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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

Zhifu Xie 1,

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.
Keywords: Predator-prey model; Reaction-diffusion system; Cross-diffusion; Turing instability; Stationary pattern; Holling type functional response..
Mathematics Subject Classification: Primary: 92C15, 35K57; Secondary: 37L15, 92D4.
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.

show all references

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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