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POSITIVE SOLUTIONS OF A REACTION-DIFFUSION SYSTEM WITH DIRICHLET BOUNDARY CONDITION

  • Ma, Zhan-Ping (School of Mathematics and Information Science Henan Polytechnic University) ;
  • Yao, Shao-Wen (School of Mathematics and Information Science Henan Polytechnic University)
  • Received : 2019.04.20
  • Accepted : 2019.09.19
  • Published : 2020.05.31

Abstract

In this article, we study a reaction-diffusion system with homogeneous Dirichlet boundary conditions, which describing a three-species food chain model. Under some conditions, the predator-prey subsystem (u1 ≡ 0) has a unique positive solution (${\bar{u_2}}$, ${\bar{u_3}}$). By using the birth rate of the prey r1 as a bifurcation parameter, a connected set of positive solutions of our system bifurcating from semi-trivial solution set (r1, (0, ${\bar{u_2}}$, ${\bar{u_3}}$)) is obtained. Results are obtained by the use of degree theory in cones and sub and super solution techniques.

References

  1. H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), no. 4, 620-709. https://doi.org/10.1137/1018114 https://doi.org/10.1137/1018114
  2. J. Blat and K. J. Brown, Bifurcation of steady-state solutions in predator-prey and competition systems, Proc. Roy. Soc. Edinburgh Sect. A 97 (1984), 21-34. https://doi.org/10.1017/S0308210500031802 https://doi.org/10.1017/S0308210500031802
  3. J. Blat and K. J. Brown, Global bifurcation of positive solutions in some systems of elliptic equations, SIAM J. Math. Anal. 17 (1986), no. 6, 1339-1353. https://doi.org/10.1137/0517094 https://doi.org/10.1137/0517094
  4. C.-H. Chiu and S.-B. Hsu, Extinction of top-predator in a three-level food-chain model, J. Math. Biol. 37 (1998), no. 4, 372-380. https://doi.org/10.1007/s002850050134 https://doi.org/10.1007/s002850050134
  5. E. Conway, R. Gardner, and J. Smoller, Stability and bifurcation of steady-state solutions for predator-prey equations, Adv. in Appl. Math. 3 (1982), no. 3, 288-334. https://doi.org/10.1016/S0196-8858(82)80009-2 https://doi.org/10.1016/S0196-8858(82)80009-2
  6. E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl. 91 (1983), no. 1, 131-151. https://doi.org/10.1016/0022-247X(83)90098-7 https://doi.org/10.1016/0022-247X(83)90098-7
  7. E. N. Dancer, On positive solutions of some pairs of differential equations. II, J. Differential Equations 60 (1985), no. 2, 236-258. https://doi.org/10.1016/0022-0396(85)90115-9 https://doi.org/10.1016/0022-0396(85)90115-9
  8. H. I. Freedman and P. Waltman, Mathematical analysis of some three-species food-chain models, Math. Biosci. 33 (1977), no. 3-4, 257-276. https://doi.org/10.1016/0025-5564(77)90142-0 https://doi.org/10.1016/0025-5564(77)90142-0
  9. M. Haque, N. Ali, and S. Chakravarty, Study of a tri-trophic prey-dependent food chain model of interacting populations, Math. Biosci. 246 (2013), no. 1, 55-71. https://doi. org/10.1016/j.mbs.2013.07.021 https://doi.org/10.1016/j.mbs.2013.07.021
  10. A. Hastings and T. Powell, Chaos in a three species food chain, Ecology 7 (1991), 896-903.
  11. L. Hei, Global bifurcation of co-existence states for a predator-prey-mutualist model with diffusion, Nonlinear Anal. Real World Appl. 8 (2007), no. 2, 619-635. https://doi.org/10.1016/j.nonrwa.2006.01.006 https://doi.org/10.1016/j.nonrwa.2006.01.006
