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. KeymerM. 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. KrakauerK. 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-LorcaJ. 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-LorcaJ. 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. SeabloomE. T. BorerK. GrossA. E. KendigC. LacroixC. E. MitchellE. 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

show all references

References:
[1]

J. E. KeymerM. 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. KrakauerK. 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-LorcaJ. 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-LorcaJ. 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. SeabloomE. T. BorerK. GrossA. E. KendigC. LacroixC. E. MitchellE. 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

Figure 2.  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
Figure 1.  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).
Figure 6.  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
Figure 3.  Coexistence of two species for $Q>1$. This scenario corresponds to the region b) of the Fig. 2. The parameters are $\kappa_1=1, $ $\kappa_2=3.5$, $\kappa_3=1.4$, $b=0.2$, $\beta_1=3.4$, $\beta_2=1.6$ $\sigma=0.1$, $p=0.5$, $d=1$, $e=1$, $u=0.18$, $c_1=0.9$, $c_2=0.2$
Figure 4.  Colonization by the specie $I_1$ for $Q>1$. This scenario corresponds to the region a) of the Fig. 2. The parameters are $\kappa_1=\kappa_2=1$, $\kappa_3=1.4$, $b=0.8$, $\beta_1=3.8$, $\beta_2=0.5$ $\sigma=0.5$, $p=1$, $d=1$, $e=1$, $u=0.4$, $c_1=0.5$, $c_2=0.5$
Figure 5.  Colonization by the specie $I_2$ for $Q>1$. This scenario corresponds to the region c) of the Fig. 2. The parameters are $\kappa_1=\kappa_2=1$, $\kappa_3=1.5$, $b=0.8$, $\beta_1=3.8$, $\beta_2=3.4$ $\sigma=0.5$, $p=0.1$, $d=1$, $e=1$, $u=0.4$, $c_1=0.5$, $c_2=0.5$
Figure 7.  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, $ $\kappa_2=1.0$, $\kappa_3=0.5$, $b=0.8$, $\beta_1=3.8$, $\beta_2=0.5$ $\sigma=0.5$, $p=0.1$, $d=1$, $e=1$, $u=0.4$, $c_1=0.9$, $c_2=0.9$
Figure 8.  Colonization by the specie $I_1$ for $Q<1$. This scenario corresponds to the region c) of the Fig. 6. The parameters are $\kappa_1=\kappa_2=1$, $\kappa_3=0.5$, $b=0.8$, $\beta_1=3.8$, $\beta_2=0.5$ $\sigma=0.5$, $p=1$, $d=1$, $e=1$, $u=0.4$, $c_1=0.5$, $c_2=0.5$
Figure 9.  Colonization by the specie $I_2$ for $Q<1$. This scenario corresponds to the region a) of the Fig. 6. The parameters are $\kappa_1=\kappa_2=1$, $\kappa_3=1.1$, $b=0.8$, $\beta_1=3.8$, $\beta_2=3.4$ $\sigma=0.5$, $p=1$, $d=1$, $e=1$, $u=0.4$, $c_1=0.5$, $c_2=0.5$
Figure 10.  Eigenvalues region for non-colonized state
[1]

Hélène Leman, Sylvie Méléard, Sepideh Mirrahimi. Influence of a spatial structure on the long time behavior of a competitive Lotka-Volterra type system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 469-493. doi: 10.3934/dcdsb.2015.20.469

[2]

Yubin Liu, Peixuan Weng. Asymptotic spreading of a three dimensional Lotka-Volterra cooperative-competitive system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 505-518. doi: 10.3934/dcdsb.2015.20.505

[3]

Xiaoyue Li, Xuerong Mao. Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 523-545. doi: 10.3934/dcds.2009.24.523

[4]

Zhaohai Ma, Rong Yuan, Yang Wang, Xin Wu. Multidimensional stability of planar traveling waves for the delayed nonlocal dispersal competitive Lotka-Volterra system. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2069-2092. doi: 10.3934/cpaa.2019093

[5]

Hao Wang, Katherine Dunning, James J. Elser, Yang Kuang. Daphnia species invasion, competitive exclusion, and chaotic coexistence. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 481-493. doi: 10.3934/dcdsb.2009.12.481

[6]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035

[7]

Yuan Lou, Dongmei Xiao, Peng Zhou. Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 953-969. doi: 10.3934/dcds.2016.36.953

[8]

Linping Peng, Zhaosheng Feng, Changjian Liu. Quadratic perturbations of a quadratic reversible Lotka-Volterra system with two centers. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4807-4826. doi: 10.3934/dcds.2014.34.4807

[9]

Xiaoli Liu, Dongmei Xiao. Bifurcations in a discrete time Lotka-Volterra predator-prey system. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 559-572. doi: 10.3934/dcdsb.2006.6.559

[10]

Fuke Wu, Yangzi Hu. Stochastic Lotka-Volterra system with unbounded distributed delay. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 275-288. doi: 10.3934/dcdsb.2010.14.275

[11]

Jong-Shenq Guo, Ying-Chih Lin. The sign of the wave speed for the Lotka-Volterra competition-diffusion system. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2083-2090. doi: 10.3934/cpaa.2013.12.2083

[12]

Qi Wang, Chunyi Gai, Jingda Yan. Qualitative analysis of a Lotka-Volterra competition system with advection. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1239-1284. doi: 10.3934/dcds.2015.35.1239

[13]

Anthony W. Leung, Xiaojie Hou, Wei Feng. Traveling wave solutions for Lotka-Volterra system re-visited. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 171-196. doi: 10.3934/dcdsb.2011.15.171

[14]

Qi Wang, Yang Song, Lingjie Shao. Boundedness and persistence of populations in advective Lotka-Volterra competition system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2245-2263. doi: 10.3934/dcdsb.2018195

[15]

Xiaoling Zou, Ke Wang. Optimal harvesting for a stochastic N-dimensional competitive Lotka-Volterra model with jumps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 683-701. doi: 10.3934/dcdsb.2015.20.683

[16]

Cheng-Hsiung Hsu, Ting-Hui Yang. Traveling plane wave solutions of delayed lattice differential systems in competitive Lotka-Volterra type. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 111-128. doi: 10.3934/dcdsb.2010.14.111

[17]

Francisco Montes de Oca, Liliana Pérez. Balancing survival and extinction in nonautonomous competitive Lotka-Volterra systems with infinite delays. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2663-2690. doi: 10.3934/dcdsb.2015.20.2663

[18]

M. R. S. Kulenović, Orlando Merino. Competitive-exclusion versus competitive-coexistence for systems in the plane. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1141-1156. doi: 10.3934/dcdsb.2006.6.1141

[19]

Juan Luis García Guirao, Marek Lampart. Transitivity of a Lotka-Volterra map. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 75-82. doi: 10.3934/dcdsb.2008.9.75

[20]

Yixiang Wu, Necibe Tuncer, Maia Martcheva. Coexistence and competitive exclusion in an SIS model with standard incidence and diffusion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1167-1187. doi: 10.3934/dcdsb.2017057

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (12)
  • HTML views (4)
  • Cited by (0)

[Back to Top]