Document Zbl 1410.37083

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Math. Comput.</a> <a href="/?q=in%3A412623" title="Articles in this Issue">264, 99-115 (2015)</a>. </div> <div class="abstract">Summary: The dynamical behaviors of an isolated population model involving delay-dependent parameters are investigated. It is shown that the positive equilibrium switches from being stable to unstable and then back to stable as the delay increases, and the Hopf bifurcation occurs finite times between the two critical values of stability changes which can be analytically determined. Moreover, the bifurcating periodic solutions are expressed analytically in an approximate form by the perturbation approach and Floquet technique. The direction and stability of the bifurcating periodic solutions are also determined. Finally, the validity of the results is shown by the consistency with the numerical simulations.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6923402">Cited in <strong>2</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A37N25" title="MSC2020">37N25</a> </td> <td class="space"> Dynamical systems in biology </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A92D25" title="MSC2020">92D25</a> </td> <td class="space"> Population dynamics (general) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34K20" title="MSC2020">34K20</a> </td> <td class="space"> Stability theory of functional-differential equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aisolated+population+model">isolated population model</a>; <a href="/?q=ut%3Adelay">delay</a>; <a href="/?q=ut%3Astability+switch">stability switch</a>; <a href="/?q=ut%3AHopf+bifurcation">Hopf bifurcation</a>; <a href="/?q=ut%3Aperturbation+approach">perturbation approach</a>; <a href="/?q=ut%3AFloquet+technique">Floquet technique</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1410.37083" data-ciurl="/ci/06923402" data-biburl="/bibtex/06923402.bib" data-amsurl="/amsrefs/06923402.bib" data-xmlurl="/xml/06923402.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06923402.pdf" title="Zbl 1410.37083 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.amc.2015.04.071" aria-label="DOI for “Stability switches and Hopf bifurcations of an isolated population model with delay-dependent parameters”" title="10.1016/j.amc.2015.04.071">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Edelstein-Keshet, L., Mathematical Models in Biology, (1988), Random House New York · <a href="/0674.92001" class="nowrap">Zbl 0674.92001</a></td> </tr><tr> <td>[2]</td> <td class="space">Merdan, H.; Gümüs, O., Stability analysis of a general discrete-time population model involving delay and allee effects, Appl. Math. Comput., 219, 1821-1832, (2012) · <a href="/1291.93027" class="nowrap">Zbl 1291.93027</a></td> </tr><tr> <td>[3]</td> <td class="space">Vargas-De-Leon, C.; Korobeinikov, A., Global stability of a population dynamics model with inhibition and negative feedback, Math. Med. Biol., 30, 65-72, (2013) · <a href="/1318.92044" class="nowrap">Zbl 1318.92044</a></td> </tr><tr> <td>[4]</td> <td class="space">Franco, D.; Logemann, H.; Perán, J., Global stability of an age-structured population model, Syst. Control Lett., 65, 30-36, (2014) · <a href="/1285.93077" class="nowrap">Zbl 1285.93077</a></td> </tr><tr> <td>[5]</td> <td class="space">Belozyorov, V. Y.; Chernyshenko, S. V., Quadratic model of inter-population interaction: investigation of stability areas, Appl. Math. Comput., 230, 43-56, (2014) · <a href="/1410.34159" class="nowrap">Zbl 1410.34159</a></td> </tr><tr> <td>[6]</td> <td class="space">Kulenovic, M. R.S.; Ladas, G.; Sficas, Y. G., Global attractivity in population-dynamics, Comput. Math. Appl., 18, 925-928, (1989) · <a href="/0686.92019" class="nowrap">Zbl 0686.92019</a></td> </tr><tr> <td>[7]</td> <td class="space">Tang, S. Y.; Chen, L. S., Global attractivity in a “food-limited” population model with impulsive effects, J. Math. Anal. Appl., 292, 211-221, (2004) · <a href="/1062.34055" class="nowrap">Zbl 1062.34055</a></td> </tr><tr> <td>[8]</td> <td class="space">El-Morshedy, H. A., Global attractivity in a population model with nonlinear death rate and distributed delays, J. Math. Anal. Appl., 410, 642-658, (2014) · <a href="/1307.92322" class="nowrap">Zbl 1307.92322</a></td> </tr><tr> <td>[9]</td> <td class="space">Gopalsamy, K.; Liu, P. Z., Persistence and global stability in a