Abstract and Applied Analysis

Effect of Diffusion and Cross-Diffusion in a Predator-Prey Model with a Transmissible Disease in the Predator Species

Guohong Zhang and Xiaoli Wang

Full-text: Open access

Abstract

We study a Lotka-Volterra type predator-prey model with a transmissible disease in the predator population. We concentrate on the effect of diffusion and cross-diffusion on the emergence of stationary patterns. We first show that both self-diffusion and cross-diffusion can not cause Turing instability from the disease-free equilibria. Then we find that the endemic equilibrium remains linearly stable for the reaction diffusion system without cross-diffusion, while it becomes linearly unstable when cross-diffusion also plays a role in the reaction-diffusion system; hence, the instability is driven solely from the effect of cross-diffusion. Furthermore, we derive some results for the existence and nonexistence of nonconstant stationary solutions when the diffusion rate of a certain species is small or large.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 167856, 12 pages.

Dates
First available in Project Euclid: 2 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412277365

Digital Object Identifier
doi:10.1155/2014/167856

Mathematical Reviews number (MathSciNet)
MR3219356

Zentralblatt MATH identifier
07021849

Citation

Zhang, Guohong; Wang, Xiaoli. Effect of Diffusion and Cross-Diffusion in a Predator-Prey Model with a Transmissible Disease in the Predator Species. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 167856, 12 pages. doi:10.1155/2014/167856. https://projecteuclid.org/euclid.aaa/1412277365


