# American Institute of Mathematical Sciences

October  2016, 21(8): 2811-2837. doi: 10.3934/dcdsb.2016074

## Spatial degeneracy vs functional response

 1 School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China 2 School of Mathematics and Statistics, Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, Gansu 730000

Received  October 2015 Revised  June 2016 Published  September 2016

In this paper, we are concerned with a predator-prey model with Beddington-DeAngelis functional response in heterogeneous environment. By the bifurcation theory and some estimates, the global bifurcation of positive stationary solution is shown. Our result shows that new stationary patterns are produced by the spatial degeneracy and the Beddington-DeAngelis functional response. Essentially different from the known results, the two factors generate two critical values for the prey growth rate $\lambda.$ As $\lambda$ crosses each critical value, the positive stationary solution set undergoes a drastic change. In particular, when $\lambda$ is suitably large, the interaction between the two factors yields nonexistence of positive stationary solutions for any $\mu,$ which is in strong contrast to the existence for suitable ranges of $\mu$ corresponding to the Lotka-Volterra or Holling-II functional response. Moreover, which one of the two factors plays a dominating role in the stationary patterns is shown. In addition, we give the asymptotic behavior of the positive stationary solutions as $\mu\rightarrow\infty$. Finally, both uniqueness and multiplicity of the positive stationary solutions are shown as well as their stability.
Citation: Yu-Xia Wang, Wan-Tong Li. Spatial degeneracy vs functional response. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2811-2837. doi: 10.3934/dcdsb.2016074
##### References:
 [1] J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44 (1975), 331-340. doi: 10.2307/3866.  Google Scholar [2] J. Blat and K. J. Brown, Global bifurcation of positive solutions in some systems of elliptic equations, SIAM J. Math. Anal., 17 (1986), 1339-1353. doi: 10.1137/0517094.  Google Scholar [3] R. S. Cantrell and C. Cosner, On the dynamics of predator-prey models with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 257 (2001), 206-222. doi: 10.1006/jmaa.2000.7343.  Google Scholar [4] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Ser. Math. Comput. Biol., John Wiley and sons, 2003. doi: 10.1002/0470871296.  Google Scholar [5] W. Y. Chen and M. X. Wang, Qualitative analysis of predator-prey models with Beddington-DeAngelis functional response and diffusion, Math. Comput. Modelling, 42 (2005), 31-44. doi: 10.1016/j.mcm.2005.05.013.  Google Scholar [6] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.  Google Scholar [7] E. N. Dancer and Y. H. Du, Effects of certain degeneracies in the predator-prey model, SIAM J. Math. Anal., 34 (2002), 292-314. doi: 10.1137/S0036141001387598.  Google Scholar [8] D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892. Google Scholar [9] D. T. Dimitrov and H. V. Kojouharov, Complete mathematical analysis of predator-prey models with linear prey growth and Beddington-DeAngelis functional response, Appl. Math. Comput., 162 (2005), 523-538. doi: 10.1016/j.amc.2003.12.106.  Google Scholar [10] Y. H. Du, Effects of a degeneracy in the competition model, part I. classical and generalized steady-state solutions, J. Differential Equations, 181 (2002), 92-132. doi: 10.1006/jdeq.2001.4074.  Google Scholar [11] Y. H. Du, Effects of a degeneracy in the competition model, part II. perturbation and dynamical behaviour, J. Differential Equations, 181 (2002), 133-164. doi: 10.1006/jdeq.2001.4075.  Google Scholar [12] Y. H. Du, Order structure and Topological Methods in Nonlinear PDEs, Vol.1: Maximum principle and Applications, World Scientific, Singapore, 2006. doi: 10.1142/9789812774446.  Google Scholar [13] Y. H. Du and S. B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Differential Equations, 203 (2004), 331-364. doi: 10.1016/j.jde.2004.05.010.  Google Scholar [14] Y. H. Du and Q. G. Huang, Blow-up solutions for a class of semilinear elliptic and parabolic equations, SIAM. J. Math. Anal., 31 (1999), 1-18. doi: 10.1137/S0036141099352844.  