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

Bifurcations and periodic orbits in variable population interactions

1. 

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

Received  November 2012 Revised  February 2013 Published  May 2013

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.
Citation: Jorge Rebaza. Bifurcations and periodic orbits in variable population interactions. Communications on Pure & 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.  doi: 10.2307/3545803.  Google Scholar

[2]

J. Addicott, Stability properties of 2-species models of mutualism: simulation studies,, Oecologia, 49 (1981), 42.  doi: 10.1007/BF00376896.  Google Scholar

[3]

D. Boucher, The ecology of mutualism,, Annual Review Ecol. Syst., 13 (1982), 315.  doi: 0.1146/annurev.es.13.110182.001531.  Google Scholar

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J. H. Cushman and J. F. Addicott, Conditional interactions in ant-plant-herbivore mutualisms,, in, 13 (1991), 92.  doi: 10.1017/S0007485300051579.  Google Scholar

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B. Ermentrout, "Simulating, Analyzing, and Animating Dynamical Systems,", SIAM (2002)., (2002).  doi: 10.1137/1.9780898718195.  Google Scholar

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M. J. Hernandez and I. Barradas, Variation in the outcome of population interactions: bifurcations and catastrophes,, Math. Biol., 46 (2003), 571.  doi: 10.1007/s00285-002-0192-4.  Google Scholar

[7]

M. J. Hernandez, Spatiotemporal dynamics in variable population interaction with density-dependent interaction coefficients,, Ecol. Modelling, 214 (2008), 3.  doi: 10.1007/s00285-002-0192-4.  Google Scholar

[8]

J. Holland and D. DeAngelis, Consumer-resource theory predicts dynamic transitions between outcomes of interspecific interactions,, Ecol. Letters, 12 (2009), 1357.  doi: 10.1111/j.1461-0248.2009.01390.x.  Google Scholar

[9]

C. Ji and D. Jiang, Persistence and non-persistence of a mutualism system with stochastic perturbation,, Disc. & Cont. Dynam. Syst., 32 (2012), 867.  doi: 10.3934/dcds.2012.32.867.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2009.07.003.  Google Scholar

[11]

K. Lan and C. Zhu, Phase portraits of predator-prey systems with harvesting rates,, Disc. & Cont. Dynam. Syst., 32 (2012), 901.  doi: 10.3934/dcds.2012.32.901.  Google Scholar

[12]

T. Lara and J. Rebaza, Dynamics of transitions in population interactions,, Nonl. Analysis: Real World Appl., 13 (2012), 1268.  doi: 10.1016/j.nonrwa.2011.10.004.  Google Scholar

[13]

B. Leard and J. Rebaza, Analysis of predator-prey models with continuous threshold harvesting,, Appl. Math. & Comp., 217 (2011), 5265.  doi: 10.1016/j.amc.2010.11.050.  Google Scholar

[14]

M. Liu and K. Wang, Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations,, Disc. & Cont. Dynam. Syst., 33 (2013), 2495.  doi: 10.3934/dcds.2013.33.2495.  Google Scholar

[15]

T. Peschak, "Currents of Contrast: Life in South Africa's Two Oceans,", Struik Publ., (2006).   Google Scholar

[16]

J. Rebaza, Dynamics of prey threshold harvesting and refuge,, J. Comp. & Appl. Math., 236 (2012), 1743.  doi: 10.1016/j.cam.2011.10.005.  Google Scholar

[17]

M. Rosenzweig, "Species Diversity in Space and Time,", \emph{Species Diversity in Space and Time}, ().  doi: 10.1017/CBO9780511623387.  Google Scholar

[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.  doi: 10.1073/pnas.94.11.5713.  Google Scholar

[19]

D. H. Wright, A simple, stable model of mutualism incorporating handling time,, The Amer. Naturalist, 194 (1989), 664.  doi: 10.1086/285003.  Google Scholar

[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.  doi: 10.1016/j.jtbi.2007.05.004.  Google Scholar

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.  doi: 10.2307/3545803.  Google Scholar

[2]

J. Addicott, Stability properties of 2-species models of mutualism: simulation studies,, Oecologia, 49 (1981), 42.  doi: 10.1007/BF00376896.  Google Scholar

[3]

D. Boucher, The ecology of mutualism,, Annual Review Ecol. Syst., 13 (1982), 315.  doi: 0.1146/annurev.es.13.110182.001531.  Google Scholar

[4]

J. H. Cushman and J. F. Addicott, Conditional interactions in ant-plant-herbivore mutualisms,, in, 13 (1991), 92.  doi: 10.1017/S0007485300051579.  Google Scholar

[5]

B. Ermentrout, "Simulating, Analyzing, and Animating Dynamical Systems,", SIAM (2002)., (2002).  doi: 10.1137/1.9780898718195.  Google Scholar

[6]

M. J. Hernandez and I. Barradas, Variation in the outcome of population interactions: bifurcations and catastrophes,, Math. Biol., 46 (2003), 571.  doi: 10.1007/s00285-002-0192-4.  Google Scholar

[7]

M. J. Hernandez, Spatiotemporal dynamics in variable population interaction with density-dependent interaction coefficients,, Ecol. Modelling, 214 (2008), 3.  doi: 10.1007/s00285-002-0192-4.  Google Scholar

[8]

J. Holland and D. DeAngelis, Consumer-resource theory predicts dynamic transitions between outcomes of interspecific interactions,, Ecol. Letters, 12 (2009), 1357.  doi: 10.1111/j.1461-0248.2009.01390.x.  Google Scholar

[9]

C. Ji and D. Jiang, Persistence and non-persistence of a mutualism system with stochastic perturbation,, Disc. & Cont. Dynam. Syst., 32 (2012), 867.  doi: 10.3934/dcds.2012.32.867.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2009.07.003.  Google Scholar

[11]

K. Lan and C. Zhu, Phase portraits of predator-prey systems with harvesting rates,, Disc. & Cont. Dynam. Syst., 32 (2012), 901.  doi: 10.3934/dcds.2012.32.901.  Google Scholar

[12]

T. Lara and J. Rebaza, Dynamics of transitions in population interactions,, Nonl. Analysis: Real World Appl., 13 (2012), 1268.  doi: 10.1016/j.nonrwa.2011.10.004.  Google Scholar

[13]

B. Leard and J. Rebaza, Analysis of predator-prey models with continuous threshold harvesting,, Appl. Math. & Comp., 217 (2011), 5265.  doi: 10.1016/j.amc.2010.11.050.  Google Scholar

[14]

M. Liu and K. Wang, Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations,, Disc. & Cont. Dynam. Syst., 33 (2013), 2495.  doi: 10.3934/dcds.2013.33.2495.  Google Scholar

[15]

T. Peschak, "Currents of Contrast: Life in South Africa's Two Oceans,", Struik Publ., (2006).   Google Scholar

[16]

J. Rebaza, Dynamics of prey threshold harvesting and refuge,, J. Comp. & Appl. Math., 236 (2012), 1743.  doi: 10.1016/j.cam.2011.10.005.  Google Scholar

[17]

M. Rosenzweig, "Species Diversity in Space and Time,", \emph{Species Diversity in Space and Time}, ().  doi: 10.1017/CBO9780511623387.  Google Scholar

[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.  doi: 10.1073/pnas.94.11.5713.  Google Scholar

[19]

D. H. Wright, A simple, stable model of mutualism incorporating handling time,, The Amer. Naturalist, 194 (1989), 664.  doi: 10.1086/285003.  Google Scholar

[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.  doi: 10.1016/j.jtbi.2007.05.004.  Google Scholar

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