• Previous Article
    Heterogeneous population dynamics and scaling laws near epidemic outbreaks
  • MBE Home
  • This Issue
  • Next Article
    Dynamical properties and tumor clearance conditions for a nine-dimensional model of bladder cancer immunotherapy
2016, 13(5): 1077-1092. doi: 10.3934/mbe.2016031

A two-sex matrix population model to represent harem structure

1. 

Department of Mathematics and Statistics, James Madison University, Harrisonburg, VA 22807, United States

2. 

Centro de Estudios Parasitológicos y de Vectores (CEPAVE, CONICET-CCT-La Plata, UNLP), Universidad Nacional de La Plata, La Plata, Prov. de Buenos Aires, Argentina, Argentina

Received  December 2014 Revised  April 2016 Published  July 2016

Population dynamic models often include males in the calculation of population change, but even in those cases males have rarely been introduced to represent polygyny (harem social structure), where it is particularly important to include males in the reproductive performance of the population. In this article we develop an adaptable matrix population modeling framework for species that have a harem-like social structure under an assumption that the transitions from newborn to juvenile and juvenile to adult both take one time step. We are able to calculate not only the growth rates and stable stage distributions, but also the mathematical expressions for harem size for this model. We then provide applications of this model to two mammal species with slightly different harem behavior.
Citation: Anthony Tongen, María Zubillaga, Jorge E. Rabinovich. A two-sex matrix population model to represent harem structure. Mathematical Biosciences & Engineering, 2016, 13 (5) : 1077-1092. doi: 10.3934/mbe.2016031
References:
[1]

J. Ballou and K. Ralls, Inbreeding and juvenile mortality in small populations of ungulates: A detailed analysis, Biological Conservation, 24 (1982), 239-272, URL http://www.sciencedirect.com/science/article/pii/0006320782900143. doi: 10.1016/0006-3207(82)90014-3.  Google Scholar

[2]

B. J. Brennan, S. M. Flaxman and S. H. Alonzo, Female alternative reproductive behaviors: The effect of female group size on mate assessment and copying, Journal of Theoretical Biology, 253 (2008), 561-569, URL http://www.sciencedirect.com/science/article/pii/S002251930800177X. doi: 10.1016/j.jtbi.2008.04.003.  Google Scholar

[3]

H. Caswell, Matrix Population Models: Construction, Analysis, and Interpretation, Sinauer Associates, Inc., 2001. Google Scholar

[4]

H. Caswell and D. E. Weeks, Two-sex models: Chaos, extinction, and other dynamic consequences of sex, The American Naturalist, 128 (1986), 707-735, URL http://www.jstor.org/stable/2461952. doi: 10.1086/284598.  Google Scholar

[5]

T. Coulson, E. J. Milner Gulland and T. Clutton Brock, The relative roles of density and climatic variation on population dynamics and fecundity rates in three contrasting ungulate species, Proceedings of the Royal Society of London B: Biological Sciences, 267 (2000), 1771-1779, URL http://rspb.royalsocietypublishing.org/content/267/1454/1771. doi: 10.1098/rspb.2000.1209.  Google Scholar

[6]

W. Franklin, Contrasting socioecologies of south america?s wild camelids: the vicu na and the guanaco, Special Publication of the American Society for Mammalogy, 7 (1983), 573-629. Google Scholar

[7]

M. Franz, O. Schülke and J. Ostner, Rapid evolution of cooperation in group-living animals, BMC Evolutionary Biology, 13 (2013), p235, URL http://www.scopus.com/inward/record.url?eid=2-s2.0-84887843907&partnerID=40&md5=f08444c4e94cae2e14e423f0d2a88abc, Cited By (since 1996)0. doi: 10.1186/1471-2148-13-235.  Google Scholar

[8]

C. Haridas, E. A. Eager, R. Rebarber and B. Tenhumberg, Frequency-dependent population dynamics: Effect of sex ratio and mating system on the elasticity of population growth rate, Theoretical Population Biology, 97 (2014), 49-56, URL http://www.sciencedirect.com/science/article/pii/S0040580914000641. doi: 10.1016/j.tpb.2014.08.003.  Google Scholar

