# American Institute of Mathematical Sciences

December  2011, 4(6): 1621-1628. doi: 10.3934/dcdss.2011.4.1621

## Turing instability in a coupled predator-prey model with different Holling type functional responses

 1 Department of Mathematics and Computer Science, Virginia State University, Petersburg, Virginia 23806

Received  April 2009 Revised  November 2009 Published  December 2010

In a reaction-diffusion system, diffusion can induce the instability of a positive equilibrium which is stable with respect to a constant perturbation, therefore, the diffusion may create new patterns when the corresponding system without diffusion fails, as shown by Turing in 1950s. In this paper we study a coupled predator-prey model with different Holling type functional responses, where cross-diffusions are included in such a way that the prey runs away from predator and the predator chase preys. We conduct the Turing instability analysis for each Holling functional response. We prove that if a positive equilibrium solution is linearly stable with respect to the ODE system of the predator-prey model, then it is also linearly stable with respect to the model. So diffusion and cross-diffusion in the predator-prey model with Holling type functional responses given in this paper can not drive Turing instability. However, diffusion and cross-diffusion can still create non-constant positive solutions for the model.
Citation: Zhifu Xie. Turing instability in a coupled predator-prey model with different Holling type functional responses. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1621-1628. doi: 10.3934/dcdss.2011.4.1621
##### References:
 [1] X. Chen, Y. Qi and M. Wang, A strongly coupled predator-prey system with non-monotonic functional response, Nonl. Anal.: TMA, 67 (2007), 1966-1979. doi: 10.1016/j.na.2006.08.022. [2] L. Chen and A. Jungel, Analysis of a parabolic cross-diffusion population model without self-diffusion, J. Differential Equations, 224 (2006), 39-59. doi: 10.1016/j.jde.2005.08.002. [3] Y. H. Du and Y. Lou, Qualitative behaviour of positive solutions of a predator-prey model: effects of saturation, Roy. Soc. Edinburgh Sect. A, 131 (2001), 321-349. doi: 10.1017/S0308210500000895. [4] Y. H. Du and J. P. Shi, Some recent results on diffusive predator-prey models in spatially heterogeneous environment, Nonlinear Dynamics and Evolution Equations, in: Fields Inst. Commun., Vol. 48, Amer. Math. Soc., Providence, RI, 2006, 95-135. [5] 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. [6] 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 (electronic). doi: 10.1090/S0002-9947-07-04262-6. [7] S. M. Fu, Z. J. Wen and S. B. Cui, On global solutions for the three-species food-chain model with cross-diffusion, ACTA Mathematica Sinica, Chinese Series, 50 (2007), 75-80. [8] L. Hei, Global bifurcation of co-existence states for a predator-prey-mutualist model with diffusion, Nonl. Ana.: RWA, 8 (2007), 619-635. doi: 10.1016/j.nonrwa.2006.01.006. [9] C. S. Holling, The components of predation as revealed by a study of small mammal predation of the European pine sawfly, Canad. Entomol., 91 (1959), 293-320. doi: 10.4039/Ent91293-5. [10] C. S. Holling, Some characteristics of simple types of predation and parasitism, Canad. Entomol., 91 (1959), 385-398. doi: 10.4039/Ent91385-7. [11] X. J. Hou and