  12. P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations 5 (1980), no. 10, 999-1030. https://doi.org/10.1080/03605308008820162 https://doi.org/10.1080/03605308008820162
  13. S.-B. Hsu, S. Ruan, and T.-H. Yang, Analysis of three species Lotka-Volterra food web models with omnivory, J. Math. Anal. Appl. 426 (2015), no. 2, 659-687. https://doi.org/10.1016/j.jmaa.2015.01.035 https://doi.org/10.1016/j.jmaa.2015.01.035
  14. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, New York, 1966.
  15. W. Ko and K. Ryu, A qualitative study on general Gause-type predator-prey models with non-monotonic functional response, Nonlinear Anal. Real World Appl. 10 (2009), no. 4, 2558-2573. https://doi.org/10.1016/j.nonrwa.2008.05.012 https://doi.org/10.1016/j.nonrwa.2008.05.012
  16. N. Krikorian, The Volterra model for three species predator-prey systems: boundedness and stability, J. Math. Biol. 7 (1979), no. 2, 117-132. https://doi.org/10.1007/BF00276925 https://doi.org/10.1007/BF00276925
  17. N. Lakos, Existence of steady-state solutions for a one-predator-two-prey system, SIAM J. Math. Anal. 21 (1990), no. 3, 647-659. https://doi.org/10.1137/0521034 https://doi.org/10.1137/0521034
  18. L. Li and Y. Liu, Spectral and nonlinear effects in certain elliptic systems of three variables, SIAM J. Math. Anal. 24 (1993), no. 2, 480-498. https://doi.org/10.1137/0524030 https://doi.org/10.1137/0524030
  19. H. Li, P. Y. H. Pang, and M. Wang, Qualitative analysis of a diffusive prey-predator model with trophic interactions of three levels, Discrete Contin. Dyn. Syst. Ser. B 17 (2012), no. 1, 127-152. https://doi.org/10.3934/dcdsb.2012.17.127
  20. Z. Lin and M. Pedersen, Stability in a diffusive food-chain model with Michaelis-Menten functional response, Nonlinear Anal. 57 (2004), no. 3, 421-433. https://doi.org/10.1016/j.na.2004.02.022 https://doi.org/10.1016/j.na.2004.02.022
  21. Z.-P. Ma, Spatiotemporal dynamics of a diffusive Leslie-Gower prey-predator model with strong Allee effect, Nonlinear Anal. Real World Appl. 50 (2019), 651-674. https://doi.org/10.1016/j.nonrwa.2019.06.008 https://doi.org/10.1016/j.nonrwa.2019.06.008
  22. C. V. Pao, Global asymptotic stability of Lotka-Volterra 3-species reaction-diffusion systems with time delays, J. Math. Anal. Appl. 281 (2003), no. 1, 186-204. https://doi.org/10.1016/S0022-247X(03)00033-7
  23. R. Peng, J. Shi, and M. Wang, Stationary pattern of a ratio-dependent food chain model with diffusion, SIAM J. Appl. Math. 67 (2007), no. 5, 1479-1503. https://doi.org/10.1137/05064624X https://doi.org/10.1137/05064624X
  24. G. T. Whyburn, Topological Analysis, Princeton University Press, Princeton, NJ, 1958.
  25. Z. Xie, Cross-diffusion induced Turing instability for a three species food chain model, J. Math. Anal. Appl. 388 (2012), no. 1, 539-547. https://doi.org/10.1016/j.jmaa.2011.10.054 https://doi.org/10.1016/j.jmaa.2011.10.054
  26. Y. Yamada, Stability of steady states for prey-predator diffusion equations with homogeneous Dirichlet conditions, SIAM J. Math. Anal. 21 (1990), no. 2, 327-345. https://doi.org/10.1137/0521018 https://doi.org/10.1137/0521018
  27. S.-W. Yao, Z.-P. Ma, and Z.-B. Cheng, Pattern formation of a diffusive predator-prey model with strong Allee effect and nonconstant death rate, Phys. A 527 (2019), 121350, 11 pp. https://doi.org/10.1016/j.physa.2019.121350 https://doi.org/10.1016/j.physa.2019.121350