January  2021, 26(1): 269-297. doi: 10.3934/dcdsb.2020140

Coexistence of competing consumers on a single resource in a hybrid model

1. 

Institute for Mathematical Sciences, Renmin University of China, Beijing, 100872, China

2. 

Department of Mathematics and Statistics, University of Ottawa, Ottawa, K1N6N5, Canada

3. 

Department of Mathematics and Statistics, Department of Biology, University of Ottawa, Ottawa, K1N6N5, Canada

* Corresponding author: Frithjof Lutscher

Received  November 2019 Revised  February 2020 Published  April 2020

The question of whether and how two competing consumers can coexist on a single limiting resource has a long tradition in ecological theory. We build on a recent seasonal (hybrid) model for one consumer and one resource, and we extend it by introducing a second consumer. Consumers reproduce only once per year, the resource reproduces throughout the"summer" season. When we use linear consumer reproduction between years, we find explicit expressions for the trivial and semi-trivial equilibria, and we prove that there is no positive equilibrium generically. When we use non-linear consumer reproduction, we determine conditions for which both semi-trivial equilibria are unstable. We prove that a unique positive equilibrium exists in this case, and we find an explicit analytical expression for it. By linear analysis and numerical simulation, we find bifurcations from the stable equilibrium to population cycles that may appear through period-doubling or Hopf bifurcations. We interpret our results in terms of climate change that changes the length of the"summer" season.

Citation: Yunfeng Geng, Xiaoying Wang, Frithjof Lutscher. Coexistence of competing consumers on a single resource in a hybrid model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 269-297. doi: 10.3934/dcdsb.2020140
References:
[1]

F. R. Adler, Coexistence of two types on a single resource in discrete time, Journal of Mathematical Biology, 28 (1990), 695-713.  doi: 10.1007/BF00160232.  Google Scholar

[2]

R. A. Armstrong and R. McGehee, Coexistence of species competing for shared resources, Theoretical Population Biology, 9 (1976), 317-328.  doi: 10.1016/0040-5809(76)90051-4.  Google Scholar

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J. P. Grover and F. B. Wang, Competition and allelopathy with resource storage: Two resources, Journal of Theoretical Biology, 351 (2014), 9-24.  doi: 10.1016/j.jtbi.2014.02.013.  Google Scholar

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S. B. HsuS. P. Hubbell and P. Waltman, Competing predators, SIAM Journal on Applied Mathematics, 35 (1978), 617-625.  doi: 10.1137/0135051.  Google Scholar

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Z. J. Jing and J. P. Yang, Bifurcation and chaos in discrete-time predator-prey system, Chaos Solitons Fractals, 27 (2006), 259-277.  doi: 10.1016/j.chaos.2005.03.040.  Google Scholar

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Y. Kang and P. Chesson, Relative nonlinearity and permanence, Theoretical Population Biology, 78 (2010), 26-35.  doi: 10.1016/j.tpb.2010.04.002.  Google Scholar

[16]

Y. Kang and H. Smith, Global dynamics of a discrete two-species lottery-Ricker competition model, Journal of Biological Dynamics, 6 (2012), 358-376.  doi: 10.1080/17513758.2011.586064.  Google Scholar

[17]

A. L. Koch, Competitive coexistence of two predators utilizing the same prey under constant environmental conditions, Journal of Theoretical Biology, 44 (1974), 387-395.  doi: 10.1016/0022-5193(74)90169-6.  Google Scholar

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[19]

J. J. Kuang and P. Chesson, Predation-competition interactions for seasonally recruiting species, The American Naturalist, 171 (2008), E119–E133. doi: 10.1086/527484.  Google Scholar

[20]

G. Ledder, Mathematics for the Life Sciences: Calculus, Modeling, Probability, and Dynamical Systems, Springer Science & Business Media, 2013. doi: 10.1007/978-1-4614-7276-6.  Google Scholar

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S. A. Levin, Community equilibria and stability, and an extension of the competitive exclusion principle, The American Naturalist, 104 (1970), 413-423.  doi: 10.1086/282676.  Google Scholar

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L. Mailleret and V. Lemesle, A note on semi-discrete modelling in the life sciences, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 367 (2009), 4779-4799.  doi: 10.1098/rsta.2009.0153.  Google Scholar

[23]

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[24]

Y. Samia and F. Lutscher, Coexistence and spread of competitors in heterogeneous landscapes, Bulletin of Mathematical Biology, 72 (2010), 2089-2112.  doi: 10.1007/s11538-010-9529-0.  Google Scholar

[25]

J. ShangB. T. Li and M. R. Barnard, Bifurcations in a discrete time model composed of Beverton-Holt function and Ricker function, Mathematical Biosciences, 263 (2015), 161-168.  doi: 10.1016/j.mbs.2015.02.014.  Google Scholar

