Spatiotemporal Dynamics in a Diffusive Predator–Prey System with Beddington–DeAngelis Functional Response

• Xiang-Ping Yan ^[1] ; Cun-Hua Zhang ^[1]
  1. [1] Lanzhou Jiaotong University

     Lanzhou Jiaotong University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 4, 2022
• Idioma: español
• DOI: 10.1007/s12346-022-00700-z
• Enlaces
  • Texto completo
• Resumen

  • This paper is concerned with a reaction–diffusion predator–prey system with a Beddington–DeAngelis functional response and subject to Neumann boundary conditions. The stability criteria and Turing instability conditions of the constant positive steady state of the system are provided. Some prior estimates, conditions to the nonexistence and the existence of non-constant positive steady-state solutions to the system are established and Turing patterns are also explored. The existence of spatially homogeneous and nonhomogeneous Hopf bifurcations of the constant positive equilibrium are discussed. Their achievements provide that the presence of the Beddington–DeAngelis functional response essentially increases the spatiotemporal complexity of the model and also leads to the appearance of Turing instability. Finally, some numerical tests are given to illustrate the theoretical findings.

• Referencias bibliográficas
  • 1. Beddington, J.R.: Mutual interference between parasites or predators and its effect on searching efficiency. J. Animal Ecol. 44, 340–412...
  • 2. Belabbas, M., Ouahab, A.F.: Rich dynamics in a stochastic predator-prey model with protection zone for the prey and multiplicative noise...
  • 3. Cantrell, R.S., Cosner, C.: On the dynamics of predator–prey models with Beddington–DeAngelis functional response. J. Math. Anal. Appl....
  • 4. Chen, W., Wang, M.: Qualitative analysis of predator-prey models with Beddington–DeAngelis functional response and diffusion. Math. Comput....
  • 5. Crandall, M.G., Rabinowitz, P.H.: The Hopf bifurcation theorem in infinite dimensions. Arch. Ration. Mech. Anal. 67, 53–72 (1977)
  • 6. DeAngelis, D.L., Goldstein, R.A., O’Neill, R.V.: A model for trophic interaction. Ecology 56, 881–892 (1975)
  • 7. Guo, G., Wu, J.: Multiplicity and uniqueness of positive solutions for a predator-prey model with B-D functional response. Nonlinear Anal....
  • 8. Guo, G., Wu, J.: The effect of mutual interference between predators on a predator–prey model with diffusion. J. Math. Anal. Appl. 389,...
  • 9. Hairston, N.G., Smith, F.E., Slobodkin, L.B.: Community structure, population control and competition. Am. Nat. 94, 421–425 (1960)
  • 10. Hassard, B.D., Kazarinoff, N.D., Wan, Y.-H.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981)
  • 11. Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math, vol. 840. Springer, Berlin, New York (1981)
  • 12. Holling, C.S.: The functional response of predators to prey density and its role in mimicry and population regulation. Mem. Entomol. Soc....
  • 13. Hsu, S.-B., Shi, J.: Relaxation oscillator profile of limit cycle in predator–prey system. Discrete Contin. Dyn. Syst. Ser. B 11, 893–911...
  • 14. Hwang, T.-W.: Global analysis of the predator-prey system with Beddington–DeAngelis functional response. J. Math. Anal. Appl. 281, 395–401...
  • 15. Hwang, T.-W.: Uniqueness of limit cycles of the predator-prey system with Beddington–DeAngelis functional response. J. Math. Anal. Appl....
  • 16. Kielhófer, H.: Bifurcation Theory: An Introduction with Applications to PDEs, Appl. Math. Sci., Vol. 156, Springer, New York (2004)
  • 17. Lin, C.S., Ni, W.M., Takagi, I.: Large amplitude stationary solutions to a chemotaxis systems. J. Differ. Equ. 72, 1–27 (1998)
  • 18. Lotka, A.J.: Elements of Physical Biology. Williams and Wilkins, New York (1925)
  • 19. Lou, Y., Ni, W.-M.: Diffusion, self-diffusion and cross-diffusion. J. Differ. Equ. 131, 79–131 (1996)
  • 20. Lou, Y., Ni, W.-M.: Diffusion vs cross-diffusion: an elliptic approach. J. Differ. Equ. 154, 157–190 (1999)
  • 21. Marsden, J.E., McCracken, M.: The Hopf Bifurcation and Its Applications. Springer, New York (1976)
  • 22. Medvinsky, A.B., Petrovskii, S.V., Tikhonova, I.A., Malchow, H., Li, B.-L.: Spatiotemporal complexity of plankton and fish dynamics. SIAM...
  • 23. Peng, R., Shi, J.-P., Wang, M.-X.: Stationary pattern of a ratio-dependent food chain model with diffusion. SIAM J. Appl. Math. 65, 1479–1503...
  • 24. Peng, R., Shi, J.-P., Wang, M.-X.: On stationary patterns of a reaction–diffusion model with autocatalysis and saturation law. Nonlinearity...
  • 25. Peng, R., Yi, F.-Q., Zhao, X.-Q.: Spatiotemporal patterns in a reaction-diffusion model with the Degn–Harrison scheme. J. Differ. Equ....
  • 26. Pao, C.-V.: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992)
  • 27. Rosenzweig, M.L.: Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science 171, 385–387 (1971)
  • 28. Shi, J.-P., Wang, X.-F.: On the global bifurcation for quasilinear elliptic systems on bounded domains. J. Differ. Equ. 246, 2788–2812...
  • 29. Souna, F., Belabbas, M., Menacer, Y.: Complex pattern formations induced by the presence of crossdiffusion in a generalized predator–prey...
  • 30. Souna, F., Lakmeche, A.: Spatiotemporal patterns in a diffusive predator-prey system with Leslie–Gower term and social behavior for the...
  • 31. Souna, F., Lakmeche, A., Djilali, S.: Spatiotemporal patterns in a diffusive predator–prey model with protection zone and predator harvesting....
  • 32. Souna, F., Lakmeche, A., Djilali, S.: Spatiotemporal behavior in a predator-prey model with herd behavior and cross-diffusion and fear...
  • 33. Souna, F., Djilali, S., Charif, F.: Mathematical analysis of a diffusive predator-prey model with herd behavior and prey escaping. Math....
  • 34. Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. Ser. B 237, 37–72 (1952)
  • 35. Volterra, V.: Variazionie fluttuazioni del numero d’individui in specie animali conviventi. Mem. Acad. Licei. 2, 31–113 (1926)
  • 36. Wang, M.-X.: Non-constant positive steady states of the Selkov model. J. Differ. Equ. 190, 600–620 (2003)
  • 37. Wang, J., Shi, J., Wei, J.: Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey. J. Differ....
  • 38. Yan, X., Zhang, C.: Stability and Turing instability in a diffusive predator–prey system with Beddington–DeAngelis functional response....
  • 39. Ye, Q.-X., Li, Z.-Y., Wang, M.-X., Wu, Y.-P.: Introduction to Reaction–Diffusion Equations, 2nd edn.Science Press, Beijing (2011).. ((in...
  • 40. Yi, F., Wei, J., Shi, J.: Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator–prey system. J. Differ. Equ. 246,...