Bifurcations and periodic orbits in variable population interactions

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November 2013, 12(6): 2997-3012. doi: 10.3934/cpaa.2013.12.2997

Bifurcations and periodic orbits in variable population interactions

Jorge Rebaza ^1,

1.	Department of Mathematics, Missouri State University, Springfield, MO 65897

Received November 2012 Revised February 2013 Published May 2013

Abstract
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Variable population interactions with harvesting on one of the species are studied. Existence and stability of equilibria and existence of periodic solutions are established, existence of some bifurcation phenomena are analytically and numerically studied, explicit threshold values are computed to determine the kind of interaction (mutualism, competition, host-parasite) between the species, and several numerical examples are provided to illustrate the main results in this work. A brief discussion on the influence of the harvesting function on the dynamics of the model is also included. Hopf bifurcations and periodic solutions are found for the first time in this kind of models.

Keywords: stability, bifurcations., Variable Population models, conditional interactions.

Mathematics Subject Classification: Primary: 37N25, 58F17; Secondary: 92D2.

Citation: Jorge Rebaza. Bifurcations and periodic orbits in variable population interactions. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2997-3012. doi: 10.3934/cpaa.2013.12.2997

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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