A. W. Leung, Traveling wave solutions for a competitive reaction-diffusion system and their asymptotics, Nonlinear Anal. Real World Appl., 9 (2008), 2196-2213. doi: 10.1016/j.nonrwa.2007.07.007. [12] S. B. Hsu and J. P. Shi, Relaxation oscillation profile of limit cycle in predator-prey system, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 893-911. doi: 10.3934/dcdsb.2009.11.893. [13] J. Von Hardenberg, E. Meron, M. Shachak and Y. Zarmi, Diversity of vegetation patterns and desertification, Phys. Rev. Lett., 87 (2001), 198101. doi: 10.1103/PhysRevLett.87.198101. [14] A. J. Lotka, "Elements of Physical Biology," Baltimore: Williams & Wilkins Co., 1925. [15] E. Meron, E. Gilad, J. von Hardenberg, M. Shachak and Y. Zarmi, Vegetation patterns along a rainfall gradient, Chaos Solitons Fractals, 19 (2004), 367-376. doi: 10.1016/S0960-0779(03)00049-3. [16] K. Ik Kim and Z. Lin, Coexistence of three species in a strongly coupled elliptic system, Nonl. Anal., 55 (2003), 313-333. doi: 10.1016/S0362-546X(03)00242-6. [17] T. Kadota and K. Kuto, Positive steady states for a prey-predator model with some nonlinear diffusion terms, J. Math. Anal. Appl., 323 (2006), 1387-1401. doi: 10.1016/j.jmaa.2005.11.065. [18] K. Kuto and Y. Yamada, Multiple coexistence states for a prey-predator system with cross-diffusion, J. Differential Equations, 197 (2004), 315-348. doi: 10.1016/j.jde.2003.08.003. [19] C. S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis systems, J. Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7. [20] J. D. Murray, "Mathematical Biology. Third Edition. I. An Introduction," Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002; II. Spatial models and biomedical applications. Interdisciplinary Applied Mathematics, 18, Springer-Verlag, New York, 2003. [21] A. Okubo, "Diffusion and Ecological Problems: Mathematical Models, An Extended Version of the Japanese Edition, Ecology and Diffusion," Translated by G. N. Parker. Biomathematics, 10, Springer-Verlag, Berlin-New York, 1980. [22] P. Y. H. Pang and M. Wang, Strategy and stationary pattern in a three-species predator-prey model, J. Differential Equations, 200 (2004), 245-273; (2004), 1065-1089. [23] J. P. Shi, Z. F. Xie and K. Little, Cross-diffusion induced instability and stability in reaction-diffusion systems, Preprint. [24] F. Yi, J. Wei and J. P. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977. doi: 10.1016/j.jde.2008.10.024. [25] A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Royal Soc. London B, 237 (1952), 37-72. doi: 10.1098/rstb.1952.0012. [26] V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Mem. R. Accad. Naz. dei Lincei. Ser. VI, 2 (1926). [27] M. Wang, Stationary patterns of strongly coupled prey-predator models, J. Math. Anal. Appl., 292 (2004), 484-505. doi: 10.1016/j.jmaa.2003.12.027. [28] X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: Effects of motility and chemotaxis and dynamics, SIAM J. Math. Anal., 31 (2000), 535-560. doi: 10.1137/S0036141098339897. [29] C. S. Zhao, Asymptotic behaviors of a class of $N$-Laplacian Neumann problems with large diffusion, Nonlinear Anal., 69 (2008), 2496-2524. doi: 10.1016/j.na.2007.08.028. [30] X. Zeng, Non-constant positive steady states of a prey-predator system with cross-diffusions, J. Math. Anal. Appl., 332 (2007), 989-1009. doi: 10.1016/j.jmaa.2006.10.075.

