## Abstract and Applied Analysis

### Global Behavior for a Strongly Coupled Predator-Prey Model with One Resource and Two Consumers

#### Abstract

We consider a strongly coupled predator-prey model with one resource and two consumers, in which the first consumer species feeds on the resource according to the Holling II functional response, while the second consumer species feeds on the resource following the Beddington-DeAngelis functional response, and they compete for the common resource. Using the energy estimates and Gagliardo-Nirenberg-type inequalities, the existence and uniform boundedness of global solutions for the model are proved. Meanwhile, the sufficient conditions for global asymptotic stability of the positive equilibrium for this model are given by constructing a Lyapunov function.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 183285, 25 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1355495806

Digital Object Identifier
doi:10.1155/2012/183285

Mathematical Reviews number (MathSciNet)
MR2959743

Zentralblatt MATH identifier
1246.35105

#### Citation

Jiao, Yujuan; Fu, Shengmao. Global Behavior for a Strongly Coupled Predator-Prey Model with One Resource and Two Consumers. Abstr. Appl. Anal. 2012 (2012), Article ID 183285, 25 pages. doi:10.1155/2012/183285. https://projecteuclid.org/euclid.aaa/1355495806

#### References

• V. Volterra, “Variations and fluctuations of the number of individuals in animal species living together,” Journal du Conseil Conseil International pour l'Exploration de la Mer, vol. 3, pp. 3–51, 1928.
• F. J. Ayala, “Experimental invalidation of the principle of competitive exclusion,” Nature, vol. 224, pp. 1076–1079, 1969.
• R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Chichester, UK, 2003.
• T. W. Schoener, “Alternatives to Lotka-Volterra competition: models of intermediate complexity,” Theoretical Population Biology, vol. 10, no. 3, pp. 309–333, 1976.
• R. S. Cantrell, C. Cosner, and S. Ruan, “Intraspecific interference and consumer-resource dynamics,” Discrete and Continuous Dynamical Systems Series B, vol. 4, no. 3, pp. 527–546, 2004.
• L.-J. Hei and Y. Yu, “Non-constant positive steady state of one resource and two consumers model with diffusion,” Journal of Mathematical Analysis and Applications, vol. 339, no. 1, pp. 566–581, 2008.
• K. Kuto, “Stability of steady-state solutions to a prey-predator system with cross-diffusion,” Journal of Differential Equations, vol. 197, no. 2, pp. 293–314, 2004.
• K. Kuto and Y. Yamada, “Multiple coexistence states for a prey-predator system with cross-diffusion,” Journal of Differential Equations, vol. 197, no. 2, pp. 315–348, 2004.
• Y. Lou and W. Ni, “Diffusion, self-diffusion and cross-diffusion,” Journal of Differential Equations, vol. 131, pp. 79–131, 1996.
• Y. Lou and W.-M. Ni, “Diffusion vs cross-diffusion: an elliptic approach,” Journal of Differential Equations, vol. 154, no. 1, pp. 157–190, 1999.
• 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.
• J. Shi, Z. Xie, and K. Little, “Cross-diffusion induced instability and stability in reaction-diffusion systems,” Journal of Applied Analysis and Computation, vol. 1, pp. 95–119, 2011.
• C. Tian, Z. Lin, and M. Pedersen, “Instability induced by cross-diffusion in reaction-diffusion systems,” Nonlinear Analysis. Real World Applications, vol. 11, no. 2, pp. 1036–1045, 2010.
• M. Wang, “Stationary patterns of strongly coupled prey-predator models,” Journal of Mathematical Analysis and Applications, vol. 292, no. 2, pp. 484–505, 2004.
• Y. S. Choi, R. Lui, and Y. Yamada, “Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion,” Discrete and Continuous Dynamical Systems Series A, vol. 9, no. 5, pp. 1193–1200, 2003.
• Y. S. Choi, R. Lui, and Y. Yamada, “Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion,” Discrete and Continuous Dynamical Systems Series A, vol. 10, no. 3, pp. 719–730, 2004.
• S. Fu, Z. Wen, and S. Cui, “Uniform boundedness and stability of global solutions in a strongly coupled three-species cooperating model,” Nonlinear Analysis: Real World Applications, vol. 55, pp. 1–18, 2006.
• D. Le, “Cross diffusion systems on $n$ spatial dimensional domains,” Indiana University Mathematics Journal, vol. 51, no. 3, pp. 625–643, 2002.
