American Institute of Mathematical Sciences

Advanced
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

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 & 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.  Google Scholar

[2]

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

[3]

D. Boucher, The ecology of mutualism, Annual Review Ecol. Syst., 13 (1982), 315-347. 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 "Ant Plant Interactions" (C. R. Huxley and D. F. Cutler eds.), Oxford University Press, 13 (1991), 92-103. doi: 10.1017/S0007485300051579.  Google Scholar

[5]

B. Ermentrout, "Simulating, Analyzing, and Animating Dynamical Systems," SIAM (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), 571-594. 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), 3-16. 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), 1357-1366. 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), 867-889. 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), 2285-2295. 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), 901-933. 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), 1268-1277. 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), 5265-5278. 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), 2495-2522. 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), 1743-1752. 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), 5713-5715. 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), 664-667. 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), 145-153. 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), 81-90. doi: 10.2307/3545803.  Google Scholar

[2]

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

[3]

D. Boucher, The ecology of mutualism, Annual Review Ecol. Syst., 13 (1982), 315-347. 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 "Ant Plant Interactions" (C. R. Huxley and D. F. Cutler eds.), Oxford University Press, 13 (1991), 92-103. doi: 10.1017/S0007485300051579.  Google Scholar

[5]

B. Ermentrout, "Simulating, Analyzing, and Animating Dynamical Systems," SIAM (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), 571-594. 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), 3-16. 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), 1357-1366. 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), 867-889. 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), 2285-2295. 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), 901-933. 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), 1268-1277. 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), 5265-5278. 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), 2495-2522. 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), 1743-1752. 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), 5713-5715. 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), 664-667. 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), 145-153. doi: 10.1016/j.jtbi.2007.05.004.  Google Scholar

[1]

Xiang-Ping Yan, Wan-Tong Li. Stability and Hopf bifurcations for a delayed diffusion system in population dynamics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 367-399. doi: 10.3934/dcdsb.2012.17.367
[2]

Rinaldo M. Colombo, Mauro Garavello. Stability and optimization in structured population models on graphs. Mathematical Biosciences & Engineering, 2015, 12 (2) : 311-335. doi: 10.3934/mbe.2015.12.311
[3]

Zhanyuan Hou. Geometric method for global stability of discrete population models. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3305-3334. doi: 10.3934/dcdsb.2020063
[4]

Jianquan Li, Zhien Ma. Stability analysis for SIS epidemic models with vaccination and constant population size. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 635-642. doi: 10.3934/dcdsb.2004.4.635
[5]

Inom Mirzaev, David M. Bortz. A numerical framework for computing steady states of structured population models and their stability. Mathematical Biosciences & Engineering, 2017, 14 (4) : 933-952. doi: 10.3934/mbe.2017049
[6]

Hal L. Smith, Horst R. Thieme. Persistence and global stability for a class of discrete time structured population models. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4627-4646. doi: 10.3934/dcds.2013.33.4627
[7]

Jingzhi Li, Masahiro Yamamoto, Jun Zou. Conditional Stability and Numerical Reconstruction of Initial Temperature. Communications on Pure & Applied Analysis, 2009, 8 (1) : 361-382. doi: 10.3934/cpaa.2009.8.361
[8]

Dan Liu, Shigui Ruan, Deming Zhu. Bifurcation analysis in models of tumor and immune system interactions. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 151-168. doi: 10.3934/dcdsb.2009.12.151
[9]

Cesare Tronci. Hybrid models for perfect complex fluids with multipolar interactions. Journal of Geometric Mechanics, 2012, 4 (3) : 333-363. doi: 10.3934/jgm.2012.4.333
[10]

Anatoly Neishtadt. On stability loss delay for dynamical bifurcations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 897-909. doi: 10.3934/dcdss.2009.2.897
[11]

Sophia R.-J. Jang. Allee effects in an iteroparous host population and in host-parasitoid interactions. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 113-135. doi: 10.3934/dcdsb.2011.15.113
[12]

Dongmei Xiao. Dynamics and bifurcations on a class of population model with seasonal constant-yield harvesting. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 699-719. doi: 10.3934/dcdsb.2016.21.699
[13]

L. Aguirre, P. Seibert. Types of change of stability and corresponding types of bifurcations. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 741-752. doi: 10.3934/dcds.1999.5.741
[14]

Mary Ballyk, Ross Staffeldt, Ibrahim Jawarneh. A nutrient-prey-predator model: Stability and bifurcations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 2975-3004. doi: 10.3934/dcdss.2020192
[15]

Zhiting Xu. Traveling waves in an SEIR epidemic model with the variable total population. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3723-3742. doi: 10.3934/dcdsb.2016118
[16]

Eduardo Liz, Alfonso Ruiz-Herrera. Delayed population models with Allee effects and exploitation. Mathematical Biosciences & Engineering, 2015, 12 (1) : 83-97. doi: 10.3934/mbe.2015.12.83
[17]

Dianmo Li, Zengxiang Gao, Zufei Ma, Baoyu Xie, Zhengjun Wang. Two general models for the simulation of insect population dynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 623-628. doi: 10.3934/dcdsb.2004.4.623
[18]

Reinhard Bürger. A survey of migration-selection models in population genetics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 883-959. doi: 10.3934/dcdsb.2014.19.883
[19]

B. E. Ainseba, W. E. Fitzgibbon, M. Langlais, J. J. Morgan. An application of homogenization techniques to population dynamics models. Communications on Pure & Applied Analysis, 2002, 1 (1) : 19-33. doi: 10.3934/cpaa.2002.1.19
[20]

Robert Carlson. Myopic models of population dynamics on infinite networks. Networks & Heterogeneous Media, 2014, 9 (3) : 477-499. doi: 10.3934/nhm.2014.9.477

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (84)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy