.
Sun et al. [6] propose novel machine learning methods for recognition cancer biomarkers in saliva. As cancer tissues can make disease-specific changes in some salivary proteins through some mediators in the pathogenesis of systemic diseases, effectively identify these salivary proteins as potential markers is one of the challenging issues. With the proposed approach, salivary secreted proteins are recognized which are considered as candidate biomarkers of cancers. SVC-KM method is used to cluster the remaining proteins, and select negative samples from each cluster in proportion. Experimental results show the proposed methods can improve the accuracy of recognition by solving the problems of unbalanced sample size and uneven distribution in training set. They analyze the gene expression data of three types of cancer, and predict that 33 genes will appear in saliva after they are translated into proteins. This study provides a computational tool to help biologists and researchers reduce the number of candidate proteins and the cost of research in saliva diagnosis.
We hope that the readers will find this Special Issue helpful in identifying tools and analysis to help them in their study of particular molecular biological problems. In addition, this Issue is also providing an insight into current developments in bioinformatics where the articles describe the strategies being employed to exploring and interpreting sophisticate biological mechanisms, inferring underling relationships and interactions, predicting consequences from disturbance and building hypothesis in molecular biological systems.
Last but not least, we thank all the authors contributing to this special issue, and editor May Zhao's help and excellent work.
[1] |
A. A. Berryman, The orgins and evolution of predator-prey theory, Ecology, 73 (1992), 1530–1535. https://doi.org/10.2307/1940005 doi: 10.2307/1940005
![]() |
[2] |
X. X. Liu, S. Y. Liu, Dynamics of a predator-prey system with inducible defense and disease in the prey, Nonlinear Anal. Real., 71 (2023), 103802. https://doi.org/10.1016/j.nonrwa.2022.103802 doi: 10.1016/j.nonrwa.2022.103802
![]() |
[3] |
X. Y. Meng, H. F. Huo, X. B. Zhang, Stability and global Hopf bifurcation in a Leslie-Gower predator-prey model with stage structure for prey, J. Appl. Math. Comput., 60 (2019), 1–25. https://doi.org/10.1007/s12190-018-1201-0 doi: 10.1007/s12190-018-1201-0
![]() |
[4] |
M. Gyllenberg, P. Yan, Y. Wang, Limit cycles for competitor-competitor-mutualist Lotka-Volterra systems, Phys. D, 221 (2006), 135–145. https://doi.org/10.1016/j.physd.2006.07.016 doi: 10.1016/j.physd.2006.07.016
![]() |
[5] |
X. Y. Meng, N. N. Qin, H. F. Huo, Dynamics of a food chain model with two infected predators, Int. J. Bifurcat. Chaos, 31 (2021), 2150019. https://doi.org/10.1142/S021812742150019X doi: 10.1142/S021812742150019X
![]() |
[6] |
Z. W. Liang, X. Y. Meng, Stability and Hopf bifurcation of a multiple delayed predator-prey system with fear effect, prey refuge and Crowley-Martin function, Chaos Solitons Fractals, 175 (2023), 113955. https://doi.org/10.1016/j.chaos.2023.113955 doi: 10.1016/j.chaos.2023.113955
![]() |
[7] |
