# American Institute of Mathematical Sciences

May  2017, 22(3): 1099-1110. doi: 10.3934/dcdsb.2017054

## On carrying-capacity construction, metapopulations and density-dependent mortality

 1 Instituto de Matemáticas, Universidad Nacional Autónoma de México, Boulevard Juriquilla No. 3001, Juriquilla, 76230, México 2 Departamento de Matemáticas Aplicadas y Sistemas, DMAS, Universidad Autónoma Metropolitana, Cuajimalpa, Av. Vasco de Quiroga 4871, Col. Santa Fe Cuajimalpa, Cuajimalpa de Morelos, 05300, México, D.F., México 3 CONACYT Research Fellow, Instituto de Matemáticas, Universidad Nacional Autónoma de México, Boulevard Juriquilla No. 3001, Juriquilla, 76230, México

Received  September 2015 Revised  April 2016 Published  January 2017

We present a mathematical model for competition between species that includes variable carrying capacity within the framework of niche construction. We make use the classical Lotka-Volterra system for species competition and introduce a new variable which contains the dynamics of the constructed niche. The paper illustrates that the total available patches at equilibrium always exceeds the constructedniche at equilibrium in the absence of species.

Citation: J. X. Velasco-Hernández, M. Núñez-López, G. Ramírez-Santiago, M. Hernández-Rosales. On carrying-capacity construction, metapopulations and density-dependent mortality. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1099-1110. doi: 10.3934/dcdsb.2017054
##### References:
 [1] J. E. Keymer, M. A. Fuentes and P. A. Marquet, Diversity emerging: From competitive exclusion to neutral coexistence in ecosystems, Theoretical Ecology, 5 (2012), 457-463. doi: 10.1007/s12080-011-0138-9. Google Scholar [2] J. E. Keymer and P. A. Marquet, he complexity of cancer ecosystems, in TMariana Benitez, Octavio Miramontes, and Alfonso Valiente, editors Frontiers in Ecology, Evolution and Complexity, Copit Arxives, Mexico City, first edition, (2014), 101–119.Google Scholar [3] D. C. Krakauer, K. M. Page and H. E. Douglas, Diversity, dilemmas, and monopolies of niche construction, The American Naturalist, 173 (2009), 26-40. doi: 10.1086/593707. Google Scholar [4] J. Mena-Lorca, J. X. Velasco-Hernández and C. Castillo-Chavez, Density-dependent dynamics and superinfection in an epidemic model, IMA Journal of Mathematics Applied to Medicine and Biology, 16 (1999), 307-317. Google Scholar [5] J. Mena-Lorca, J. X. Velasco-Hernández and P. A. Marquet, Coexistence in metacommunities: A three-species model, Mathematical Biosciences, 202 (2006), 42-56. doi: 10.1016/j.mbs.2006.04.005. Google Scholar [6] F. J. Odling-Smee, K. N. Laland and M. W. Feldman, Niche Construction: The Neglected Process in Evolution, Princeton University Press, 2003.Google Scholar [7] E. W. Seabloom, E. T. Borer, K. Gross, A. E. Kendig, C. Lacroix, C. E. Mitchell, E. A. Mordecai and A. G. Power, The community ecology of pathogens: Coinfection, coexistence and community composition, Ecology Letters, 18 (2015), 401-415. doi: 10.1111/ele.12418. Google Scholar [8] D. Tilman, Competition and Biodiversity in spatially structured habitats, Ecology, 75 (1994), 2-16. doi: 10.2307/1939377. Google Scholar