population model, J. Math. Anal. Appl., 224, 59-80, (1998) · <a href="/0912.34040" class="nowrap">Zbl 0912.34040</a></td> </tr><tr> <td>[10]</td> <td class="space">Cheng, C. P.; Li, W. T.; Wang, Z. C., Persistence of bistable waves in a delayed population model with stage structure on a two-dimensional spatial lattice, Nonlinear Anal. Real World Appl., 13, 1873-1890, (2012) · <a href="/1254.92074" class="nowrap">Zbl 1254.92074</a></td> </tr><tr> <td>[11]</td> <td class="space">Freedman, H. I.; Pinho, S. T.R., Persistence and extinction in a mathematical model of cell populations affected by radiation, Period. Math. Hung., 56, 25-35, (2008) · <a href="/1164.92002" class="nowrap">Zbl 1164.92002</a></td> </tr><tr> <td>[12]</td> <td class="space">Wang, Q.; Wang, Y.; Dai, B., Existence and uniqueness of positive periodic solutions for a neutral logarithmic population model, Appl. Math. Comput., 213, 137-147, (2009) · <a href="/1177.34093" class="nowrap">Zbl 1177.34093</a></td> </tr><tr> <td>[13]</td> <td class="space">Chen, F., Periodic solutions and almost periodic solutions for a delay multispecies logarithmic population model, Appl. Math. Comput., 171, 760-770, (2005) · <a href="/1089.92038" class="nowrap">Zbl 1089.92038</a></td> </tr><tr> <td>[14]</td> <td class="space">Zhao, Y.; Wang, L.; Zhao, H., Existence and exponential stability of almost periodic solutions for a neutral multi-species logarithmic population model, Appl. Math. Comput., 218, 5346-5356, (2012) · <a href="/1244.92047" class="nowrap">Zbl 1244.92047</a></td> </tr><tr> <td>[15]</td> <td class="space">Wang, Q.; Dai, B., Existence of positive periodic solutions for a neutral population model with delays and impulse, Nonlinear Anal. Real World Appl., 69, 3919-3930, (2008) · <a href="/1166.34047" class="nowrap">Zbl 1166.34047</a></td> </tr><tr> <td>[16]</td> <td class="space">Alzabut, J. O.; Stamov, G. T.; Sermutlu, E., On almost periodic solutions for an impulsive delay logarithmic population model, Math. Comput. Model., 51, 625-631, (2010) · <a href="/1190.34087" class="nowrap">Zbl 1190.34087</a></td> </tr><tr> <td>[17]</td> <td class="space">Zhang, Y.; Jia, J., Hopf bifurcation of an epidemic model with a nonlinear birth in population and vertical transmission, Appl. Math. Comput., 230, 164-173, (2014) · <a href="/1410.92146" class="nowrap">Zbl 1410.92146</a></td> </tr><tr> <td>[18]</td> <td class="space">Ma, Z. P., Stability and Hopf bifurcation for a three component reaction diffusion population model with delay effect, Appl. Math. Model., 37, 5984-6007, (2013) · <a href="/1288.35053" class="nowrap">Zbl 1288.35053</a></td> </tr><tr> <td>[19]</td> <td class="space">Yan, X. P.; Li, W. T., Stability of bifurcating periodic solutions in a delayed reaction-diffusion population model, Nonlinearity, 23, 1413, (2010) · <a href="/1198.37080" class="nowrap">Zbl 1198.37080</a></td> </tr><tr> <td>[20]</td> <td class="space">Lan, Y.; Li, Q. G., Control of Hopf bifurcation in a simple plankton population model with a non-integer exponent of closure, Appl. Math. Comput., 200, 220-230, (2008) · <a href="/1137.92031" class="nowrap">Zbl 1137.92031</a></td> </tr><tr> <td>[21]</td> <td class="space">Liz, E.; Ruiz-Herrera, A., Chaos in discrete structured population models, SIAM J. Dyn. Syst., 11, 1200-1214, (2012) · <a href="/1260.37021" class="nowrap">Zbl 1260.37021</a></td> </tr><tr> <td>[22]</td> <td class="space">Liz, E.; Ruiz-Herrera, A., The hydra effect, bubbles, and chaos in a simple discrete population model with constant effort harvesting, J. Math. Biol., 65, 997-1016, (2012) · <a href="/1251.92044" class="nowrap">Zbl 1251.92044</a></td> </tr><tr> <td>[23]</td> <td class="space">Duarte, J.; Januário, C.; Martins, N.; Sardanyés, J., On chaos, transient chaos and ghosts in single population models with allee effects, Nonlinear Anal. Real World Appl., 13, 1647-1661, (2012) · <a href="/1254.92076" class="nowrap">Zbl 1254.92076</a></td> </tr><tr> <td>[24]</td> <td class="space">Petráš, I., Chaos in fractional-order population model, Int. J. Bifur. Chaos, 22, 1250072, (2012) · <a href="/1258.34103" class="nowrap">Zbl 1258.34103</a></td> </tr><tr> <td>[25]</td> <td class="space">Aiello, W. G.; Freedman, H. I.; Wu, J., Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52, 855-869, (1992) · <a href="/0760.92018" class="nowrap">Zbl 0760.92018</a></td> </tr><tr> <td>[26]</td> <td class="space">Brauer, F.; Castillo-Châavez, C., Mathematical