• D. Le, L. V. Nguyen, and T. T. Nguyen, “Shigesada-Kawasaki-Teramoto model on higher dimensional domains,” Electronic Journal of Differential Equations, vol. 72, pp. 1–12, 2003.
• Y. Lou, W.-M. Ni, and Y. Wu, “On the global existence of a cross-diffusion system,” Discrete and Continuous Dynamical Systems, vol. 4, no. 2, pp. 193–203, 1998.
• P. V. Tuoc, “Global existence of solutions to Shigesada-Kawasaki-Teramoto cross-diffusion systems on domains of arbitrary dimensions,” Proceedings of the American Mathematical Society, vol. 135, no. 12, pp. 3933–3941, 2007.
• P. V. Tuoc, “On global existence of solutions to a cross-diffusion system,” Journal of Mathematical Analysis and Applications, vol. 343, no. 2, pp. 826–834, 2008.
• Y. Yamada, “Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with cross-diffusion,” Mathematical and Analytic Institute Communications in Waseda University, vol. 1358, pp. 24–33, 2004.
• J. W. Barrett and J. F. Blowey, “Finite element approximation of a nonlinear cross-diffusion population model,” Numerische Mathematik, vol. 98, no. 2, pp. 195–221, 2004.
• J. W. Barrett, J. F. Blowey, and H. Garcke, “Finite element approximation of a fourth order nonlinear degenerate parabolic equation,” Numerische Mathematik, vol. 80, no. 4, pp. 525–556, 1998.
• J. W. Barrett, H. Garcke, and R. Nürnberg, “Finite element approximation of surfactant spreading on a thin film,” SIAM Journal on Numerical Analysis, vol. 41, no. 4, pp. 1427–1464, 2003.
• L. Chen and A. Jüngel, “Analysis of a parabolic cross-diffusion population model without self-diffusion,” Journal of Differential Equations, vol. 224, no. 1, pp. 39–59, 2006.
• F. Filbet, “A finite volume scheme for the Patlak-Keller-Segel chemotaxis model,” Numerische Mathematik, vol. 104, no. 4, pp. 457–488, 2006.
• G. Galiano, M. L. Garzón, and A. Jüngel, “Semi-discretization in time and numerical convergence of solutions of a nonlinear cross-diffusion population model,” Numerische Mathematik, vol. 93, no. 4, pp. 655–673, 2003.
• G. Gilardi and U. Stefanelli, “Time-discretization and global solution for a doubly nonlinear Volterra equation,” Journal of Differential Equations, vol. 228, no. 2, pp. 707–736, 2006.
• Y.-H. Fan, L.-L. Wang, and M.-X. Wang, “Notes on multiple bifurcations in a delayed predator-prey model with nonmonotonic functional response,” International Journal of Biomathematics, vol. 2, no. 2, pp. 129–138, 2009.
• Z. Luo and Z.-R. He, “Optimal harvesting problem for an age-dependent $n$-dimensional competing system with diffusion,” International Journal of Biomathematics, vol. 1, no. 2, pp. 133–145, 2008.
• B. Dubey and B. Hussain, “A predator-prey interaction model with self and cross-diffusion,” Ecological Modelling, vol. 141, pp. 67–76, 2001.
• H. Amann, “Dynamic theory of quasilinear parabolic equations–-I. Abstract evolution equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 12, no. 9, pp. 895–919, 1988.
• H. Amann, “Dynamic theory of quasilinear parabolic equations–-II. Reaction-diffusion systems,” Differential and Integral Equations, vol. 3, no. 1, pp. 13–75, 1990.
• H. Amann, “Dynamic theory of quasilinear parabolic systems–-III. Global existence,” Mathematische Zeitschrift, vol. 202, no. 2, pp. 219–250, 1989.
• M. Wang, Nonlinear Partial Differential Equations of Parabolic Type, Science Press, Beijing, China, 1993.
• S.-A. Shim, “Uniform boundedness and convergence of solutions to cross-diffusion systems,” Journal of Differential Equations, vol. 185, no. 1, pp. 281–305, 2002.
• S.-A. Shim, “Uniform boundedness and convergence of solutions to the systems with cross-diffusions dominated by self-diffusions,” Nonlinear Analysis. Real World Applications, vol. 4, no. 1, pp. 65–86, 2003.
• S.-A. Shim, “Uniform boundedness and convergence of solutions to the systems with a single nonzero cross-diffusion,” Journal of Mathematical Analysis and Applications, vol. 279, no. 1, pp. 1–21, 2003.
• L. Nirenberg, “On elliptic partial differential equations,” Annali della Scuola Normale Superiore di Pisa, vol. 13, pp. 115–162, 1959.
• Z. Lin and M. Pedersen, “Stability in a diffusive food-chain model with Michaelis-Menten functional response,” Nonlinear Analysis. Theory, Methods & Applications, vol. 57, no. 3, pp. 421–433, 2004.