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##### References:
 [1] J. E. Keymer, M. A. Fuentes and P. A. Marquet, Diversity emerging: From competitive exclusion to neutral coexistence in ecosystems, Theoretical Ecology, 5 (2012), 457-463. doi: 10.1007/s12080-011-0138-9. Google Scholar [2] J. E. Keymer and P. A. Marquet, he complexity of cancer ecosystems, in TMariana Benitez, Octavio Miramontes, and Alfonso Valiente, editors Frontiers in Ecology, Evolution and Complexity, Copit Arxives, Mexico City, first edition, (2014), 101–119.Google Scholar [3] D. C. Krakauer, K. M. Page and H. E. Douglas, Diversity, dilemmas, and monopolies of niche construction, The American Naturalist, 173 (2009), 26-40. doi: 10.1086/593707. Google Scholar [4] J. Mena-Lorca, J. X. Velasco-Hernández and C. Castillo-Chavez, Density-dependent dynamics and superinfection in an epidemic model, IMA Journal of Mathematics Applied to Medicine and Biology, 16 (1999), 307-317. Google Scholar [5] J. Mena-Lorca, J. X. Velasco-Hernández and P. A. Marquet, Coexistence in metacommunities: A three-species model, Mathematical Biosciences, 202 (2006), 42-56. doi: 10.1016/j.mbs.2006.04.005. Google Scholar [6] F. J. Odling-Smee, K. N. Laland and M. W. Feldman, Niche Construction: The Neglected Process in Evolution, Princeton University Press, 2003.Google Scholar [7] E. W. Seabloom, E. T. Borer, K. Gross, A. E. Kendig, C. Lacroix, C. E. Mitchell, E. A. Mordecai and A. G. Power, The community ecology of pathogens: Coinfection, coexistence and community composition, Ecology Letters, 18 (2015), 401-415. doi: 10.1111/ele.12418. Google Scholar [8] D. Tilman, Competition and Biodiversity in spatially structured habitats, Ecology, 75 (1994), 2-16. doi: 10.2307/1939377. Google Scholar
Possible scenarios for coexistence and extinction regions when $Q>1$. These are regions where an equilibrium point is feasible. The conditions that define them are necessary but not sufficient for their existence. Region e) is dashed to indicate that it is not feasible, given that not exist one or two species
Qualitative behavior of $Q$ as a function of $(\kappa_2 ,\kappa_3 )\in [0,5]\times [0,5]$ for a) $\omega_2/\omega_1>1$, b) $\omega_2/\omega_1<1$. Note the narrowing of the range where $Q>1$ when going from a) to b).
Possible scenarios for coexistence and extinction regions when $Q<1$. These are regions where an equilibrium point is feasible. The conditions that define them are necessary but not sufficient for their existence. Region e) is dashed to indicate that it is not feasible, given that not exist one or two species
Coexistence of two species for $Q>1$. This scenario corresponds to the region b) of the Fig. 2. The parameters are $\kappa_1=1,$ $\kappa_2=3.5$, $\kappa_3=1.4$, $b=0.2$, $\beta_1=3.4$, $\beta_2=1.6$ $\sigma=0.1$, $p=0.5$, $d=1$, $e=1$, $u=0.18$, $c_1=0.9$, $c_2=0.2$
Colonization by the specie $I_1$ for $Q>1$. This scenario corresponds to the region a) of the Fig. 2. The parameters are $\kappa_1=\kappa_2=1$, $\kappa_3=1.4$, $b=0.8$, $\beta_1=3.8$, $\beta_2=0.5$ $\sigma=0.5$, $p=1$, $d=1$, $e=1$, $u=0.4$, $c_1=0.5$, $c_2=0.5$
Colonization by the specie $I_2$ for $Q>1$. This scenario corresponds to the region c) of the Fig. 2. The parameters are $\kappa_1=\kappa_2=1$, $\kappa_3=1.5$, $b=0.8$, $\beta_1=3.8$, $\beta_2=3.4$ $\sigma=0.5$, $p=0.1$, $d=1$, $e=1$, $u=0.4$, $c_1=0.5$, $c_2=0.5$
Colonization by the specie $I_1$ for $Q<1$. This scenario corresponds to the region c) of the Fig. 6. The parameters are $\kappa_1=1,$ $\kappa_2=1.0$, $\kappa_3=0.5$, $b=0.8$, $\beta_1=3.8$, $\beta_2=0.5$ $\sigma=0.5$, $p=0.1$, $d=1$, $e=1$, $u=0.4$, $c_1=0.9$, $c_2=0.9$
Colonization by the specie $I_1$ for $Q<1$. This scenario corresponds to the region c) of the Fig. 6. The parameters are $\kappa_1=\kappa_2=1$, $\kappa_3=0.5$, $b=0.8$, $\beta_1=3.8$, $\beta_2=0.5$ $\sigma=0.5$, $p=1$, $d=1$, $e=1$, $u=0.4$, $c_1=0.5$, $c_2=0.5$
Colonization by the specie $I_2$ for $Q<1$. This scenario corresponds to the region a) of the Fig. 6. The parameters are $\kappa_1=\kappa_2=1$, $\kappa_3=1.1$, $b=0.8$, $\beta_1=3.8$, $\beta_2=3.4$ $\sigma=0.5$, $p=1$, $d=1$, $e=1$, $u=0.4$, $c_1=0.5$, $c_2=0.5$
Eigenvalues region for non-colonized state
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