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http://dx.doi.org/10.4134/BKMS.b190416

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)
Publication Information
Bulletin of the Korean Mathematical Society / v.57, no.3, 2020 , pp. 677-690 More about this Journal
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.
Keywords
Reaction-diffusion; food chain model; positive solutions; bifurcation; degree theory;
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1 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.   DOI
2 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   DOI
3 G. T. Whyburn, Topological Analysis, Princeton University Press, Princeton, NJ, 1958.
4 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   DOI
5 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   DOI
6 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   DOI
7 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   DOI
8 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   DOI
9 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   DOI
10 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   DOI
11 A. Hastings and T. Powell, Chaos in a three species food chain, Ecology 7 (1991), 896-903.   DOI
12 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   DOI
13 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   DOI
14 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   DOI
15 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   DOI
16 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   DOI
17 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   DOI
18 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   DOI
19 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   DOI
20 T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, New York, 1966.
21 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   DOI
22 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   DOI
23 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   DOI
24 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   DOI
25 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   DOI
26 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
27 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   DOI