Gan, Wenzhen; Tian, Canrong; Zhang, Qunying; Lin, Zhigui Stability in a simple food chain system with Michaelis-Menten functional response and nonlocal delays. (English) Zbl 1470.35181 Abstr. Appl. Anal. 2013, Article ID 936952, 14 p. (2013). Summary: This paper is concerned with the asymptotical behavior of solutions to the reaction-diffusion system under homogeneous Neumann boundary condition. By taking food ingestion and species’ moving into account, the model is further coupled with Michaelis-Menten type functional response and nonlocal delay. Sufficient conditions are derived for the global stability of the positive steady state and the semitrivial steady state of the proposed problem by using the Lyapunov functional. Our results show that intraspecific competition benefits the coexistence of prey and predator. Furthermore, the introduction of Michaelis-Menten type functional response positively affects the coexistence of prey and predator, and the nonlocal delay is harmless for stabilities of all nonnegative steady states of the system. Numerical simulations are carried out to illustrate the main results. MSC: 35K51 Initial-boundary value problems for second-order parabolic systems 35B35 Stability in context of PDEs 92D25 Population dynamics (general) × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Xu, R.; Chaplain, M. A. J., Persistence and global stability in a delayed predator-prey system with Michaelis-Menten type functional response, Applied Mathematics and Computation, 130, 2-3, 441-455 (2002) · Zbl 1030.34069 · doi:10.1016/S0096-3003(01)00111-4 [2] Huo, H. F.; Li, W. T., Periodic solution of a delayed predator-prey system with Michaelis-Menten type functional response, Journal of Computational and Applied Mathematics, 166, 2, 453-463 (2004) · Zbl 1047.34081 · doi:10.1016/j.cam.2003.08.042 [3] Lin, Z.; Pedersen, M., Stability in a diffusive food-chain model with Michaelis-Menten functional response, Nonlinear Analysis: Theory, Methods & Applications, 57, 3, 421-433 (2004) · Zbl 1053.35026 · doi:10.1016/j.na.2004.02.022 [4] Dai, B. X.; Zhang, N.; Zou, J. Z., Permanence for the Michaelis-Menten type discrete three-species ratio-dependent food chain model with delay, Journal of Mathematical Analysis and Applications, 324, 1, 728-738 (2006) · Zbl 1101.92048 · doi:10.1016/j.jmaa.2005.12.060 [5] Britton, N. F., Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM Journal on Applied Mathematics, 50, 6, 1663-1688 (1990) · Zbl 0723.92019 · doi:10.1137/0150099 [6] Gourley, S. A.; Britton, N. F., Instability of travelling wave solutions of a population model with nonlocal effects, IMA Journal of Applied Mathematics, 51, 3, 299-310 (1993) · Zbl 0832.35068 · doi:10.1093/imamat/51.3.299 [7] Zhang, X.; Xu, R., Traveling waves of a diffusive predator-prey model with nonlocal delay and stage structure, Journal of Mathematical Analysis and Applications, 373, 2, 475-484 (2011) · Zbl 1208.34108 · doi:10.1016/j.jmaa.2010.07.044 [8] Wang, Z. C.; Li, W. T.; Ruan, S. G., Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, Journal of Differential Equations, 238, 1, 153-200 (2007) · Zbl 1124.35089 · doi:10.1016/j.jde.2007.03.025 [9] Gourley, S. A.; Ruan, S. G., Convergence and travelling fronts in functional differential equations with nonlocal terms: a competition model, SIAM Journal on Mathematical Analysis, 35, 3, 806-822 (2003) · Zbl 1040.92045 · doi:10.1137/S003614100139991 [10] Lai, X. H.; Yao, T. X., Exponential stability of impulsive delayed reaction-diffusion cellular neural networks via Poincaré integral inequality, Abstract and Applied Analysis, 2013 (2013) · Zbl 1273.35287 · doi:10.1155/2013/131836 [11] Xu, R.; Ma, Z. E., Global stability of a reaction-diffusion predator-prey model with a nonlocal delay, Mathematical and Computer Modelling, 50, 1-2, 194-206 (2009) · Zbl 1185.35130 · doi:10.1016/j.mcm.2009.02.011 [12] Li, Z.; Chen, F. D.; He, M. X., Asymptotic behavior of the reaction-diffusion model of plankton allelopathy with nonlocal delays, Nonlinear Analysis: Real World Applications, 12, 3, 1748-1758 (2011) · Zbl 1223.35069 · doi:10.1016/j.nonrwa.2010.11.007 [13] Kuang, Y.; Smith, H. L., Global stability in diffusive delay Lotka-Volterra systems, Differential and Integral Equations, 4, 1, 117-128 (1991) · Zbl 0752.34041 [14] Lin, Z. G.; Pedersen, M.; Zhang, L., A predator-prey system with stage-structure for predator and nonlocal delay, Nonlinear Analysis: Theory, Methods & Applications, 72, 3-4, 2019-2030 (2010) · Zbl 1201.35041 · doi:10.1016/j.na.2009.10.002 [15] Xu, R., A reaction-diffusion predator-prey model with stage structure and nonlocal delay, Applied Mathematics and Computation, 175, 2, 984-1006 (2006) · Zbl 1099.92081 · doi:10.1016/j.amc.2005.08.014 [16] Chen, S. S.; Shi, J. P., Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect, Journal of Differential Equations, 253, 12, 3440-3470 (2012) · Zbl 1256.35177 · doi:10.1016/j.jde.2012.08.031 [17] Freedman, H. I., Deterministic Mathematical Models in Population Ecology. Deterministic Mathematical Models in Population Ecology, Monographs and Textbooks in Pure and Applied Mathematics, 57 (1980), New York, NY, USA: Marcel Dekker, New York, NY, USA · Zbl 0448.92023 [18] Abrams, P. A.; Ginzburg, L. R., The nature of predation: prey depedent, ratio dependent or neither, Trends in Ecology and Evolution, 15, 8, 337-341 (2000) · doi:10.1016/S0169-5347(00)01908-X [19] DeAngelis, D. L.; Goldstein, R. A.; O’Neill, R. V., A model for trophic iteration, Ecology, 56, 4, 881-892 (1975) [20] Pao, C. V., Dynamics of nonlinear parabolic systems with time delays, Journal of Mathematical Analysis and Applications, 198, 3, 751-779 (1996) · Zbl 0860.35138 · doi:10.1006/jmaa.1996.0111 [21] Xu, F.; Hou, Y.; Lin, Z. G., Time delay parabolic system in a predator-prey model with stage structure, Acta Mathematica Sinica, 48, 6, 1121-1130 (2005) · Zbl 1124.35330 [22] Pang, P. Y. H.; Wang, M. X., Strategy and stationary pattern in a three-species predator-prey model, Journal of Differential Equations, 200, 2, 245-273 (2004) · Zbl 1106.35016 · doi:10.1016/j.jde.2004.01.004 [23] Henry, D., Geometric Theory of Semilinear Parabolic Equations. Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840 (1981), New York, NY, USA: Springer, New York, NY, USA · Zbl 0456.35001 [24] Brown, K. J.; Dunne, P. C.; Gardner, R. A., A semilinear parabolic system arising in the theory of superconductivity, Journal of Differential Equations, 40, 2, 232-252 (1981) · Zbl 0431.35054 · doi:10.1016/0022-0396(81)90020-6 [25] Gourley, S. A.; So, J. W. H., Dynamics of a food-limited population model incorporating nonlocal delays on a finite domain, Journal of Mathematical Biology, 44, 1, 49-78 (2002) · Zbl 0993.92027 · doi:10.1007/s002850100109 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.