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Bifurcations and periodic orbits in variable population interactions
1. | Department of Mathematics, Missouri State University, Springfield, MO 65897 |
References:
[1] |
Z. Abramsky, M. Rosenzweig and A. Subach, Gerbils under threat of owl predation: isoclines and isodars, Oikos, 78 (1997), 81-90.
doi: 10.2307/3545803. |
[2] |
J. Addicott, Stability properties of 2-species models of mutualism: simulation studies, Oecologia, 49 (1981), 42-49.
doi: 10.1007/BF00376896. |
[3] |
D. Boucher, The ecology of mutualism, Annual Review Ecol. Syst., 13 (1982), 315-347.
doi: 0.1146/annurev.es.13.110182.001531. |
[4] |
J. H. Cushman and J. F. Addicott, Conditional interactions in ant-plant-herbivore mutualisms, in "Ant Plant Interactions" (C. R. Huxley and D. F. Cutler eds.), Oxford University Press, 13 (1991), 92-103.
doi: 10.1017/S0007485300051579. |
[5] |
B. Ermentrout, "Simulating, Analyzing, and Animating Dynamical Systems," SIAM (2002).
doi: 10.1137/1.9780898718195. |
[6] |
M. J. Hernandez and I. Barradas, Variation in the outcome of population interactions: bifurcations and catastrophes, Math. Biol., 46 (2003), 571-594.
doi: 10.1007/s00285-002-0192-4. |
[7] |
M. J. Hernandez, Spatiotemporal dynamics in variable population interaction with density-dependent interaction coefficients, Ecol. Modelling, 214 (2008), 3-16.
doi: 10.1007/s00285-002-0192-4. |
[8] |
J. Holland and D. DeAngelis, Consumer-resource theory predicts dynamic transitions between outcomes of interspecific interactions, Ecol. Letters, 12 (2009), 1357-1366.
doi: 10.1111/j.1461-0248.2009.01390.x. |
[9] |
C. Ji and D. Jiang, Persistence and non-persistence of a mutualism system with stochastic perturbation, Disc. & Cont. Dynam. Syst., 32 (2012), 867-889.
doi: 10.3934/dcds.2012.32.867. |
[10] |
L. Ji and C. Wu, Qualitative analysis of a predator-prey model with constant-rate prey harvesting incorporating a constant prey refuge, Nonl. Anal: Real World Appl., 11 (2010), 2285-2295.
doi: 10.1016/j.nonrwa.2009.07.003. |
[11] |
K. Lan and C. Zhu, Phase portraits of predator-prey systems with harvesting rates, Disc. & Cont. Dynam. Syst., 32 (2012), 901-933.
doi: 10.3934/dcds.2012.32.901. |
[12] |
T. Lara and J. Rebaza, Dynamics of transitions in population interactions, Nonl. Analysis: Real World Appl., 13 (2012), 1268-1277.
doi: 10.1016/j.nonrwa.2011.10.004. |
[13] |
B. Leard and J. Rebaza, Analysis of predator-prey models with continuous threshold harvesting, Appl. Math. & Comp., 217 (2011), 5265-5278.
doi: 10.1016/j.amc.2010.11.050. |
[14] |
M. Liu and K. Wang, Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations, Disc. & Cont. Dynam. Syst., 33 (2013), 2495-2522.
doi: 10.3934/dcds.2013.33.2495. |
[15] |
T. Peschak, "Currents of Contrast: Life in South Africa's Two Oceans," Struik Publ., 2006. |
[16] |
J. Rebaza, Dynamics of prey threshold harvesting and refuge, J. Comp. & Appl. Math., 236 (2012), 1743-1752.
doi: 10.1016/j.cam.2011.10.005. |
[17] |
M. Rosenzweig, "Species Diversity in Space and Time," Species Diversity in Space and Time
doi: 10.1017/CBO9780511623387. |
[18] |
M. Rosenzweig, Z. Abramsky and A. Subach, Safety in numbers: Sophisticated vigilance by Allenby抯 gerbil, Annual Review Ecol. Syst. Ecology, 94 (1997), 5713-5715.
doi: 10.1073/pnas.94.11.5713. |
[19] |
D. H. Wright, A simple, stable model of mutualism incorporating handling time, The Amer. Naturalist, 194 (1989), 664-667.
doi: 10.1086/285003. |
[20] |
B. Zhang, Z. Zhang, Z. Li and Y. Tao, Stability analysis of a two-species model with transitions between population interactions, J. Theor. Biol., 248 (2007), 145-153.