[26]

R. J. Smith and G. S. K. Wolkowicz., Analysis of a model of the nutrient driven self-cycling fermentation process, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 11 (2004), 239-266.   Google Scholar

[27]

K. StaňkováA. Abate and M. W. Sabelis, Irreversible prey diapause as an optimal strategy of a physiologically extended Lotka-Volterra model, Journal of Mathematical Biology, 66 (2013), 767-794.  doi: 10.1007/s00285-012-0599-5.  Google Scholar

[28]

A. Szilágyi and G. Meszéna, Coexistence in a fluctuating environment by the effect of relative nonlinearity: A minimal model, Journal of Theoretical Biology, 267 (2010), 502-512.  doi: 10.1016/j.jtbi.2010.09.020.  Google Scholar

[29]

M. Turelli, A reexamination of stability in randomly varying versus deterministic environments with comments on stochastic theory of limiting similarity, Theoretical Population Biology, 13 (1978), 244-267.  doi: 10.1016/0040-5809(78)90045-X.  Google Scholar

[30]

V. Volterra, Variazioni e Fluttuazioni del Numero D'individui in Specie Animali Conviventi, C. Ferrari, 1927. Google Scholar

[31]

V. Volterra, Variations and fluctuations of the number of individuals in animal species living together, ICES Journal of Marine Science, 3 (1928), 3-51.  doi: 10.1093/icesjms/3.1.3.  Google Scholar

[32]

F. Y. Wang and G. P. Pang, Competition in a chemostat with Beddington-DeAngelis growth rates and periodic pulsed nutrient, Journal of Mathematical Chemistry, 44 (2008), 691-710.  doi: 10.1007/s10910-008-9346-y.  Google Scholar

[33]

X. Y. Wang and F. Lutscher, Turing patterns in a predator-prey model with seasonality, Journal of Mathematical Biology, 78 (2019), 711-737.  doi: 10.1007/s00285-018-1289-8.  Google Scholar

[34]

S. L. Yuan, Y. Zhao and A. F. Xiao, Competition between plasmid-bearing and plasmid-free organisms in a chemostat with pulsed input and washout, Mathematical Problems in Engineering, 2009 (2009), 204632. doi: 10.1155/2009/204632.  Google Scholar

[35]

S. L. YuanW. G. Zhang and Y. Zhao, Bifurcation analysis of a model of plasmid-bearing, plasmid-free competition in a pulsed chemostat with an internal inhibitor, IMA Journal of Applied Mathematics, 76 (2011), 277-297.  doi: 10.1093/imamat/hxq036.  Google Scholar

[36]

X. ZhouX. Song and X. Shi, Analysis of competitive chemostat models with the Beddington-DeAngelis functional response and impulsive effect, Applied Mathematical Modelling, 31 (2007), 2299-2312.   Google Scholar

show all references

References:
[1]

F. R. Adler, Coexistence of two types on a single resource in discrete time, Journal of Mathematical Biology, 28 (1990), 695-713.  doi: 10.1007/BF00160232.  Google Scholar

[2]

R. A. Armstrong and R. McGehee, Coexistence of species competing for shared resources, Theoretical Population Biology, 9 (1976), 317-328.  doi: 10.1016/0040-5809(76)90051-4.  Google Scholar

[3]

R. A. Armstrong and R. McGehee, Competitive exclusion, The American Naturalist, 115 (1980), 151-170.  doi: 10.1086/283553.  Google Scholar

[4]

R. J. H. Beverton and S. J. Holt, On the Dynamics of Exploited Fish Populations, Springer Science & Business Media, 2012. doi: 10.1007/978-94-011-2106-4.  Google Scholar

[5]

W. Ebenhöh, Coexistence of an unlimited number of algal species in a model system, Theoretical Population Biology, 34 (1988), 130-144.  doi: 10.1016/0040-5809(88)90038-X.  Google Scholar

[6]

H. T. Eskola and K. Parvinen, On the mechanistic underpinning of discrete-time population models with Allee effect, Theoretical Population Biology, 72 (2007), 41-51.  doi: 10.1016/j.tpb.2007.03.004.  Google Scholar

[7]

E. Funasaki and M. Kot, Invasion and chaos in a periodically pulsed mass-action chemostat, Theoretical Population Biology, 44 (1993), 203-224.  doi: 10.1006/tpbi.1993.1026.  Google Scholar

[8]

S. A. H. Geritz and É. Kisdi, On the mechanistic underpinning of discrete-time population models with complex dynamics, Journal of Theoretical Biology, 228 (2004), 261-269.  doi: 10.1016/j.jtbi.2004.01.003.  Google Scholar

[9]