Google Scholar [15] Y. H. Du and X. Liang, A diffusive competition model with a protection zone, J. Differential Equations, 244 (2008), 61-86. doi: 10.1016/j.jde.2007.10.005.  Google Scholar [16] Y. H. Du and Y. Lou, Qualitative behaviour of positive solutions of a predator-prey model: Effects of saturation, Proc. Roy. Soc. Edinburgh A, 131 (2001), 321-349. doi: 10.1017/S0308210500000895.  Google Scholar [17] Y. H. Du and Y. Lou, S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model, J. Differential Equations, 144 (1998), 390-440. doi: 10.1006/jdeq.1997.3394.  Google Scholar [18] Y. H. Du and L. Ma, Logistic type equations on $\mathbbR^N$ by a squeezing method involving boundary blow-up solutions, J. London Math. Soc., 64 (2001), 107-124. doi: 10.1017/S0024610701002289.  Google Scholar [19] Y. H. Du, R. Peng and M. X. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Differential Equations, 246 (2009), 3932-3956. doi: 10.1016/j.jde.2008.11.007.  Google Scholar [20] Y. H. Du and J. P. 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Biol., 37 (1998), 61-83. doi: 10.1007/s002850050120.  Google Scholar [25] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, $2^{nd}$ edition, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar [26] G. H. Guo and J. H. Wu, Multiplicity and uniqueness of positive solutions for a predator-prey model with B-D functional response, Nonlinear Anal., 72 (2010), 1632-1646. doi: 10.1016/j.na.2009.09.003.  Google Scholar [27] G. H. Guo and J. H. Wu, The effect of mutual interference between predators on a predator-prey model with diffusion, J. Math. Anal. Appl., 389 (2012), 179-194. doi: 10.1016/j.jmaa.2011.11.044.  Google Scholar [28] V. Hutson, Y. Lou, K. Mischaikow and P. Poláčik, Competing species near a degenerate limit, SIAM. J. Math. Anal., 35 (2003), 453-491. doi: 10.1137/S0036141002402189.  Google Scholar [29] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin/New York, 1966.  Google Scholar [30] K. Kuto, Bifurcation branch of stationary solutions for a Lotka-Volterra cross-diffusion system in a spatially heterogeneous environment, Nonlinear Anal. RWA, 10 (2009), 943-965. doi: 10.1016/j.nonrwa.2007.11.015.  Google Scholar [31] C. S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7.  Google Scholar [32] J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, in Research Notes in Mathematics, vol. 426, Chapman and Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420035506.  Google Scholar [33] Y. Lou, S. Martínez and P. Poláčik, Loops and branches of coexistence states in a Lotka-Volterra competition model, J. Differential Equations, 230 (2006), 720-742. doi: 10.1016/j.jde.2006.04.005.  Google Scholar [34] K. Oeda, Effect of cross-diffusion on the stationary problem of a prey-predator model with a protection zone, J. Differential Equations, 250 (2011), 3988-4009. doi: 10.1016/j.jde.2011.01.026.  Google Scholar [35] T. Ouyang, On the positive solutions of semilinear equations $\Delta u+\lambda u-hu^p=0$ on the compact manifolds, Trans. Amer. Math. Soc., 331 (1992), 503-527. doi: 10.2307/2154124.  Google Scholar [36] R. Peng and M. X. Wang, Uniqueness and stability of steady states for a predator-prey model in heterogeneous environment, Proc. Amer. Math. Soc., 136 (2008), 859-865. doi: 10.1090/S0002-9939-07-09061-2.  Google Scholar [37] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9.  Google Scholar [38] J. P. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531. doi: 10.1006/jfan.1999.3483.  Google Scholar [39] M. X. Wang, Nonlinear Elliptic Equations (in Chinese), Science Press, Beijing, 2010. Google Scholar [40] M. X. Wang, P. Y. H. Pang and W. Y. Chen, Sharp spatial patterns of the diffusive Holling-Tanner prey-predator model in heterogeneous environment, IMA J. Appl. Math., 73 (2008), 815-835. doi: 10.1093/imamat/hxn016.  Google Scholar [41] Y. X. Wang and W. T. Li, Effects of cross-diffusion and heterogeneous environment on positive steady states of a prey-predator system, Nonlinear Anal. RWA, 14 (2013), 1235-1246. doi: 10.1016/j.nonrwa.2012.09.015.  Google Scholar [42] Y. X. Wang and W. T. Li, Fish-hook shaped global bifurcation branch of a spatially heterogeneous cooperative system with cross-diffusion, J. Differential Equations, 251 (2011), 1670-1695. doi: 10.1016/j.jde.2011.03.009.  Google Scholar [43] J. Zhou and J. P. Shi, The existence, bifurcation and stability of positive stationary solutions of a diffusive Leslie-Gower predator-prey model with Holling-type II functional responses, J. Math. Anal. Appl., 405 (2013), 618-630. doi: 10.1016/j.jmaa.2013.03.064.  Google Scholar