Y. S. Chen, T. Giletti, J. S. Guo, Persistence of preys in a diffusive three species predator-prey system with a pair of strong-weak competing preys, J. Differ. Equations, 281 (2021), 341–378. https://doi.org/10.1016/j.jde.2021.02.013 doi: 10.1016/j.jde.2021.02.013
![]() |
[8] | R. E. Gaines, J. L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, New York, 2006. https://doi.org/10.1007/bfb0089537 |
[9] |
M. Sen, M. Banerjee, A. Morozov, Bifurcation analysis of a ratio-dependent prey-predator model with the Allee effect, Ecol. Complex., 11 (2012), 12–27. https://doi.org/10.1016/j.ecocom.2012.01.002 doi: 10.1016/j.ecocom.2012.01.002
![]() |
[10] |
Y. Song, W. Xiao, X. Y. Qi, Stability and Hopf bifurcation of a predator-prey model with stage structure and time delay for the prey, Nonlinear Dyn., 83 (2016), 1409–1418. http://dx.doi.org/10.1007/s11071-015-2413-6 doi: 10.1007/s11071-015-2413-6
![]() |
[11] |
M. Cai, S. L. Yan, Z. J. Du, Positive periodic solutions of an eco-epidemic model with Crowley-Martin type functional response and disease in the prey, Qual. Theor. Dyn. Syst., 19 (2020), 1–20. https://doi.org/10.1007/s12346-020-00392-3 doi: 10.1007/s12346-020-00392-3
![]() |
[12] |
J. B. Zhang, H. Fang, Multiple periodic solutions for a discrete time model of plankton allelopathy, Adv. Differ. Equation, 2006 (2006), 1–14. https://doi.org/10.1155/ade/2006/90479 doi: 10.1155/ade/2006/90479
![]() |
[13] |
X. Y. Meng, Y. Q. Wu, Dynamical analysis of a fuzzy phytoplankton-zooplankton model with refuge, fishery protection and harvesting, J. Appl. Math. Comput., 63 (2020), 361–389. https://doi.org/10.1007/s12190-020-01321-y doi: 10.1007/s12190-020-01321-y
![]() |
[14] |
X. S. Xiong, Z. Q. Zhang, Periodic solutions of a discrete two-species competitive model with stage structure, Math. Comput. Model., 48 (2008), 333–343. https://doi.org/10.1016/j.mcm.2007.10.004 doi: 10.1016/j.mcm.2007.10.004
![]() |
[15] |
W. P. Zhang, D. M. Zhu, P. Bi, Multiple positive periodic solutions of a delayed discrete predator-prey system with type IV functional responses, Appl. Math. Lett., 20 (2007), 1031–1038. https://doi.org/10.1016/j.aml.2006.11.005 doi: 10.1016/j.aml.2006.11.005
![]() |
[16] |
Z. Q. Zhang, J. B. Luo, Multiple periodic solutions of a delayed predator-prey system with stage structure for the predator, Nonlinear Anal. Real., 11 (2010), 4109–4120. https://doi.org/10.1016/j.nonrwa.2010.03.015 doi: 10.1016/j.nonrwa.2010.03.015
![]() |
[17] |
Y. K. Li, K. H. Zhao, Y. Ye, Multiple positive periodic solutions of n-species delay competition systems with harvesting terms, Nonlinear Anal. Real., 12 (2011), 1013–1022. https://doi.org/10.1016/j.nonrwa.2010.08.024 doi: 10.1016/j.nonrwa.2010.08.024
![]() |
[18] |
Y. G. Sun, S. H. Saker, Positive periodic solutions of discrete three-level food-chain model of Holling type II, Appl. Math. Comput., 180 (2006), 353–365. https://doi.org/10.1016/j.amc.2005.12.015 doi: 10.1016/j.amc.2005.12.015
![]() |
[19] |
X. H. Ding, C. Lu, Existence of positive periodic solution for ratio-dependent n-species difference system, Appl. Math. Model., 33 (2009), 2748–2756. https://doi.org/10.1016/j.apm.2008.08.008 doi: 10.1016/j.apm.2008.08.008
![]() |
[20] |