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##### References:
 [1] X. Chen, Y. Qi and M. Wang, A strongly coupled predator-prey system with non-monotonic functional response, Nonl. Anal.: TMA, 67 (2007), 1966-1979. doi: 10.1016/j.na.2006.08.022. [2] L. Chen and A. Jungel, Analysis of a parabolic cross-diffusion population model without self-diffusion, J. Differential Equations, 224 (2006), 39-59. doi: 10.1016/j.jde.2005.08.002. [3] Y. H. Du and Y. Lou, Qualitative behaviour of positive solutions of a predator-prey model: effects of saturation, Roy. Soc. Edinburgh Sect. A, 131 (2001), 321-349. doi: 10.1017/S0308210500000895. [4] Y. H. Du and J. P. Shi, Some recent results on diffusive predator-prey models in spatially heterogeneous environment, Nonlinear Dynamics and Evolution Equations, in: Fields Inst. Commun., Vol. 48, Amer. Math. Soc., Providence, RI, 2006, 95-135. [5] 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. [6] 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 (electronic). doi: 10.1090/S0002-9947-07-04262-6. [7] S. M. Fu, Z. J. Wen and S. B. Cui, On global solutions for the three-species food-chain model with cross-diffusion, ACTA Mathematica Sinica, Chinese Series, 50 (2007), 75-80. [8] L. Hei, Global bifurcation of co-existence states for a predator-prey-mutualist model with diffusion, Nonl. Ana.: RWA, 8 (2007), 619-635. doi: 10.1016/j.nonrwa.2006.01.006. [9] C. S. Holling, The components of predation as revealed by a study of small mammal predation of the European pine sawfly, Canad. Entomol., 91 (1959), 293-320. doi: 10.4039/Ent91293-5. [10] C. S. Holling, Some characteristics of simple types of predation and parasitism, Canad. Entomol., 91 (1959), 385-398. doi: 10.4039/Ent91385-7. [11] X. J. Hou and A. W. Leung, Traveling wave solutions for a competitive reaction-diffusion system and their asymptotics, Nonlinear Anal. Real World Appl., 9 (2008), 2196-2213. doi: 10.1016/j.nonrwa.2007.07.007. [12] S. B. Hsu and J. P. Shi, Relaxation oscillation profile of limit cycle in predator-prey system, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 893-911. doi: 10.3934/dcdsb.2009.11.893. [13] J. Von Hardenberg, E. Meron, M. Shachak and Y. Zarmi, Diversity of vegetation patterns and desertification, Phys. Rev. Lett., 87 (2001), 198101. doi: 10.1103/PhysRevLett.87.198101. [14] A. J. Lotka, "Elements of Physical Biology," Baltimore: Williams & Wilkins Co., 1925. [15] E. Meron, E. Gilad, J. von Hardenberg, M. Shachak and Y. Zarmi, Vegetation patterns along a rainfall gradient, Chaos Solitons Fractals, 19 (2004), 367-376. doi: 10.1016/S0960-0779(03)00049-3. [16] K. Ik Kim and Z. Lin, Coexistence of three species in a strongly coupled elliptic system, Nonl. Anal., 55 (2003), 313-333. doi: 10.1016/S0362-546X(03)00242-6. [17] T. Kadota and K. Kuto, Positive steady states for a prey-predator model with some nonlinear diffusion terms, J. Math. Anal. Appl., 323 (2006), 1387-1401. doi: 10.1016/j.jmaa.2005.11.065. [18] K. Kuto and Y. Yamada, Multiple coexistence states for a prey-predator system with cross-diffusion, J. Differential Equations, 197 (2004), 315-348. doi: 10.1016/j.jde.2003.08.003. [19] C. S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis systems, J. Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7. [20] J. D. Murray, "Mathematical Biology. Third Edition. I. An Introduction," Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002; II. Spatial models and biomedical applications. Interdisciplinary Applied Mathematics, 18, Springer-Verlag, New York, 2003. [21] A. Okubo, "Diffusion and Ecological Problems: Mathematical Models, An Extended Version of the Japanese Edition, Ecology and Diffusion," Translated by G. N. Parker. Biomathematics, 10, Springer-Verlag, Berlin-New York, 1980. [22] P. Y. H. Pang and M. Wang, Strategy and stationary pattern in a three-species predator-prey model, J. Differential Equations, 200 (2004), 245-273; (2004), 1065-1089. [23] J. P. Shi, Z. F. Xie and K. Little, Cross-diffusion induced instability and stability in reaction-diffusion systems, Preprint. [24] F. Yi, J. Wei and J. P. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977. doi: 10.1016/j.jde.2008.10.024. [25] A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Royal Soc. London B, 237 (1952), 37-72. doi: 10.1098/rstb.1952.0012. [26] V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Mem. R. Accad. Naz. dei Lincei. Ser. VI, 2 (1926). [27] M. Wang, Stationary patterns of strongly coupled prey-predator models, J. Math. Anal. Appl., 292 (2004), 484-505. doi: 10.1016/j.jmaa.2003.12.027. [28] X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: Effects of motility and chemotaxis and dynamics, SIAM J. Math. Anal., 31 (2000), 535-560. doi: 10.1137/S0036141098339897. [29] C. S. Zhao, Asymptotic behaviors of a class of $N$-Laplacian Neumann problems with large diffusion, Nonlinear Anal., 69 (2008), 2496-2524. doi: 10.1016/j.na.2007.08.028. [30] X. Zeng, Non-constant positive steady states of a prey-predator system with cross-diffusions, J. Math. Anal. Appl., 332 (2007), 989-1009. doi: 10.1016/j.jmaa.2006.10.075.
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