Models in Population Biology and Epidemiology, (2012), Springer New York · <a href="/1302.92001" class="nowrap">Zbl 1302.92001</a></td> </tr><tr> <td>[27]</td> <td class="space">Cooke, K.; van den Driessche, P.; Zhou, X., Interaction of maturation delay and nonlinear birth in population and epidemic model, J. Math. Biol., 39, 332-352, (1999) · <a href="/0945.92016" class="nowrap">Zbl 0945.92016</a></td> </tr><tr> <td>[28]</td> <td class="space">Nisbet, R. M.; Gurney, W. S.C., Modelling Fluctuating Populations, (1982), John Wiley and Sons New York · <a href="/0593.92013" class="nowrap">Zbl 0593.92013</a></td> </tr><tr> <td>[29]</td> <td class="space">Velasco-Hernández, J. X., A model for chagas disease involving transmission by vectors and blood transfusion, Theor. Pop. Biol., 46, 1-31, (1994) · <a href="/0804.92024" class="nowrap">Zbl 0804.92024</a></td> </tr><tr> <td>[30]</td> <td class="space">Carletti, M.; Beretta, E., Numerical detection of instability regions for delay models with delay-dependent parameters, J. Comput. Appl. Math., 205, 835-848, (2007) · <a href="/1124.34053" class="nowrap">Zbl 1124.34053</a></td> </tr><tr> <td>[31]</td> <td class="space">Ma, Z. P.; Huo, H. F.; Liu, C. Y., Stability and Hopf bifurcation analysis on a predator-prey model with discrete and distributed delays, Nonlinear Anal. Real World Appl., 10, 1160-1172, (2009) · <a href="/1167.34382" class="nowrap">Zbl 1167.34382</a></td> </tr><tr> <td>[32]</td> <td class="space">Xiao, M.; Zheng, W. X.; Cao, J., Hopf bifurcation of an (n+1)-neuron bidirectional associative memory neural network model with delays, IEEE Trans. Neural Netw. Learn. Syst., 24, 118-132, (2013)</td> </tr><tr> <td>[33]</td> <td class="space">Xu, W.; Cao, J.; Xiao, M., Bifurcation analysis and control in exponential red algorithm, Neurocomputing, 129, 232-245, (2014)</td> </tr><tr> <td>[34]</td> <td class="space">Shen, Z.; Zhang, C., Double Hopf bifurcation of coupled dissipative stuart-Landau oscillators with delay, Appl. Math. Comput., 227, 553-566, (2014) · <a href="/1364.37110" class="nowrap">Zbl 1364.37110</a></td> </tr><tr> <td>[35]</td> <td class="space">Xiao, M.; Zheng, W. X.; Cao, J., Bifurcation and control in a neural network with small and large delays, Neural Netw., 44, 132-142, (2013) · <a href="/1296.93065" class="nowrap">Zbl 1296.93065</a></td> </tr><tr> <td>[36]</td> <td class="space">Adimy, M.; Crauste, F.; Hbid, M. L.; Qesmi, R., Stability and Hopf bifurcation for a cell population model with state-dependent delay, SIAM J. Appl. Math., 70, 1611-1633, (2010) · <a href="/1206.34102" class="nowrap">Zbl 1206.34102</a></td> </tr><tr> <td>[37]</td> <td class="space">Beretta, E.; Kuang, Y., Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33, 1144-1165, (2002) · <a href="/1013.92034" class="nowrap">Zbl 1013.92034</a></td> </tr><tr> <td>[38]</td> <td class="space">Li, F.; Shi, Z. K.; Tang, S. Y., Critical values of stability and Hopf bifurcations for a delayed population model with delay-dependent parameters, Nonlinear Anal. Real World Appl., 11, 341-355, (2010) · <a href="/1190.34106" class="nowrap">Zbl 1190.34106</a></td> </tr><tr> <td>[39]</td> <td class="space">Casal, A.; Freedman, M., A poincare-Lindstedt approach to bifurcation problems for differential-delay equations, IEEE Trans. Automat. Control, 25, 967-973, (1980) · <a href="/0437.34058" class="nowrap">Zbl 0437.34058</a></td> </tr><tr> <td>[40]</td> <td class="space">Gopalsamy, K., Delay induced periodicity in a neural netlet of excitation and inhibition, Physica D, 89, 395-426, (1996) · <a href="/0883.68108" class="nowrap">Zbl 0883.68108</a></td> </tr><tr> <td>[41]</td> <td class="space">Xiao, M.; Cao, J., Approximate expressions of the bifurcating periodic solutions in a neuron model with delay-dependent parameters by perturbation approach, Cogn. Neurodyn., 4, 241-250, (2010)</td> </tr><tr> <td>[42]</td> <td class="space">Gopalsamy, K., Stability and Oscillations in Delay Differential Equations of Population Dynamics, (1992), Kluwer Academic Publishers Dordrecht · <a href="/0752.34039" class="nowrap">Zbl 0752.34039</a></td> </tr><tr> <td>[43]</td> <td class="space">Stokes, A., On the stability of a limit cycle of autonomous functional differential equations, Contr. Differ. Equ., 3, 121-140, (1963)</td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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