K. Chakraborty, M. Chakraborty, T. K. Kar, Bifurcation and control of a bioeconomic model of a prey-predator system with a time delay, Nonlinear Anal. Hyb., 5 (2011), 613–625. https://doi.org/10.1016/j.nahs.2011.05.004 doi: 10.1016/j.nahs.2011.05.004
![]() |
[21] |
Sajan, B. Dubey, S. K. Sasmal, Chaotic dynamics of a plankton-fish system with fear and its carry over effects in the presence of a discrete delay, Chaos Solitons Fractals, 160 (2022), 112245. https://doi.org/10.1016/j.chaos.2022.112245 doi: 10.1016/j.chaos.2022.112245
![]() |
[22] |
J. G. Wang, X. Y. Meng, L. Lv, J. Li, Stability and bifurcation analysis of a Beddington-DeAngelis prey-predator model with fear effect, prey refuge and harvesting, Int. J. Bifurcat. Chaos, 33 (2023), 2350013. https://dx.doi.org/10.1142/S021812742350013X doi: 10.1142/S021812742350013X
![]() |
[23] | K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Springer Science and Business Media, Netherland, 1992. |
[24] | Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993. |
[25] |
L. Fan, Z. K. Shi, S. Y. Tang, Critical values of stability and Hopf bifurcations for a delayed population model with delay-dependent parameters, Nonlinear Anal. Real., 11 (2010), 341–355. https://doi.org/10.1016/j.nonrwa.2008.11.016 doi: 10.1016/j.nonrwa.2008.11.016
![]() |
[26] |
J. B. Geng, Y. H. Xia, Almost periodic solutions of a nonlinear ecological model, Commun. Nonlinear. Sci., 16 (2011), 2575–2597. https://doi.org/10.1016/j.cnsns.2010.09.033 doi: 10.1016/j.cnsns.2010.09.033
![]() |
[27] | J. C. Holmes, W. M. Bethel, Modification of intermediate host behaviour by parasites, Zoolog. J. Linnean Soc., 51 (1972), 123–149. |
[28] | R. M. Anderson, R. M. May, Infectious Diseases of Humans: Dynamics and Control, London, Oxford University Press, 1991. |
[29] |
E. Venturino, Epidemics in predator-prey models: disease in the predators, Math. Medic. Biolog., 19 (2002), 185–205. https://doi.org/10.1093/imammb/19.3.185 doi: 10.1093/imammb/19.3.185
![]() |
[30] |
M. Haque, A predator-prey model with disease in the predator species only, Nonlinear Anal. Real., 11 (2010), 2224–2236. https://doi.org/10.1016/j.nonrwa.2009.06.012 doi: 10.1016/j.nonrwa.2009.06.012
![]() |
[31] |
A. Pal, A. Bhattacharyya, A. Mondal, Qualitative analysis and control of predator switching on an eco-epidemiological model with prey refuge and harvesting, Result. Control. Opt., 7 (2022), 100099. https://doi.org/10.1016/j.rico.2022.100099 doi: 10.1016/j.rico.2022.100099
![]() |
[32] |
Y. Zhang, S. J. Gao, S. H. Chen, A stochastic predator-prey eco-epidemiological model with the fear effect, Appl. Math. Lett., 134 (2022), 108300. https://doi.org/10.1016/j.aml.2022.108300 doi: 10.1016/j.aml.2022.108300
![]() |
[33] |
Z. K. Guo, W. L. Li, L. H. Cheng, Z. Z. Li, Eco-epidemiological model with epidemic and response function in the predator, J. Lanzhou Univ., 45 (2009), 117–121. https://doi.org/10.1360/972009-1650 doi: 10.1360/972009-1650
![]() |
[34] |
Y. N. Zeng, P. Yu, Complex dynamics of predator-prey systems with {Allee Effect}, Int. J. Bifurcat. Chaos, 32 (2022), 2250203. https://doi.org/10.1142/S0218127422502030 doi: 10.1142/S0218127422502030
![]() |
[35] |
