Analysis on Steady States of a Competition System with Nonlinear Diffusion Terms

Abstract

Competition is a fundamental force shaping population size and structure as a result of limited availability of resources. In biomathematics, the biological models with competitive interactions exist widely. Furthermore, the nonlinear-diffusion (including self- and cross-diffusions) terms are incorporated to the biological models to better simulate the actual movement of species. Therefore, better compatibility with reality can be achieved by introducing nonlinear-diffusion into biological models with competitive interactions. As a result, a competition system with nonlinear-diffusion and nonlinear functional response is proposed and analyzed in this paper. We first briefly discuss the stability of trivial and semi-trivial solutions by spectrum analysis. Then the boundedness and the non-existence of steady states are studied. Based on the boundedness of the solutions, the existence of the steady states is also investigated by the fixed point index theory in a positive cone. The result shows that the two species can coexist when their diffusion and inter-specific competition pressures are controlled in a certain range.

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References

  1. 1.

    Chesson, P., Kuang, J.J.: The interaction between predation and competition. Nature 456, 235–238 (2008)

    Article  Google Scholar 

  2. 2.

    Lou, Y., Tao, Y., Winkler, M.: Approaching the ideal free distribution in two-species competition models with fitness-dependent dispersal. SIAM J. Math. Anal. 46, 1228–1262 (2014)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Dellal, M., Lakrib, M., Sari, T.: The operating diagram of a model of two competitors in a chemostat with an external inhibitor. Math. Biosci. 302, 27–45 (2018)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Yamada, Y.: Positive solutions for Lotka-Volterra competition system with diffusion. Nonlinear Anal. 47, 6085–6096 (2001)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Dancer, E.N., Zhang, Z.: Dynamics of Lotka-Volterra competition system with large interaction. J. Differ. Equ. 182, 470–489 (2002)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Jia, Y., Wu, J., Xu, H.-K.: Positive solutions of a Lotka-Volterra competition model with cross-diffusion. Comput. Math. Appl. 68, 1220–1228 (2014)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Pao, C.V.: Dynamics of Lotka-Volterra competition reaction diffusion systems with degenerate diffusion. J. Math. Anal. Appl. 421, 1721–1742 (2015)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Alhasanat, A., Ou, C.: Minimal-speed selection of traveling waves to the Lotka-Volterra competition model. J. Differ. Equ. 266, 7357–7378 (2019)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Sakthivel, K., Baranibalan, N., Kim, J.-H., et al.: Erratum to: Stability of diffusion coefficients in an inverse problem for the Lotka-Volterra competition system. Acta Appl. Math. 111, 149–152 (2010)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Gopalsamy, K.: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic, Boston (1992)

    Google Scholar 

  11. 11.

    Liu, Z., Tan, R., Chen, Y.: Modeling and analysis of a delayed competitive system with impulsive perturbations. Rocky Mt. J. Math. 38, 1505–1524 (2008)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Barabanova, A.: On the global existence of solutions of a reaction-diffusion equation with exponential nonlinearity. Proc. Am. Math. Soc. 122, 827–831 (1994)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Jia, Y.: Analysis on dynamics of a population model with predator-prey-dependent functional response. Appl. Math. Lett. 80, 64–70 (2018)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Douaifia, R., Abdelmalek, S., Bendoukha, S.: Global existence and asymptotic stability for a class of coupled reaction-diffusion systems on growing domains. Acta Appl. Math. 171, 17, 13 pages (2021)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Du, Y., Brown, K.J.: Bifurcation and monotonicity in competition reaction-diffusion systems. Nonlinear Anal. 23, 707–720 (1994)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Hsu, S.-B., Mei, L., Wang, F.-B.: On a nonlocal reaction-diffusion-advection system modelling the growth of phytoplankton with cell quota structure. J. Differ. Equ. 259, 5353–5378 (2015)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Kerner, E.H.: A statistical mechanics of interacting biological species. Bull. Math. Biol. 19, 121–146 (1957)

    MathSciNet  Google Scholar 

  18. 18.

    Shigesada, N., Kawasaki, K., Teramoto, E.: Spatial segregation of interacting species. J. Theor. Biol. 79, 83–99 (1979)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Shi, J., Xie, Z., Little, K.: Cross-diffusion induced instability and stability in reaction-diffusion systems. J. Appl. Anal. Comput. 1, 95–119 (2011)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Bendahmane, M.: Weak and classical solutions to predator-prey system with cross-diffusion. Nonlinear Anal. 73, 2489–2503 (2010)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Haile, D., Xie, Z.: Long-time behavior and Turing instability induced by cross-diffusion in a three species food chain model with a Holling type-II functional response. Math. Biosci. 267, 134–148 (2015)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Lou, Y., Tao, Y., Winkler, M.: Nonexistence of nonconstant steady-state solutions in a triangular cross-diffusion model. J. Differ. Equ. 262, 5160–5178 (2017)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Tulumello, E., Lombardo, M.C., Sammartino, M.: Cross-diffusion driven instability in a predator-prey system with cross-diffusion. Acta Appl. Math. 132, 621–633 (2014)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Li, S., Yamada, Y.: Effect of cross-diffusion in the diffusion prey-predator model with a protection zone II. J. Math. Anal. Appl. 461, 971–992 (2018)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Kuto, K.: Stability of steady-state solutions to a prey-predator system with cross-diffusion. J. Differ. Equ. 197, 293–314 (2004)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Iida, M., Mimura, M., Ninomiya, H.: Diffusion, cross-diffusion and competitive interaction. J. Math. Biol. 53, 617–641 (2006)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Oeda, K.: Effect of cross-diffusion on the stationary problem of a prey-predator model with a protection zone. J. Differ. Equ. 250, 3988–4009 (2011)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Li, Q., Liu, Z., Yuan, S.: Cross-diffusion induced Turing instability for a competition model with saturation effect. Appl. Math. Comput. 347, 64–77 (2019)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Choi, Y.S., Lui, R., Yamada, Y.: Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion. Discrete Contin. Dyn. Syst. 10, 719–730 (2004)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Jia, Y., Xue, P.: Effects of the self- and cross-diffusion on positive steady states for a generalized predator-prey system. Nonlinear Anal., Real World Appl. 32, 229–241 (2016)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Ryu, K., Ahn, I.: Coexistence theorem of steady states for nonlinear self-cross diffusion systems with competitive dynamics. J. Math. Anal. Appl. 283, 46–65 (2003)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Li, L., Ghoreishi, A.: On positive solutions of general nonlinear elliptic symbiotic interacting systems. Appl. Anal. 40, 281–295 (1991)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Li, L.: Coexistence theorems of steady states for predator-prey interacting systems. Trans. Am. Math. Soc. 305, 143–166 (1988)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18, 620–709 (1976)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Yamada, Y.: Stability of steady states for prey-predator diffusion equations with homogeneous Dirichlet conditions. SIAM J. Math. Anal. 21, 327–345 (1990)

    MathSciNet  Article  Google Scholar 

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Correspondence to Jingjing Wang.

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The work is supported in part by the Natural Science Foundations of China (11771262), and by the Natural Science Basic Research Plan in Shaanxi Province of China (2018JM1020).

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Wang, J., Zheng, H. Analysis on Steady States of a Competition System with Nonlinear Diffusion Terms. Acta Appl Math 171, 26 (2021). https://doi.org/10.1007/s10440-021-00393-7

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Keywords

  • Steady states
  • Competition model
  • Nonlinear-diffusion
  • Boundedness
  • Existence

Mathematics Subject Classification (2000)

  • 35K57
  • 92D25
  • 93C20