J. P. Grover and F. B. Wang, Competition and allelopathy with resource storage: Two resources, Journal of Theoretical Biology, 351 (2014), 9-24.  doi: 10.1016/j.jtbi.2014.02.013.  Google Scholar

[10]

G. Hardin, The competitive exclusion principle, Science, 131 (1960), 1292-1297.  doi: 10.1126/science.131.3409.1292.  Google Scholar

[11]

Z. M. He and B. Li, Complex dynamic behavior of a discrete-time predator-prey system of Holling-III type, Advances in Difference Equations, 2014 (2014), 13 pp. doi: 10.1186/1687-1847-2014-180.  Google Scholar

[12]

S. B. HsuS. P. Hubbell and P. Waltman, A contribution to the theory of competing predators, Ecological Monographs, 48 (1978), 337-349.   Google Scholar

[13]

S. B. HsuS. P. Hubbell and P. Waltman, Competing predators, SIAM Journal on Applied Mathematics, 35 (1978), 617-625.  doi: 10.1137/0135051.  Google Scholar

[14]

Z. J. Jing and J. P. Yang, Bifurcation and chaos in discrete-time predator-prey system, Chaos Solitons Fractals, 27 (2006), 259-277.  doi: 10.1016/j.chaos.2005.03.040.  Google Scholar

[15]

Y. Kang and P. Chesson, Relative nonlinearity and permanence, Theoretical Population Biology, 78 (2010), 26-35.  doi: 10.1016/j.tpb.2010.04.002.  Google Scholar

[16]

Y. Kang and H. Smith, Global dynamics of a discrete two-species lottery-Ricker competition model, Journal of Biological Dynamics, 6 (2012), 358-376.  doi: 10.1080/17513758.2011.586064.  Google Scholar

[17]

A. L. Koch, Competitive coexistence of two predators utilizing the same prey under constant environmental conditions, Journal of Theoretical Biology, 44 (1974), 387-395.  doi: 10.1016/0022-5193(74)90169-6.  Google Scholar

[18] M. Kot, Elements of Mathematical Ecology, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511608520.  Google Scholar
[19]

J. J. Kuang and P. Chesson, Predation-competition interactions for seasonally recruiting species, The American Naturalist, 171 (2008), E119–E133. doi: 10.1086/527484.  Google Scholar

[20]

G. Ledder, Mathematics for the Life Sciences: Calculus, Modeling, Probability, and Dynamical Systems, Springer Science & Business Media, 2013. doi: 10.1007/978-1-4614-7276-6.  Google Scholar

[21]

S. A. Levin, Community equilibria and stability, and an extension of the competitive exclusion principle, The American Naturalist, 104 (1970), 413-423.  doi: 10.1086/282676.  Google Scholar

[22]

L. Mailleret and V. Lemesle, A note on semi-discrete modelling in the life sciences, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 367 (2009), 4779-4799.  doi: 10.1098/rsta.2009.0153.  Google Scholar

[23]

E. PachepskyR. M. Nisbet and W. W. Murdoch, Between discrete and continuous: Consumer-resource dynamics with synchronized reproduction, Ecology, 89 (2008), 280-288.  doi: 10.1890/07-0641.1.  Google Scholar

[24]

Y. Samia and F. Lutscher, Coexistence and spread of competitors in heterogeneous landscapes, Bulletin of Mathematical Biology, 72 (2010), 2089-2112.  doi: 10.1007/s11538-010-9529-0.  Google Scholar

[25]

J. ShangB. T. Li and M. R. Barnard, Bifurcations in a discrete time model composed of Beverton-Holt function and Ricker function, Mathematical Biosciences, 263 (2015), 161-168.  doi: 10.1016/j.mbs.2015.02.014.  Google Scholar

[26]

R. J. Smith and G. S. K. Wolkowicz., Analysis of a model of the nutrient driven self-cycling fermentation process, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 11 (2004), 239-266.   Google Scholar

[27]

K. StaňkováA. Abate and M. W. Sabelis, Irreversible prey diapause as an optimal strategy of a physiologically extended Lotka-Volterra model, Journal of Mathematical Biology, 66 (2013), 767-794.  doi: 10.1007/s00285-012-0599-5.  Google Scholar

[28]

A. Szilágyi and G. Meszéna, Coexistence in a fluctuating environment by the effect of relative nonlinearity: A minimal model, Journal of Theoretical Biology, 267 (2010), 502-512.  doi: 10.1016/j.jtbi.2010.09.020.  Google Scholar

[29]

M. Turelli, A reexamination of stability in randomly varying versus deterministic environments with comments on stochastic theory of limiting similarity, Theoretical Population Biology, 13 (1978), 244-267.  doi: 10.1016/0040-5809(78)90045-X.  Google Scholar

[30]

V. Volterra, Variazioni e Fluttuazioni del Numero D'individui in Specie Animali Conviventi, C. Ferrari, 1927. Google Scholar

