December  2021, 41(12): 5743-5764. doi: 10.3934/dcds.2021094

Extinction or coexistence in periodic Kolmogorov systems of competitive type

1. 

Centro de Matemática Computacional e Estocástica, Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Emídio Navarro, 1, 1950-062 Lisboa, Portugal

2. 

Centro de Matemática Computacional e Estocástica, Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edificio C6, piso 2, 1749-016, Lisboa, Portugal

3. 

École des Hautes Études en Sciences Sociales, Centre d'Analyse et de Mathématique Sociales (CAMS), CNRS, 54 Boulevard Raspail, 75006, Paris, France

* Corresponding author: C. Rebelo

Received  March 2021 Revised  May 2021 Published  December 2021 Early access  June 2021

Fund Project: I. Coelho and C. Rebelo have been supported by FCT projects UIDB/04621/2020 and UIDP/04621/2020 of CEMAT at FC-Universidade de Lisboa. E. Sovrano has been supported by the Fondation Sciences Mathématiques de Paris (FSMP)

We study a periodic Kolmogorov system describing two species nonlinear competition. We discuss coexistence and extinction of one or both species, and describe the domain of attraction of nontrivial periodic solutions in the axes, under conditions that generalise Gopalsamy conditions. Finally, we apply our results to a model of microbial growth and to a model of phytoplankton competition under the effect of toxins.

Citation: Isabel Coelho, Carlota Rebelo, Elisa Sovrano. Extinction or coexistence in periodic Kolmogorov systems of competitive type. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5743-5764. doi: 10.3934/dcds.2021094
References:
[1]

C. Alvarez and A. Lazer, An application of topological degree to the periodic competing species problem, J. Austral. Math. Soc. Ser. B, 28 (1986), 202-219.  doi: 10.1017/S0334270000005300.

[2]

H. Amann, Ordinary Differential Equations, vol. 13, De Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 1990. doi: 10.1515/9783110853698.

[3]

Z. Amine and R. Ortega, A periodic prey-predator system, J. Math. Anal. Appl., 185 (1994), 477-489.  doi: 10.1006/jmaa.1994.1262.

[4]

J. Chattopadhyay, Effect of toxic substances on a two-species competitive system, Ecological Modelling, 84 (1996), 287-289.  doi: 10.1016/0304-3800(94)00134-0.

[5]

P. de Mottoni and A. Schiaffino, Competition systems with periodic coefficients: A geometric approach, J. Math. Biol., 11 (1981), 319-335.  doi: 10.1007/BF00276900.

[6]

M. E. Gilpin and F. J. Ayala, Global models of growth and competition, Proc. Nat. Acad. Sci. USA, 70 (1973), 3590-3593.  doi: 10.1073/pnas.70.12.3590.

[7]

K. Gopalsamy, Exchange of equilibria in two-species Lotka-Volterra competition models, J. Austral. Math. Soc., 24 (1982/83), 160-170.  doi: 10.1017/S0334270000003659.

[8]

K. Gopalsamy, Global asymptotic stability in a periodic Lotka-Volterra system, J. Austral. Math. Soc., 27 (1985), 66-72.  doi: 10.1017/S0334270000004768.

[9] M. Kot, Elements of Mathematical Ecology, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511608520.
[10]

Z. Li and F. Chen, Extinction in two dimensional nonautonomous Lotka-Volterra systems with the effect of toxic substances, Appl. Math. Comput., 182 (2006), 684-690.  doi: 10.1016/j.amc.2006.04.034.

[11]

R. Ortega, Some applications of the topological degree to stability theory, in Topological Methods in Differential Equations and Inclusions (Montreal, PQ, 1994), vol. 472 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, 1995,377–409.

[12]

R. Ortega and A. Tineo, An exclusion principle for periodic competitive systems in three dimensions, Nonlinear Anal., 31 (1998), 883-893.  doi: 10.1016/S0362-546X(97)00445-8.

[13]

R. Ortega, Periodic Differential Equations in the Plane, vol. 29 of De Gruyter Series in Nonlinear Analysis and Applications, De Gruyter, Berlin, 2019. doi: 10.1515/9783110551167.

[14]

Y. Ram, Predicting microbial growth in a mixed culture from growth data, PNAS, 116 (2019), 14698-14707.  doi: 10.1073/pnas.1902217116.

[15]

W. Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill Book Co., 1976.

[16]

A. Ruiz-Herrera, Permanence of two species and fixed point index, Nonlinear Analysis, 74 (2011), 146-153.  doi: 10.1016/j.na.2010.08.028.

[17]

H. L. Smith, Periodic competitive differential equations and the discrete dynamics of competitive maps, J. Differential Equations, 64 (1986), 165-194.  doi: 10.1016/0022-0396(86)90086-0.

[18]

H. L. Smith, Dynamics of competition,, in Mathematics Inspired by Biology (Martina Franca, 1997), vol. 1714 of Lecture Notes in Math., Springer, Berlin, 1999,191–240. doi: 10.1007/BFb0092378.

[19]

J. SoléGa rcía-LadonaP. E. Ruardij and M. Estrada, Modelling allelopathy among marine algae, Ecological Modelling, 183 (2005), 373-384. 

[20]

Z. Teng and L. Chen, On the extinction of periodic Lotka-Volterra competition systems, Appl. Anal., 72 (1999), 275-285.  doi: 10.1080/00036819908840742.

[21]

A. Tineo, On the asymptotic behavior of some population models, J. Math. Anal. Appl., 167 (1992), 516-529.  doi: 10.1016/0022-247X(92)90222-Y.