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##### References:
 [1] J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44 (1975), 331-340. doi: 10.2307/3866.  Google Scholar [2] J. Blat and K. J. Brown, Global bifurcation of positive solutions in some systems of elliptic equations, SIAM J. Math. Anal., 17 (1986), 1339-1353. doi: 10.1137/0517094.  Google Scholar [3] R. S. Cantrell and C. Cosner, On the dynamics of predator-prey models with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 257 (2001), 206-222. doi: 10.1006/jmaa.2000.7343.  Google Scholar [4] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Ser. Math. Comput. Biol., John Wiley and sons, 2003. doi: 10.1002/0470871296.  Google Scholar [5] W. Y. Chen and M. X. Wang, Qualitative analysis of predator-prey models with Beddington-DeAngelis functional response and diffusion, Math. Comput. Modelling, 42 (2005), 31-44. doi: 10.1016/j.mcm.2005.05.013.  Google Scholar [6] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.  Google Scholar [7] E. N. Dancer and Y. H. Du, Effects of certain degeneracies in the predator-prey model, SIAM J. Math. Anal., 34 (2002), 292-314. doi: 10.1137/S0036141001387598.  Google Scholar [8] D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892. Google Scholar [9] D. T. Dimitrov and H. V. Kojouharov, Complete mathematical analysis of predator-prey models with linear prey growth and Beddington-DeAngelis functional response, Appl. Math. Comput., 162 (2005), 523-538. doi: 10.1016/j.amc.2003.12.106.  Google Scholar [10] Y. H. Du, Effects of a degeneracy in the competition model, part I. classical and generalized steady-state solutions, J. Differential Equations, 181 (2002), 92-132. doi: 10.1006/jdeq.2001.4074.  Google Scholar [11] Y. H. Du, Effects of a degeneracy in the competition model, part II. perturbation and dynamical behaviour, J. Differential Equations, 181 (2002), 133-164. doi: 10.1006/jdeq.2001.4075.  Google Scholar [12] Y. H. Du, Order structure and Topological Methods in Nonlinear PDEs, Vol.1: Maximum principle and Applications, World Scientific, Singapore, 2006. doi: 10.1142/9789812774446.  Google Scholar [13] Y. H. Du and S. B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Differential Equations, 203 (2004), 331-364. doi: 10.1016/j.jde.2004.05.010.  Google Scholar [14] Y. H. Du and Q. G. Huang, Blow-up solutions for a class of semilinear elliptic and parabolic equations, SIAM. J. Math. Anal., 31 (1999), 1-18. doi: 10.1137/S0036141099352844.  Google Scholar [15] Y. H. Du and X. Liang, A diffusive competition model with a protection zone, J. Differential Equations, 244 (2008), 61-86. doi: 10.1016/j.jde.2007.10.005.  Google Scholar [16] Y. H. Du and Y. Lou, Qualitative behaviour of positive solutions of a predator-prey model: Effects of saturation, Proc. Roy. Soc. Edinburgh A, 131 (2001), 321-349. doi: 10.1017/S0308210500000895.  Google Scholar [17] Y. H. Du and Y. Lou, S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model, J. Differential Equations, 144 (1998), 390-440. doi: 10.1006/jdeq.1997.3394.  Google Scholar [18] Y. H. Du and L. Ma, Logistic type equations on $\mathbbR^N$ by a squeezing method involving boundary blow-up solutions, J. London Math. Soc., 64 (2001), 107-124. doi: 10.1017/S0024610701002289.  Google Scholar [19] Y. H. Du, R. Peng and M. X. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Differential Equations, 246 (2009), 3932-3956. doi: 10.1016/j.jde.2008.11.007.  Google Scholar [20] Y. H. Du and J. P. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Amer. Math. Soc., 359 (2007), 4557-4593. doi: 10.1090/S0002-9947-07-04262-6.  Google Scholar [21] Y. H. Du and J. P. Shi, A diffusive predator-prey model with a protection zone, J. Differential Equations, 229 (2006), 63-91. doi: 10.1016/j.jde.2006.01.013.  