# American Institute of Mathematical Sciences

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  January 2021 Early access  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 and 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. [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. [3] R. A. Armstrong and R. McGehee, Competitive exclusion, The American Naturalist, 115 (1980), 151-170.  doi: 10.1086/283553. [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. [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. [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. [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. [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. [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. [10] G. Hardin, The competitive exclusion principle, Science, 131 (1960), 1292-1297.  doi: 10.1126/science.131.3409.1292. [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. [12] S. B. Hsu, S. P. Hubbell and P. Waltman, A contribution to the theory of competing predators, Ecological Monographs, 48 (1978), 337-349. [13] S. B. Hsu, S. P. Hubbell and P. Waltman, Competing predators, SIAM Journal on Applied Mathematics, 35 (1978), 617-625.  doi: 10.1137/0135051. [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. [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. [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. [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. [18] M. Kot, Elements of Mathematical Ecology, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511608520. [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. [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. [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. [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. [23] E. Pachepsky, R. 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. [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. [25] J. Shang, B. 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. [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. [27] K. Staňková, A. 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. [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. [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. [30] V. Volterra, Variazioni e Fluttuazioni del Numero D'individui in Specie Animali Conviventi, C. Ferrari, 1927. [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. [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. [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. [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. [35] S. L. Yuan, W. 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. [36] X. Zhou, X. 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.