[9]

A. Horev, R. Yosef, P. Tryjanowski and O. Ovadia, Consequences of variation in male harem size to population persistence: Modeling poaching and extinction risk of bengal tigers (panthera tigris), Biological Conservation, 147 (2012), 22-31, URL http://www.sciencedirect.com/science/article/pii/S0006320712000250. doi: 10.1016/j.biocon.2012.01.012.  Google Scholar

[10]

R. Jefferson, Size and Spacing of the Sedentary Guanaco Family Groups, Master's thesis, Iowa State University, Ames, Iowa, 1980. Google Scholar

[11]

A. Jensen, Sex and age structured matrix model applied to harvesting a white tailed deer population, Ecological Modelling, 128 (2000), 245-249, URL http://www.sciencedirect.com/science/article/pii/S0304380000001988. doi: 10.1016/S0304-3800(00)00198-8.  Google Scholar

[12]

R. Lancia, K. Pollock, J. Bishir and M. Conner, A white-tailed deer harvesting strategy, Journal of Wildlife Management, 52 (1988), 589-595, URL http://www.scopus.com/inward/record.url?eid=2-s2.0-0024264311&partnerID=40&md5=ccb5d2f1622664e0163043753ba4d542, Cited By (since 1996)8. doi: 10.2307/3800912.  Google Scholar

[13]

R. Langvatn and A. Loison, Consequences of harvesting on age structure, sex ratio and population dynamics of red deer Cervus elaphus in central norway, Wildlife Biology, 5 (1999), 213-223. Google Scholar

[14]

K. G. Magnusson and T. Kasuya, Mating strategies in whale populations: Searching strategy vs. harem strategy, Ecological Modelling, 102 (1997), 225-242, URL http://www.sciencedirect.com/science/article/pii/S0304380097000586. doi: 10.1016/S0304-3800(97)00058-6.  Google Scholar

[15]

L. Marescot, O. Gimenez, C. Duchamp, E. Marboutin and G. Chapron, Reducing matrix population models with application to social animal species, Ecological Modelling, 232 (2012), 91-96, URL http://www.sciencedirect.com/science/article/pii/S0304380012000932. doi: 10.1016/j.ecolmodel.2012.02.017.  Google Scholar

[16]

A. Marino and R. Baldi, Ecological correlates of group-size variation in a resource-defense ungulate, the sedentary guanaco, PLoS ONE, 9 (2014), e89060. doi: 10.1371/journal.pone.0089060.  Google Scholar

[17]

E. Milner-Gulland, A dynamic game model for the decision to join an aggregation, Ecological Modelling, 145 (2001), 85-99, URL http://www.sciencedirect.com/science/article/pii/S0304380001003817. Google Scholar

[18]

A. Moller, Sexual selection and extinction: why sex matters and why asexual models are insufficient, Annales Zoologici Fennici, 40 (2003), 221-230. Google Scholar

[19]

S. Puig and F. Videla, Comportamiento y organizacion social del guanaco, in Tecnicas para el Manejo de Guanacos (ed. Santiago), UICN, Suiza, 1995, chapter 7, 97-118. Google Scholar

[20]

J. Pérez-González and J. Carranza, Female aggregation interacts with population structure to influence the degree of polygyny in red deer, Animal Behaviour, 82 (2011), 957-970, URL http://www.sciencedirect.com/science/article/pii/S0003347211003034. Google Scholar

[21]

J. Rabinovich and M. Zubillaga, Informe Final del proyecto: "Modelo de manejo de poblaciones de guanacos para la Provincia del Chubut", 2012,, URL , ().  ,+2012,+" target="_new" title="Go to article in Google Scholar"> Google Scholar

[22]