[31]

V. Volterra, Variations and fluctuations of the number of individuals in animal species living together, ICES Journal of Marine Science, 3 (1928), 3-51.  doi: 10.1093/icesjms/3.1.3.  Google Scholar

[32]

F. Y. Wang and G. P. Pang, Competition in a chemostat with Beddington-DeAngelis growth rates and periodic pulsed nutrient, Journal of Mathematical Chemistry, 44 (2008), 691-710.  doi: 10.1007/s10910-008-9346-y.  Google Scholar

[33]

X. Y. Wang and F. Lutscher, Turing patterns in a predator-prey model with seasonality, Journal of Mathematical Biology, 78 (2019), 711-737.  doi: 10.1007/s00285-018-1289-8.  Google Scholar

[34]

S. L. Yuan, Y. Zhao and A. F. Xiao, Competition between plasmid-bearing and plasmid-free organisms in a chemostat with pulsed input and washout, Mathematical Problems in Engineering, 2009 (2009), 204632. doi: 10.1155/2009/204632.  Google Scholar

[35]

S. L. YuanW. G. Zhang and Y. Zhao, Bifurcation analysis of a model of plasmid-bearing, plasmid-free competition in a pulsed chemostat with an internal inhibitor, IMA Journal of Applied Mathematics, 76 (2011), 277-297.  doi: 10.1093/imamat/hxq036.  Google Scholar

[36]

X. ZhouX. Song and X. Shi, Analysis of competitive chemostat models with the Beddington-DeAngelis functional response and impulsive effect, Applied Mathematical Modelling, 31 (2007), 2299-2312.   Google Scholar

Figure 1.  Illustration of the unique positive impulsive periodic orbit, corresponding to the unique positive steady state of the discrete map. Parameter values are: $ \mu_1 = 0.7,\; \mu_2 = 0.3813,\; \eta_1 = 10,\; \eta_2 = 7.5811,\; \varepsilon_1 = \varepsilon_2 = 0.8711,\; \xi_1 = \xi_2 = 0.2941,\; \alpha = 5.3063,\; \rho = 0.8324 $
Figure 2.  The existence range of the positive equilibrium with respect to parameters $ \mu_1 $ and $ \eta_1 $. The upper boundary curve corresponds to condition (39), while the lower curve corresponds to (45). Parameters are as in Figure 1, unless otherwise noted
Figure 3.  The stability region of the semi-trivial equilibrium in the $ u $-$ v $ plane. Parameters are $ \varepsilon_1 = 0.5307,\; \xi_1 = 0.2941,\; \alpha = 20.3063,\; \rho = 0.8324. $
Figure 4.  Bifurcation diagram with respect to parameters $ \mu_1 $ and $ \eta_1 $. Other parameters are as in Figure 2, except $ \varepsilon_1 = \varepsilon_2 = 0.5307 $
Figure 5.  Orbit diagram corresponding to fixing $ \eta_1 = 10 $, see the dashed line in Figure 4. Other parameters are as in Figure 4
Figure 6.  Bifurcation diagram with respect to parameters $ \mu_1 $ and $ \eta_1 $. Other parameters are as in Figure 4, except $ \alpha = 20.3063 $
Figure 7.  Orbit diagram corresponding to fixing $ \eta_1 = 30 $, see the dashed line in Figure 5. Other parameters are as in Figure 5
Figure 8.  Bifurcation diagram with respect to parameters $ \mu_1 $ and $ \eta_1 $. Other parameters are as in Figure 4, except $ \varepsilon_1 = \varepsilon_2 = 0.2307 $
Figure 9.  Orbit diagram corresponding to fixing $ \eta_1 = 10 $, see the dashed line in Figure 6. Other parameters are as in Figure 6
Figure 10.  Three species densities with different summer length. Parameters are: $ m_1 = 0.6,\; m_2 = 0.3813,\; \theta_1 = 2,\; \theta_2 = 1.516,\; \delta_1 = \delta_2 = 0.5307,\; \xi_1 = \xi_2 = 0.2941,\; r = 5.3063,\; \rho = 0.8324,\; K = 5,\; a_1 = a_2 = 1 $
Figure 11.  Three species densities with different summer length. $ \rho,\; \xi_1,\; \xi_2 $ increasing in $ T $ with $ \omega = 0.7 $, other parameters are as in Figure 7, except $ \delta_1 = \delta_2 = 0.0711 $ in right plot
Figure 12.  Three species coexist with linear reproduction. Parameters are $ \mu_1 = 0.7,\; \mu_2 = 0.5378,\; \eta_1 = 5.9,\; \eta_2 = 7.2284,\; \xi_1 = 0.6681,\; \xi_2 = 0.1788,\; \alpha = 55.0495,\; \rho = 0.9599. $
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