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. [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. [3] R. A. Armstrong and R. McGehee, Competitive exclusion, The American Naturalist, 115 (1980), 151-170.  doi: 10.1086/283553. [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. [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. [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. [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. [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. [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. [10] G. Hardin, The competitive exclusion principle, Science, 131 (1960), 1292-1297.  doi: 10.1126/science.131.3409.1292. [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. [12] S. B. Hsu, S. P. Hubbell and P. Waltman, A contribution to the theory of competing predators, Ecological Monographs, 48 (1978), 337-349. [13] S. B. Hsu, S. P. Hubbell and P. Waltman, Competing predators, SIAM Journal on Applied Mathematics, 35 (1978), 617-625.  doi: 10.1137/0135051. [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. [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. [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. [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. [18] M. Kot, Elements of Mathematical Ecology, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511608520. [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. [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. [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. [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. [23] E. Pachepsky, R. 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. [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. [25] J. Shang, B. 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. [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. [27] K. Staňková, A. 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. [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. [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. [30] V. Volterra, Variazioni e Fluttuazioni del Numero D'individui in Specie Animali Conviventi, C. Ferrari, 1927. [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. [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. [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. [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. [35] S. L. Yuan, W. 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. [36] X. Zhou, X. 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.
Illustration of the unique positive impulsive periodic orbit, corresponding to the unique positive steady state of the discrete map. Parameter values are: $\mu_1 = 0.7,\; \mu_2 = 0.3813,\; \eta_1 = 10,\; \eta_2 = 7.5811,\; \varepsilon_1 = \varepsilon_2 = 0.8711,\; \xi_1 = \xi_2 = 0.2941,\; \alpha = 5.3063,\; \rho = 0.8324$
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
The stability region of the semi-trivial equilibrium in the $u$-$v$ plane. Parameters are $\varepsilon_1 = 0.5307,\; \xi_1 = 0.2941,\; \alpha = 20.3063,\; \rho = 0.8324.$
Bifurcation diagram with respect to parameters $\mu_1$ and $\eta_1$. Other parameters are as in Figure 2, except $\varepsilon_1 = \varepsilon_2 = 0.5307$
Orbit diagram corresponding to fixing $\eta_1 = 10$, see the dashed line in Figure 4. Other parameters are as in Figure 4
Bifurcation diagram with respect to parameters $\mu_1$ and $\eta_1$. Other parameters are as in Figure 4, except $\alpha = 20.3063$
Orbit diagram corresponding to fixing $\eta_1 = 30$, see the dashed line in Figure 5. Other parameters are as in Figure 5
Bifurcation diagram with respect to parameters $\mu_1$ and $\eta_1$. Other parameters are as in Figure 4, except $\varepsilon_1 = \varepsilon_2 = 0.2307$
Orbit diagram corresponding to fixing $\eta_1 = 10$, see the dashed line in Figure 6. Other parameters are as in Figure 6
Three species densities with different summer length. Parameters are: $m_1 = 0.6,\; m_2 = 0.3813,\; \theta_1 = 2,\; \theta_2 = 1.516,\; \delta_1 = \delta_2 = 0.5307,\; \xi_1 = \xi_2 = 0.2941,\; r = 5.3063,\; \rho = 0.8324,\; K = 5,\; a_1 = a_2 = 1$
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
Three species coexist with linear reproduction. Parameters are $\mu_1 = 0.7,\; \mu_2 = 0.5378,\; \eta_1 = 5.9,\; \eta_2 = 7.2284,\; \xi_1 = 0.6681,\; \xi_2 = 0.1788,\; \alpha = 55.0495,\; \rho = 0.9599.$
 [1] Yunfei Lv, Yongzhen Pei, Rong Yuan. On a non-linear size-structured population model. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3111-3133. doi: 10.3934/dcdsb.2020053 [2] Hamza Khalfi, Amal Aarab, Nour Eddine Alaa. Energetics and coarsening analysis of a simplified non-linear surface growth model. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 161-177. doi: 10.3934/dcdss.2021014 [3] Kurt Falk, Marc Kesseböhmer, Tobias Henrik Oertel-Jäger, Jens D. M. Rademacher, Tony Samuel. Preface: Diffusion on fractals and non-linear dynamics. Discrete and Continuous Dynamical Systems - S, 2017, 10 (2) : i-iv. doi: 10.3934/dcdss.201702i [4] Dmitry Dolgopyat. Bouncing balls in non-linear potentials. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 165-182. doi: 10.3934/dcds.2008.22.165 [5] Dorin Ervin Dutkay and Palle E. T. Jorgensen. Wavelet constructions in non-linear dynamics. Electronic Research Announcements, 2005, 11: 21-33. [6] Armin Lechleiter. Explicit characterization of the support of non-linear inclusions. Inverse Problems and Imaging, 2011, 5 (3) : 675-694. doi: 10.3934/ipi.2011.5.675 [7] Denis Serre. Non-linear electromagnetism and special relativity. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 435-454. doi: 10.3934/dcds.2009.23.435 [8] Feng-Yu Wang. Exponential convergence of non-linear monotone SPDEs. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5239-5253. doi: 10.3934/dcds.2015.35.5239 [9] Anugu Sumith Reddy, Amit Apte. Stability of non-linear filter for deterministic dynamics. Foundations of Data Science, 2021, 3 (3) : 647-675. doi: 10.3934/fods.2021025 [10] Ahmad El Hajj, Aya Oussaily. Continuous solution for a non-linear eikonal system. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3795-3823. doi: 10.3934/cpaa.2021131 [11] Paolo Buttà, Franco Flandoli, Michela Ottobre, Boguslaw Zegarlinski. A non-linear kinetic model of self-propelled particles with multiple equilibria. Kinetic and Related Models, 2019, 12 (4) : 791-827. doi: 10.3934/krm.2019031 [12] Francesca Biagini, Katharina Oberpriller. Reduced-form setting under model uncertainty with non-linear affine intensities. Probability, Uncertainty and Quantitative Risk, 2021, 6 (3) : 159-188. doi: 10.3934/puqr.2021008 [13] Tommi Brander, Joonas Ilmavirta, Manas Kar. Superconductive and insulating inclusions for linear and non-linear conductivity equations. Inverse Problems and Imaging, 2018, 12 (1) : 91-123. doi: 10.3934/ipi.2018004 [14] Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084 [15] Shuhua Zhang, Longzhou Cao, Zuliang Lu. An EOQ inventory model for deteriorating items with controllable deterioration rate under stock-dependent demand rate and non-linear holding cost. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021156 [16] Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424 [17] Pablo Ochoa. Approximation schemes for non-linear second order equations on the Heisenberg group. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1841-1863. doi: 10.3934/cpaa.2015.14.1841 [18] Eugenio Aulisa, Akif Ibragimov, Emine Yasemen Kaya-Cekin. Stability analysis of non-linear plates coupled with Darcy flows. Evolution Equations and Control Theory, 2013, 2 (2) : 193-232. doi: 10.3934/eect.2013.2.193 [19] Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302 [20] Olusola Kolebaje, Ebenezer Bonyah, Lateef Mustapha. The first integral method for two fractional non-linear biological models. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 487-502. doi: 10.3934/dcdss.2019032

2020 Impact Factor: 1.327