Google Scholar [22] Y. H. Du and J. P. Shi, Some recent results on diffusive predator-prey models in spatially heterogeneous environment, in Nonlinear dynamics and evolution equations (eds. H. Brummer, X. Zhao and X. Zou), Fields Inst. Commun. 48, AMS, Providence, RI, (2006), 95-135.  Google Scholar [23] Y. H. Du and M. X. Wang, Asymptotic behaviour of positive steady states to a predator-prey model, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 759-778. doi: 10.1017/S0308210500004704.  Google Scholar [24] J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction diffusion model, J. Math. Biol., 37 (1998), 61-83. doi: 10.1007/s002850050120.  Google Scholar [25] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, $2^{nd}$ edition, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar [26] G. H. Guo and J. H. Wu, Multiplicity and uniqueness of positive solutions for a predator-prey model with B-D functional response, Nonlinear Anal., 72 (2010), 1632-1646. doi: 10.1016/j.na.2009.09.003.  Google Scholar [27] G. H. Guo and J. H. Wu, The effect of mutual interference between predators on a predator-prey model with diffusion, J. Math. Anal. Appl., 389 (2012), 179-194. doi: 10.1016/j.jmaa.2011.11.044.  Google Scholar [28] V. Hutson, Y. Lou, K. Mischaikow and P. Poláčik, Competing species near a degenerate limit, SIAM. J. Math. Anal., 35 (2003), 453-491. doi: 10.1137/S0036141002402189.  Google Scholar [29] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin/New York, 1966.  Google Scholar [30] K. Kuto, Bifurcation branch of stationary solutions for a Lotka-Volterra cross-diffusion system in a spatially heterogeneous environment, Nonlinear Anal. RWA, 10 (2009), 943-965. doi: 10.1016/j.nonrwa.2007.11.015.  Google Scholar [31] C. S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7.  Google Scholar [32] J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, in Research Notes in Mathematics, vol. 426, Chapman and Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420035506.  Google Scholar [33] Y. Lou, S. Martínez and P. Poláčik, Loops and branches of coexistence states in a Lotka-Volterra competition model, J. Differential Equations, 230 (2006), 720-742. doi: 10.1016/j.jde.2006.04.005.  Google Scholar [34] K. Oeda, Effect of cross-diffusion on the stationary problem of a prey-predator model with a protection zone, J. Differential Equations, 250 (2011), 3988-4009. doi: 10.1016/j.jde.2011.01.026.  Google Scholar [35] T. Ouyang, On the positive solutions of semilinear equations $\Delta u+\lambda u-hu^p=0$ on the compact manifolds, Trans. Amer. Math. Soc., 331 (1992), 503-527. doi: 10.2307/2154124.  Google Scholar [36] R. Peng and M. X. Wang, Uniqueness and stability of steady states for a predator-prey model in heterogeneous environment, Proc. Amer. Math. Soc., 136 (2008), 859-865. doi: 10.1090/S0002-9939-07-09061-2.  Google Scholar [37] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9.  Google Scholar [38] J. P. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531. doi: 10.1006/jfan.1999.3483.  Google Scholar [39] M. X. Wang, Nonlinear Elliptic Equations (in Chinese), Science Press, Beijing, 2010. Google Scholar [40] M. X. Wang, P. Y. H. Pang and W. Y. Chen, Sharp spatial patterns of the diffusive Holling-Tanner prey-predator model in heterogeneous environment, IMA J. Appl. Math., 73 (2008), 815-835. doi: 10.1093/imamat/hxn016.  Google Scholar [41] Y. X. Wang and W. T. Li, Effects of cross-diffusion and heterogeneous environment on positive steady states of a prey-predator system, Nonlinear Anal. RWA, 14 (2013), 1235-1246. doi: 10.1016/j.nonrwa.2012.09.015.  Google Scholar [42] Y. X. Wang and W. T. Li, Fish-hook shaped global bifurcation branch of a spatially heterogeneous cooperative system with cross-diffusion, J. Differential Equations, 251 (2011), 1670-1695. doi: 10.1016/j.jde.2011.03.009.  Google Scholar [43] J. Zhou and J. P. Shi, The existence, bifurcation and stability of positive stationary solutions of a diffusive Leslie-Gower predator-prey model with Holling-type II functional responses, J. Math. Anal. Appl., 405 (2013), 618-630. doi: 10.1016/j.jmaa.2013.03.064.  Google Scholar
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