D. J. Rankin and H. Kokko, Do males matter? the role of males in population dynamics, Oikos, 116 (2007), 335-348, URL http://search.ebscohost.com/login.aspx?direct=true&db=a9h&AN=23729292&site=ehost-live&scope=site. doi: 10.1111/j.0030-1299.2007.15451.x.  Google Scholar

[23]

A. O. Shelton, The ecological and evolutionary drivers of female-biased sex ratios: Two-sex models of perennial seagrasses, American Naturalist, 175 (2010), 302-315, URL http://search.ebscohost.com/login.aspx?direct=true&db=eih&AN=48656779&site=ehost-live&scope=site. doi: 10.1086/650374.  Google Scholar

[24]

A. Skonhoft, N. G. Yoccoz, N. C. Stenseth, J.-M. Gaillard and A. Loison, Management of chamois (rupicapra rupicapra) moving between a protected core area and a hunting area, Ecological Applications, 12 (2002), 1199-1211, URL http://www.jstor.org/stable/3061046. Google Scholar

[25]

A. Sundelöf and P. Âberg, Birth functions in stage structured two-sex models, Ecological Modelling, 193 (2006), 787-795, URL http://www.sciencedirect.com/science/article/pii/S0304380005003996. Google Scholar

[26]

C. Vanpé, P. Kjellander, M. Galan, J.-F. Cosson, S. Aulagnier, O. Liberg and A. J. M. Hewison, Mating system, sexual dimorphism, and the opportunity for sexual selection in a territorial ungulate, Behavioral Ecology, 19 (2008), 309-316, URL http://beheco.oxfordjournals.org/content/19/2/309.abstract. Google Scholar

[27]

M. J. Wade and S. M. Shuster, Sexual selection: Harem size and the variance in male reproductive success, The American Naturalist, 164 (2004), E83-E89, URL http://www.jstor.org/stable/10.1086/424531. doi: 10.1086/424531.  Google Scholar

[28]

M. Wade, S. Shuster and J. Demuth, Sexual selection favors female-biased sex ratios: The balance between the opposing forces of sex-ratio selection and sexual selection, American Naturalist, 162 (2003), 403-414, URL http://www.scopus.com/inward/record.url?eid=2-s2.0-0242625210&partnerID=40&md5=31fadf3308fdcd4293a968f4e0bf6a19, Cited By (since 1996)20. doi: 10.1086/378211.  Google Scholar

[29]

J. F. Wittenberger, Group size and polygamy in social mammals, The American Naturalist, 115 (1980), 197-222, URL http://www.jstor.org/stable/2460594. doi: 10.1086/283555.  Google Scholar

show all references

References:
[1]

J. Ballou and K. Ralls, Inbreeding and juvenile mortality in small populations of ungulates: A detailed analysis, Biological Conservation, 24 (1982), 239-272, URL http://www.sciencedirect.com/science/article/pii/0006320782900143. doi: 10.1016/0006-3207(82)90014-3.  Google Scholar

[2]

B. J. Brennan, S. M. Flaxman and S. H. Alonzo, Female alternative reproductive behaviors: The effect of female group size on mate assessment and copying, Journal of Theoretical Biology, 253 (2008), 561-569, URL http://www.sciencedirect.com/science/article/pii/S002251930800177X. doi: 10.1016/j.jtbi.2008.04.003.  Google Scholar

[3]

H. Caswell, Matrix Population Models: Construction, Analysis, and Interpretation, Sinauer Associates, Inc., 2001. Google Scholar

[4]

H. Caswell and D. E. Weeks, Two-sex models: Chaos, extinction, and other dynamic consequences of sex, The American Naturalist, 128 (1986), 707-735, URL http://www.jstor.org/stable/2461952. doi: 10.1086/284598.  Google Scholar

[5]

T. Coulson, E. J. Milner Gulland and T. Clutton Brock, The relative roles of density and climatic variation on population dynamics and fecundity rates in three contrasting ungulate species, Proceedings of the Royal Society of London B: Biological Sciences, 267 (2000), 1771-1779, URL http://rspb.royalsocietypublishing.org/content/267/1454/1771. doi: 10.1098/rspb.2000.1209.  Google Scholar