G. H. Lin, L. Wang, J. S. Yu, Basins of attraction and paired Hopf bifurcations for delay differential equations with bistable nonlinearity and delay-dependent coefficient, J. Differ. Equations, 354 (2023), 183–206. https://doi.org/10.1016/j.jde.2023.01.015 doi: 10.1016/j.jde.2023.01.015
![]() |
[36] |
R. Xu, S. H. Zhang, Modelling and analysis of a delayed predator-prey model with disease in the predator, Appl. Math. Comput., 224 (2013), 372–386. https://doi.org/10.1016/j.amc.2013.08.067 doi: 10.1016/j.amc.2013.08.067
![]() |
[37] |
A. K. Verma, S. Gupta, S. P. Singh, N. S. Nagpure, An update on mechanism of entry of white spot syndrome virus into shrimps, Fish Shel. Immun., 67 (2017), 141–146. https://doi.org/10.1016/j.fsi.2017.06.007 doi: 10.1016/j.fsi.2017.06.007
![]() |
[38] |
C. F. Lo, C. H. Ho, C. H. Chen, K. F. Liu, Y. L. Chiu, P. Y. Yeh, et al., Detection and tissue tropism of white spot syndrome baculovirus (WSBV) in captured brooders of Penaeus monodon with a special emphasis on reproductive organs, Dis. Aquat. Organ., 30 (1997), 53–72. https://doi.org/10.3354/dao030053 doi: 10.3354/dao030053
![]() |
[39] | A. P. Sangamaheswaran, Jeyaseelan, White spot viral disease in penaeid shrimp–A review, Naga, 24 (2001), 16–22. |
[40] |
K. Pada Das, K. Kundu, J. Chattopadhyay, A predator–prey mathematical model with both the populations affected by diseases, Ecol. Complex., 8 (2011), 68–80. https://doi.org/10.1016/j.ecocom.2010.04.001 doi: 10.1016/j.ecocom.2010.04.001
![]() |
[41] |
S. Durand, D. Lightner, R. Redman, J. Bonami, Ultrastructure and morphogenesis of white spot syndrome baculovirus, Dis. Aquat. Organ., 29 (1997), 205–211. https://doi.org/10.3354/dao029205 doi: 10.3354/dao029205
![]() |
[42] |
M. E. Megahed, A comparison of the severity of white spot disease in cultured shrimp (Fenneropenaeus indicus) at a farm level in Egypt. I-Molecular, histopathological and field observations, Egypt. J. Aquat. Biol. Fish., 23 (2019), 613–637. https://doi.org/10.21608/ejabf.2019.47301 doi: 10.21608/ejabf.2019.47301
![]() |
[43] |
W. Warapond, A. Chitchanok, K. Panmile, J. Wachira, Effect of dietary Pediococcus pentosaceus MR001 on intestinal bacterial diversity and white spot syndrome virus protection in Pacific white shrimp, Aquacult. Rep., 30 (2023), 101570. https://doi.org/10.1016/j.aqrep.2023.101570 doi: 10.1016/j.aqrep.2023.101570
![]() |
[44] |
X. H. Wang, C. X. Lu, F. X. Wan, M. M. Onchari, X. Yin, B. Tian, et al., Enhance the biocontrol efficiency of Bacillus velezensis Bs916 for white spot syndrome virus in crayfish by overproduction of cyclic lipopeptide locillomycin, Aquaculture, 573 (2023), 739596. https://doi.org/10.1016/j.aquaculture.2023.739596 doi: 10.1016/j.aquaculture.2023.739596
![]() |
[45] |
X. B. Gao, Q. H. Pan, M. F. He, Y. B. Kang, A predator-prey model with diseases in both prey and predator, Physica A, 392 (2013), 5898–5906. https://doi.org/10.1016/j.physa.2013.07.077 doi: 10.1016/j.physa.2013.07.077
![]() |
[46] |
X. D. Ding, Global attractivity of Nicholson's blowflies system with patch structure and multiple pairs of distinct time-varying delays, Int. J. Biomat., 16 (2023), 2250081. https://doi.org/10.1142/S1793524522500814 doi: 10.1142/S1793524522500814
![]() |
[47] |