doi: 10.1016/j.jtbi.2007.05.004. |
show all references
References:
[1] |
Z. Abramsky, M. Rosenzweig and A. Subach, Gerbils under threat of owl predation: isoclines and isodars, Oikos, 78 (1997), 81-90.
doi: 10.2307/3545803. |
[2] |
J. Addicott, Stability properties of 2-species models of mutualism: simulation studies, Oecologia, 49 (1981), 42-49.
doi: 10.1007/BF00376896. |
[3] |
D. Boucher, The ecology of mutualism, Annual Review Ecol. Syst., 13 (1982), 315-347.
doi: 0.1146/annurev.es.13.110182.001531. |
[4] |
J. H. Cushman and J. F. Addicott, Conditional interactions in ant-plant-herbivore mutualisms, in "Ant Plant Interactions" (C. R. Huxley and D. F. Cutler eds.), Oxford University Press, 13 (1991), 92-103.
doi: 10.1017/S0007485300051579. |
[5] |
B. Ermentrout, "Simulating, Analyzing, and Animating Dynamical Systems," SIAM (2002).
doi: 10.1137/1.9780898718195. |
[6] |
M. J. Hernandez and I. Barradas, Variation in the outcome of population interactions: bifurcations and catastrophes, Math. Biol., 46 (2003), 571-594.
doi: 10.1007/s00285-002-0192-4. |
[7] |
M. J. Hernandez, Spatiotemporal dynamics in variable population interaction with density-dependent interaction coefficients, Ecol. Modelling, 214 (2008), 3-16.
doi: 10.1007/s00285-002-0192-4. |
[8] |
J. Holland and D. DeAngelis, Consumer-resource theory predicts dynamic transitions between outcomes of interspecific interactions, Ecol. Letters, 12 (2009), 1357-1366.
doi: 10.1111/j.1461-0248.2009.01390.x. |
[9] |
C. Ji and D. Jiang, Persistence and non-persistence of a mutualism system with stochastic perturbation, Disc. & Cont. Dynam. Syst., 32 (2012), 867-889.
doi: 10.3934/dcds.2012.32.867. |
[10] |
L. Ji and C. Wu, Qualitative analysis of a predator-prey model with constant-rate prey harvesting incorporating a constant prey refuge, Nonl. Anal: Real World Appl., 11 (2010), 2285-2295.
doi: 10.1016/j.nonrwa.2009.07.003. |
[11] |
K. Lan and C. Zhu, Phase portraits of predator-prey systems with harvesting rates, Disc. & Cont. Dynam. Syst., 32 (2012), 901-933.
doi: 10.3934/dcds.2012.32.901. |
[12] |
T. Lara and J. Rebaza, Dynamics of transitions in population interactions, Nonl. Analysis: Real World Appl., 13 (2012), 1268-1277.
doi: 10.1016/j.nonrwa.2011.10.004. |
[13] |
B. Leard and J. Rebaza, Analysis of predator-prey models with continuous threshold harvesting, Appl. Math. & Comp., 217 (2011), 5265-5278.
doi: 10.1016/j.amc.2010.11.050. |
[14] |
M. Liu and K. Wang, Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations, Disc. & Cont. Dynam. Syst., 33 (2013), 2495-2522.
doi: 10.3934/dcds.2013.33.2495. |
[15] |
T. Peschak, "Currents of Contrast: Life in South Africa's Two Oceans," Struik Publ., 2006. |
[16] |
J. Rebaza, Dynamics of prey threshold harvesting and refuge, J. Comp. & Appl. Math., 236 (2012), 1743-1752.
doi: 10.1016/j.cam.2011.10.005. |
[17] |
M. Rosenzweig, "Species Diversity in Space and Time," Species Diversity in Space and Time
doi: 10.1017/CBO9780511623387. |
[18] |
M. Rosenzweig, Z. Abramsky and A. Subach, Safety in numbers: Sophisticated vigilance by Allenby抯 gerbil, Annual Review Ecol. Syst. Ecology, 94 (1997), 5713-5715.
doi: 10.1073/pnas.94.11.5713. |
[19] |
D. H. Wright, A simple, stable model of mutualism incorporating handling time, The Amer. Naturalist, 194 (1989), 664-667.
doi: 10.1086/285003. |
[20] |
B. Zhang, Z. Zhang, Z. Li and Y. Tao, Stability analysis of a two-species model with transitions between population interactions, J. Theor. Biol., 248 (2007), 145-153.
doi: 10.1016/j.jtbi.2007.05.004. |
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