[6]

W. Franklin, Contrasting socioecologies of south america?s wild camelids: the vicu na and the guanaco, Special Publication of the American Society for Mammalogy, 7 (1983), 573-629. Google Scholar

[7]

M. Franz, O. Schülke and J. Ostner, Rapid evolution of cooperation in group-living animals, BMC Evolutionary Biology, 13 (2013), p235, URL http://www.scopus.com/inward/record.url?eid=2-s2.0-84887843907&partnerID=40&md5=f08444c4e94cae2e14e423f0d2a88abc, Cited By (since 1996)0. doi: 10.1186/1471-2148-13-235.  Google Scholar

[8]

C. Haridas, E. A. Eager, R. Rebarber and B. Tenhumberg, Frequency-dependent population dynamics: Effect of sex ratio and mating system on the elasticity of population growth rate, Theoretical Population Biology, 97 (2014), 49-56, URL http://www.sciencedirect.com/science/article/pii/S0040580914000641. doi: 10.1016/j.tpb.2014.08.003.  Google Scholar

[9]

A. Horev, R. Yosef, P. Tryjanowski and O. Ovadia, Consequences of variation in male harem size to population persistence: Modeling poaching and extinction risk of bengal tigers (panthera tigris), Biological Conservation, 147 (2012), 22-31, URL http://www.sciencedirect.com/science/article/pii/S0006320712000250. doi: 10.1016/j.biocon.2012.01.012.  Google Scholar

[10]

R. Jefferson, Size and Spacing of the Sedentary Guanaco Family Groups, Master's thesis, Iowa State University, Ames, Iowa, 1980. Google Scholar

[11]

A. Jensen, Sex and age structured matrix model applied to harvesting a white tailed deer population, Ecological Modelling, 128 (2000), 245-249, URL http://www.sciencedirect.com/science/article/pii/S0304380000001988. doi: 10.1016/S0304-3800(00)00198-8.  Google Scholar

[12]

R. Lancia, K. Pollock, J. Bishir and M. Conner, A white-tailed deer harvesting strategy, Journal of Wildlife Management, 52 (1988), 589-595, URL http://www.scopus.com/inward/record.url?eid=2-s2.0-0024264311&partnerID=40&md5=ccb5d2f1622664e0163043753ba4d542, Cited By (since 1996)8. doi: 10.2307/3800912.  Google Scholar

[13]

R. Langvatn and A. Loison, Consequences of harvesting on age structure, sex ratio and population dynamics of red deer Cervus elaphus in central norway, Wildlife Biology, 5 (1999), 213-223. Google Scholar

[14]

K. G. Magnusson and T. Kasuya, Mating strategies in whale populations: Searching strategy vs. harem strategy, Ecological Modelling, 102 (1997), 225-242, URL http://www.sciencedirect.com/science/article/pii/S0304380097000586. doi: 10.1016/S0304-3800(97)00058-6.  Google Scholar

[15]

L. Marescot, O. Gimenez, C. Duchamp, E. Marboutin and G. Chapron, Reducing matrix population models with application to social animal species, Ecological Modelling, 232 (2012), 91-96, URL http://www.sciencedirect.com/science/article/pii/S0304380012000932. doi: 10.1016/j.ecolmodel.2012.02.017.  Google Scholar

[16]

A. Marino and R. Baldi, Ecological correlates of group-size variation in a resource-defense ungulate, the sedentary guanaco, PLoS ONE, 9 (2014), e89060. doi: 10.1371/journal.pone.0089060.  Google Scholar

[17]

E. Milner-Gulland, A dynamic game model for the decision to join an aggregation, Ecological Modelling, 145 (2001), 85-99, URL http://www.sciencedirect.com/science/article/pii/S0304380001003817. Google Scholar

[18]

A. Moller, Sexual selection and extinction: why sex matters and why asexual models are insufficient, Annales Zoologici Fennici, 40 (2003), 221-230. Google Scholar