X. Long, S. H. Gong, New results on stability of Nicholson's blowflies equation with multiple pairs of time-varying delays, Appl. Math. Lett., 100 (2020), 106027. https://doi.org/10.1016/j.aml.2019.106027 doi: 10.1016/j.aml.2019.106027
![]() |
[48] |
S. Gao, K. Y. Peng, C. R. Zhang, Existence and global exponential stability of periodic solutions for feedback control complex dynamical networks with time-varying delays, Chaos Soliton. Fract., 152 (2021), 111483. https://doi.org/10.1016/j.chaos.2021.111483 doi: 10.1016/j.chaos.2021.111483
![]() |
[49] |
C. J. Xu, P. L. Li, Y. Guo, Global asymptotical stability of almost periodic solutions for a non-autonomous competing model with time-varying delays and feedback controls, J. Biolog. Dyn., 13 (2019), 407–421. https://doi.org/10.1080/17513758.2019.1610514 doi: 10.1080/17513758.2019.1610514
![]() |
[50] |
X. Y. Zhou, X. Y. Shi, X. Y. Song, Analysis of nonautonomous predator-prey model with nonlinear diffusion and time delay, J. Appl. Math. Comput., 196 (2008), 129–136. https://doi.org/10.1016/j.amc.2007.05.041 doi: 10.1016/j.amc.2007.05.041
![]() |
[51] |
X. P. Yan, C. H. Zhang, Hopf bifurcation in a delayed Lokta-Volterra predator-prey system, Nonlinear Anal. Real., 9 (2008), 114–127. https://doi.org/10.1016/j.nonrwa.2006.09.007 doi: 10.1016/j.nonrwa.2006.09.007
![]() |
[52] |
K. Li, J. J. Wei, Stability and Hopf bifurcation analysis of a prey-predator system with two delays, Chaos Soliton. Fract., 42 (2009), 2606–2613. https://doi.org/10.1016/j.chaos.2009.04.001 doi: 10.1016/j.chaos.2009.04.001
![]() |
[53] |
X. Lv, S. Lu, P. Yan, Existence and global attractivity of positive periodic solutions of Lotka-Volterra predator-prey systems with deviatin arguments, Nonlinear Anal. Real., 11 (2010), 574–583. https://doi.org/10.1016/j.nonrwa.2009.09.004 doi: 10.1016/j.nonrwa.2009.09.004
![]() |
[54] |
F. D. Chen, Z. Li, Y. J. Huang, Note on the permanence of a competitive system with infinite delay and feedback controls, Nonlinear Anal. Real., 8 (2007), 680–687. https://doi.org/10.1016/j.nonrwa.2006.02.006 doi: 10.1016/j.nonrwa.2006.02.006
![]() |
[55] | H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, United States of America, 1995. http://dx.doi.org/10.1090/surv/041/03 |
[56] | J. K. Hale, S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. |
[57] |
F. Montes de Oca, M. Vivas, Extinction in two dimensional Lotka-Volterra system with infinite delay, Nonlinear Anal. Real., 7 (2006), 1042–1047. https://doi.org/10.1016/j.nonrwa.2005.09.005 doi: 10.1016/j.nonrwa.2005.09.005
![]() |
[58] | T. Yoshizawa, Stability Theory by Liapunov's Second Method, Mathematical Society of Japan, Tokyo, 1966. |
1. | Igor Pantic, Jovana Paunovic, Snezana Pejic, Dunja Drakulic, Ana Todorovic, Sanja Stankovic, Danijela Vucevic, Jelena Cumic, Tatjana Radosavljevic, Artificial intelligence approaches to the biochemistry of oxidative stress: Current state of the art, 2022, 358, 00092797, 109888, 10.1016/j.cbi.2022.109888 | |
2. | Arindam Ghosh, Aritri Bir, Evaluating ChatGPT's Ability to Solve Higher-Order Questions on the Competency-Based Medical Education Curriculum in Medical Biochemistry, 2023, 2168-8184, 10.7759/cureus.37023 |