[22]

A. Tineo, Iterative schemes for some population models, Nonlinear World, 3 (1996), 695-708. 

[23]

X. Xie, Y. Xue, R. Wu and L. Zhao, Extinction of a two species competitive system with nonlinear inter-inhibition terms and one toxin producing phytoplankton, Adv. Difference Equ., 2016 (2016), Paper No. 258, 13pp. doi: 10.1186/s13662-016-0974-4.

[24]

F. Zanolin, Permanence and positive periodic solutions for Kolmogorov competing species systems, Results Math., 21 (1992), 224-250.  doi: 10.1007/BF03323081.

[25]

F. Zanolin, Continuation theorems for the periodic problem via the translation operator, Rend. Sem. Mat. Univ. Poi. Torino, 54 (1996), 1-23. 

show all references

References:
[1]

C. Alvarez and A. Lazer, An application of topological degree to the periodic competing species problem, J. Austral. Math. Soc. Ser. B, 28 (1986), 202-219.  doi: 10.1017/S0334270000005300.

[2]

H. Amann, Ordinary Differential Equations, vol. 13, De Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 1990. doi: 10.1515/9783110853698.

[3]

Z. Amine and R. Ortega, A periodic prey-predator system, J. Math. Anal. Appl., 185 (1994), 477-489.  doi: 10.1006/jmaa.1994.1262.

[4]

J. Chattopadhyay, Effect of toxic substances on a two-species competitive system, Ecological Modelling, 84 (1996), 287-289.  doi: 10.1016/0304-3800(94)00134-0.

[5]

P. de Mottoni and A. Schiaffino, Competition systems with periodic coefficients: A geometric approach, J. Math. Biol., 11 (1981), 319-335.  doi: 10.1007/BF00276900.

[6]

M. E. Gilpin and F. J. Ayala, Global models of growth and competition, Proc. Nat. Acad. Sci. USA, 70 (1973), 3590-3593.  doi: 10.1073/pnas.70.12.3590.

[7]

K. Gopalsamy, Exchange of equilibria in two-species Lotka-Volterra competition models, J. Austral. Math. Soc., 24 (1982/83), 160-170.  doi: 10.1017/S0334270000003659.

[8]

K. Gopalsamy, Global asymptotic stability in a periodic Lotka-Volterra system, J. Austral. Math. Soc., 27 (1985), 66-72.  doi: 10.1017/S0334270000004768.

[9] M. Kot, Elements of Mathematical Ecology, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511608520.
[10]

Z. Li and F. Chen, Extinction in two dimensional nonautonomous Lotka-Volterra systems with the effect of toxic substances, Appl. Math. Comput., 182 (2006), 684-690.  doi: 10.1016/j.amc.2006.04.034.

[11]

R. Ortega, Some applications of the topological degree to stability theory, in Topological Methods in Differential Equations and Inclusions (Montreal, PQ, 1994), vol. 472 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, 1995,377–409.

[12]

R. Ortega and A. Tineo, An exclusion principle for periodic competitive systems in three dimensions, Nonlinear Anal., 31 (1998), 883-893.  doi: 10.1016/S0362-546X(97)00445-8.

[13]

R. Ortega, Periodic Differential Equations in the Plane, vol. 29 of De Gruyter Series in Nonlinear Analysis and Applications, De Gruyter, Berlin, 2019. doi: 10.1515/9783110551167.

[14]

Y. Ram, Predicting microbial growth in a mixed culture from growth data, PNAS, 116 (2019), 14698-14707.  doi: 10.1073/pnas.1902217116.

[15]

W. Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill Book Co., 1976.

[16]

A. Ruiz-Herrera, Permanence of two species and fixed point index, Nonlinear Analysis, 74 (2011), 146-153.  doi: 10.1016/j.na.2010.08.028.

[17]

H. L. Smith, Periodic competitive differential equations and the discrete dynamics of competitive maps, J. Differential Equations, 64 (1986), 165-194.  doi: 10.1016/0022-0396(86)90086-0.

[18]

H. L. Smith, Dynamics of competition,, in Mathematics Inspired by Biology (Martina Franca, 1997), vol. 1714 of Lecture Notes in Math., Springer, Berlin, 1999,191–240. doi: 10.1007/BFb0092378.

[19]

J. SoléGa rcía-LadonaP. E. Ruardij and M. Estrada, Modelling allelopathy among marine algae, Ecological Modelling, 183 (2005), 373-384. 

[20]

Z. Teng and L. Chen, On the extinction of periodic Lotka-Volterra competition systems, Appl. Anal., 72 (1999), 275-285.  doi: 10.1080/00036819908840742.

[21]

A. Tineo, On the asymptotic behavior of some population models, J. Math. Anal. Appl., 167 (1992), 516-529.  doi: 10.1016/0022-247X(92)90222-Y.

[22]

A. Tineo, Iterative schemes for some population models, Nonlinear World, 3 (1996), 695-708. 

[23]

X. Xie, Y. Xue, R. Wu and L. Zhao, Extinction of a two species competitive system with nonlinear inter-inhibition terms and one toxin producing phytoplankton, Adv. Difference Equ., 2016 (2016), Paper No. 258, 13pp. doi: 10.1186/s13662-016-0974-4.

[24]

F. Zanolin, Permanence and positive periodic solutions for Kolmogorov competing species systems, Results Math., 21 (1992), 224-250.  doi: 10.1007/BF03323081.

[25]

F. Zanolin, Continuation theorems for the periodic problem via the translation operator, Rend. Sem. Mat. Univ. Poi. Torino, 54 (1996), 1-23. 

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