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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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[21]

J. Rabinovich and M. Zubillaga, Informe Final del proyecto: "Modelo de manejo de poblaciones de guanacos para la Provincia del Chubut", 2012, URL http://figshare.com/articles/Wild_Guanaco_Population_Management_Report/779840.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[21]

J. Rabinovich and M. Zubillaga, Informe Final del proyecto: "Modelo de manejo de poblaciones de guanacos para la Provincia del Chubut", 2012, URL http://figshare.com/articles/Wild_Guanaco_Population_Management_Report/779840.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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