[19]

S. Puig and F. Videla, Comportamiento y organizacion social del guanaco, in Tecnicas para el Manejo de Guanacos (ed. Santiago), UICN, Suiza, 1995, chapter 7, 97-118. Google Scholar

[20]

J. Pérez-González and J. Carranza, Female aggregation interacts with population structure to influence the degree of polygyny in red deer, Animal Behaviour, 82 (2011), 957-970, URL http://www.sciencedirect.com/science/article/pii/S0003347211003034. Google Scholar

[21]

J. Rabinovich and M. Zubillaga, Informe Final del proyecto: "Modelo de manejo de poblaciones de guanacos para la Provincia del Chubut", 2012,, URL , ().  ,+2012,+" target="_new" title="Go to article in Google Scholar"> Google Scholar

[22]

D. J. Rankin and H. Kokko, Do males matter? the role of males in population dynamics, Oikos, 116 (2007), 335-348, URL http://search.ebscohost.com/login.aspx?direct=true&db=a9h&AN=23729292&site=ehost-live&scope=site. doi: 10.1111/j.0030-1299.2007.15451.x.  Google Scholar

[23]

A. O. Shelton, The ecological and evolutionary drivers of female-biased sex ratios: Two-sex models of perennial seagrasses, American Naturalist, 175 (2010), 302-315, URL http://search.ebscohost.com/login.aspx?direct=true&db=eih&AN=48656779&site=ehost-live&scope=site. doi: 10.1086/650374.  Google Scholar

[24]

A. Skonhoft, N. G. Yoccoz, N. C. Stenseth, J.-M. Gaillard and A. Loison, Management of chamois (rupicapra rupicapra) moving between a protected core area and a hunting area, Ecological Applications, 12 (2002), 1199-1211, URL http://www.jstor.org/stable/3061046. Google Scholar

[25]

A. Sundelöf and P. Âberg, Birth functions in stage structured two-sex models, Ecological Modelling, 193 (2006), 787-795, URL http://www.sciencedirect.com/science/article/pii/S0304380005003996. Google Scholar

[26]

C. Vanpé, P. Kjellander, M. Galan, J.-F. Cosson, S. Aulagnier, O. Liberg and A. J. M. Hewison, Mating system, sexual dimorphism, and the opportunity for sexual selection in a territorial ungulate, Behavioral Ecology, 19 (2008), 309-316, URL http://beheco.oxfordjournals.org/content/19/2/309.abstract. Google Scholar

[27]

M. J. Wade and S. M. Shuster, Sexual selection: Harem size and the variance in male reproductive success, The American Naturalist, 164 (2004), E83-E89, URL http://www.jstor.org/stable/10.1086/424531. doi: 10.1086/424531.  Google Scholar

[28]

M. Wade, S. Shuster and J. Demuth, Sexual selection favors female-biased sex ratios: The balance between the opposing forces of sex-ratio selection and sexual selection, American Naturalist, 162 (2003), 403-414, URL http://www.scopus.com/inward/record.url?eid=2-s2.0-0242625210&partnerID=40&md5=31fadf3308fdcd4293a968f4e0bf6a19, Cited By (since 1996)20. doi: 10.1086/378211.  Google Scholar

[29]

J. F. Wittenberger, Group size and polygamy in social mammals, The American Naturalist, 115 (1980), 197-222, URL http://www.jstor.org/stable/2460594. doi: 10.1086/283555.  Google Scholar

[1]

Agnieszka Ulikowska. An age-structured two-sex model in the space of radon measures: Well posedness. Kinetic & Related Models, 2012, 5 (4) : 873-900. doi: 10.3934/krm.2012.5.873

[2]

Yacouba Simporé, Oumar Traoré. Null controllability of a nonlinear age, space and two-sex structured population dynamics model. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021052

[3]

Bruno Buonomo, Deborah Lacitignola. On the stabilizing effect of cannibalism in stage-structured population models. Mathematical Biosciences & Engineering, 2006, 3 (4) : 717-731. doi: 10.3934/mbe.2006.3.717