Export citation

References

  • A. Casal, J. C. Eilbeck, and J. López-Gómez, “Existence and uniqueness of coexistence states for a predator-prey model with diffusion,” Differential and Integral Equations, vol. 7, no. 2, pp. 411–439, 1994.
  • R. Peng and J. P. Shi, “Non-existence of non-constant positive steady states of two Holling type-II predator-prey systems: strong interaction case,” Journal of Differential Equations, vol. 247, no. 3, pp. 866–886, 2009.
  • P. Y. H. Pang and M. X. Wang, “Non-constant positive steady states of a predator-prey system with non-monotonic functional response and diffusion,” Proceedings of the London Mathematical Society, vol. 88, no. 1, pp. 135–157, 2004.
  • Y. H. Du and Y. Lou, “Qualitative behaviour of positive solutions of a predator-prey model: effects of saturation,” Proceedings of the Royal Society of Edinburgh A. Mathematics, vol. 131, no. 2, pp. 321–349, 2001.
  • Y. H. Du and Y. Lou, “Some uniqueness and exact multiplicity results for a predator-prey model,” Transactions of the American Mathematical Society, vol. 349, no. 6, pp. 2443–2475, 1997.
  • M. X. Wang, “Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion,” Physica D, vol. 196, no. 1-2, pp. 172–192, 2004.
  • M. X. Wang, “Stationary patterns of strongly coupled prey-predator models,” Journal of Mathematical Analysis and Applications, vol. 292, no. 2, pp. 484–505, 2004.
  • W. Ko and K. Ryu, “Coexistence states of a predator-prey system with non-monotonic functional response,” Nonlinear Analysis. Real World Applications, vol. 8, no. 3, pp. 769–786, 2007.
  • L. Li, “Coexistence theorems of steady states for predator-prey interacting systems,” Transactions of the American Mathematical Society, vol. 305, no. 1, pp. 143–166, 1988.
  • E. N. Dancer, “On positive solutions of some pairs of differential equations,” Transactions of the American Mathematical Society, vol. 284, no. 2, pp. 729–743, 1984.
  • J. Zhou and C. L. Mu, “Coexistence states of a Holling type-II predator-prey system,” Journal of Mathematical Analysis and Applications, vol. 369, no. 2, pp. 555–563, 2010.
  • M. Haque, “Ratio-dependent predator-prey models of interacting populations,” Bulletin of Mathematical Biology, vol. 71, no. 2, pp. 430–452, 2009.
  • M. Haque, “A predator-prey model with disease in the predator species only,” Nonlinear Analysis. Real World Applications, vol. 11, no. 4, pp. 2224–2236, 2010.
  • M. Haque and E. Venturino, “The role of transmissible diseases in Holling-Tanner predatorprey model,” Theoretical Population Biology, vol. 70, pp. 273–288, 2006.
  • M. Haque, J. Zhen, and E. Venturino, “An ecoepidemiological predator-prey model with standard disease incidence,” Mathematical Methods in the Applied Sciences, vol. 32, no. 7, pp. 875–898, 2009.
  • Y. Xiao and L. Chen, “Analysis of a three species eco-epidemiological model,” Journal of Mathematical Analysis and Applications, vol. 258, no. 2, pp. 733–754, 2001.
  • Y. Xiao and L. Chen, “Modeling and analysis of a predator-prey model with disease in the prey,” Mathematical Biosciences, vol. 171, no. 1, pp. 59–82, 2001.
  • H. W. Hethcote, W. Wang, and Z. Ma, “A predator-prey model with infected prey,” Journal of Theoretical Population Biology, vol. 66, pp. 259–268, 2004.
  • H. Malchow, S. V. Petrovskii, and E. Venturino, Spatiotemporal Patterns in Ecology and Epidemiology, CRC, 2008.
  • E. Venturino, “The effects of diseases on competing species,” Mathematical Biosciences, vol. 174, no. 2, pp. 111–131, 2001.
  • E. Venturino, “The influence of diseases on Lotka-Volterra systems,” The Rocky Mountain Journal of Mathematics, vol. 24, no. 1, pp. 381–402, 1994.
  • E. Venturino, “Epidemics in predator-prey models: disease in the predators,” IMA Journal of Mathematics Applied in Medicine and Biology, vol. 19, pp. 185–205, 2002.
  • J. Chattopadhyay and O. Arino, “A predator-prey model with disease in the prey,” Nonlinear Analysis. Theory, Methods & Applications, vol. 36, no. 6, pp. 747–766, 1999.
  • N. Shigesada, K. Kawasaki, and E. Teramoto, “Spatial segregation of interacting species,” Journal of Theoretical Biology, vol. 79, no. 1, pp. 83–99, 1979.
  • C. Tian, Z. Ling, and Z. Lin, “Turing pattern formation in a predator-prey-mutualist system,” Nonlinear Analysis. Real World Applications, vol. 12, no. 6, pp. 3224–3237, 2011.
  • Z. Xie, “Turing instability in a coupled predator-prey model with different Holling type functional responses,” Discrete and Continuous Dynamical Systems S, vol. 4, no. 6, pp. 1621–1628, 2011.
  • Z. Xie, “Cross-diffusion induced Turing instability for a three species food chain model,” Journal of Mathematical Analysis and Applications, vol. 388, no. 1, pp. 539–547, 2012.
  • R. Peng, M. Wang, and G. Yang, “Stationary patterns of the Holling-Tanner prey-predator model with diffusion and cross-diffusion,” Applied Mathematics and Computation, vol. 196, no. 2, pp. 570–577, 2008.
  • M. Haque, S. Sarwardi, S. Preston, and E. Venturino, “Effect of delay in a Lotka-Volterra type predator-prey model with a transmissible disease in the predator species,” Mathematical Biosciences, vol. 234, no. 1, pp. 47–57, 2011.
  • P. Y. H. Pang and M. Wang, “Strategy and stationary pattern in a three-species predator-prey model,” Journal of Differential Equations, vol. 200, no. 2, pp. 245–273, 2004.
  • M. X. Wang, “Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion,” Physica D, vol. 196, no. 1-2, pp. 172–192, 2004.
  • M. X. Wang, “Stationary patterns caused by cross-diffusion for a three-species prey-predator model,” Computers & Mathematics with Applications, vol. 52, no. 5, pp. 707–720, 2006.
  • W. Chen and R. Peng, “Stationary patterns created by cross-diffusion for the competitor-competitor-mutualist model,” Journal of Mathematical Analysis and Applications, vol. 291, no. 2, pp. 550–564, 2004.
  • J.-F. Zhang, W.-T. Li, and Y.-X. Wang, “Turing patterns of a strongly coupled predator-prey system with diffusion effects,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 3, pp. 847–858, 2011.
  • Y. H. Xia and M. Han, “New conditions on the existence and stability of periodic solution in Lotka-Volterra's population system,” SIAM Journal on Applied Mathematics, vol. 69, no. 6, pp. 1580–1597, 2009.
  • Y. H. Xia, J. Cao, and M. Han, “A new analytical method for the linearization of dynamic equation on measure chains,” Journal of Differential Equations, vol. 235, no. 2, pp. 527–543, 2007.
  • D. Henry, Geometric Theory of Semilinear Parabolic Equations, vol. 840 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1981.
  • Y. Lou and W.-M. Ni, “Diffusion, self-diffusion and cross-diffusion,” Journal of Differential Equations, vol. 131, no. 1, pp. 79–131, 1996.
  • C.-S. Lin, W.-M. Ni, and I. Takagi, “Large amplitude stationary solutions to a chemotaxis system,” Journal of Differential Equations, vol. 72, no. 1, pp. 1–27, 1988.
  • L. Nirenberg, Topics in Nonlinear Functional Analysis, vol. 6, American Mathematical Society, Providence, RI, USA, 2001.
  • P. Y. H. Pang and M. X. Wang, “Qualitative analysis of a ratio-dependent predator-prey system with diffusion,” Proceedings of the Royal Society of Edinburgh A. Mathematics, vol. 133, no. 4, pp. 919–942, 2003. \endinput