[4]

Wei Feng, Michael T. Cowen, Xin Lu. Coexistence and asymptotic stability in stage-structured predator-prey models. Mathematical Biosciences & Engineering, 2014, 11 (4) : 823-839. doi: 10.3934/mbe.2014.11.823

[5]

Gigi Thomas, Edward M. Lungu. A two-sex model for the influence of heavy alcohol consumption on the spread of HIV/AIDS. Mathematical Biosciences & Engineering, 2010, 7 (4) : 871-904. doi: 10.3934/mbe.2010.7.871

[6]

Shangbing Ai, Jia Li, Jianshe Yu, Bo Zheng. Stage-structured models for interactive wild and periodically and impulsively released sterile mosquitoes. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021172

[7]

Lorenzo Mari, Marino Gatto, Renato Casagrandi. Local resource competition and the skewness of the sex ratio: a demographic model. Mathematical Biosciences & Engineering, 2008, 5 (4) : 813-830. doi: 10.3934/mbe.2008.5.813

[8]

Jia Li. Malaria model with stage-structured mosquitoes. Mathematical Biosciences & Engineering, 2011, 8 (3) : 753-768. doi: 10.3934/mbe.2011.8.753

[9]

Jinping Fang, Guang Lin, Hui Wan. Analysis of a stage-structured dengue model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 4045-4061. doi: 10.3934/dcdsb.2018125

[10]

Shangzhi Li, Shangjiang Guo. Dynamics of a two-species stage-structured model incorporating state-dependent maturation delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1393-1423. doi: 10.3934/dcdsb.2017067

[11]

Shangzhi Li, Shangjiang Guo. Dynamics of a stage-structured population model with a state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3523-3551. doi: 10.3934/dcdsb.2020071

[12]

Guirong Jiang, Qishao Lu, Linping Peng. Impulsive Ecological Control Of A Stage-Structured Pest Management System. Mathematical Biosciences & Engineering, 2005, 2 (2) : 329-344. doi: 10.3934/mbe.2005.2.329

[13]

Tufail Malik, Abba Gumel, Elamin H. Elbasha. Qualitative analysis of an age- and sex-structured vaccination model for human papillomavirus. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2151-2174. doi: 10.3934/dcdsb.2013.18.2151

[14]

Fengqing Chao, Ajit Kumar Yadav. Levels and trends in the sex ratio at birth and missing female births for 29 states and union territories in India 1990–2016: A Bayesian modeling study. Foundations of Data Science, 2019, 1 (2) : 177-196. doi: 10.3934/fods.2019008

[15]

Xinyu Song, Liming Cai, U. Neumann. Ratio-dependent predator-prey system with stage structure for prey. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 747-758. doi: 10.3934/dcdsb.2004.4.747

[16]

Seong Lee, Inkyung Ahn. Diffusive predator-prey models with stage structure on prey and beddington-deangelis functional responses. Communications on Pure & Applied Analysis, 2017, 16 (2) : 427-442. doi: 10.3934/cpaa.2017022

[17]

Qing Zhu, Huaqin Peng, Xiaoxiao Zheng, Huafeng Xiao. Bifurcation analysis of a stage-structured predator-prey model with prey refuge. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2195-2209. doi: 10.3934/dcdss.2019141

[18]

Thazin Aye, Guanyu Shang, Ying Su. On a stage-structured population model in discrete periodic habitat: III. unimodal growth and delay effect. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1763-1781. doi: 10.3934/dcdsb.2021005

[19]

Chenchen Zu, Xiaoqi Yang, Carisa Kwok Wai Yu. Sparse minimax portfolio and Sharpe ratio models. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021111

[20]

Jaume Llibre, Claudio Vidal. Hopf periodic orbits for a ratio--dependent predator--prey model with stage structure. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1859-1867. doi: 10.3934/dcdsb.2016026

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (132)
  • HTML views (0)
  • Cited by (0)

[Back to Top]