TRAVELLING WAVE SOLUTIONS FOR A DIFFUSIVE
PREY-PREDATOR MODEL WITH ONE
PREDATOR AND TWO PREYS
N.B. Sharmila, C. Gunasundari
Department of Mathematics
SRM Institute of Science and Technology
Kattankulathur-603203, INDIA

Abstract

The paper discusses about a three-species diffusive prey predator system, in which the predator is a generalist that can thrive on two distinct prey species. The occurrence of invasion waves in the model is investigated in this paper. Schauder's fixed point theorem proves the existence of travelling semi fronts $\omega>\omega^{*}$, where $\omega$ is the minimal wave speed. Harnack's inequality for positive super solutions on $\mathbb{R}$ is established to solve the reducibility problem that arises in the proofs. The boundedness is demonstrated and LaSalle's invariance principle proves that such waves connect coexistence equilibrium. The rescaling method and limit arguments are used to determine the existence of a travelling front with $\omega=\omega^{*}$. The Laplace transform demonstrates the nonexistence of moving fronts with speed $0<\omega<\omega*$.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 35
Issue: 5
Year: 2022

DOI: 10.12732/ijam.v35i5.3

Download Section



Download the full text of article from here.

You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.

References

  1. [1] A. J. Lotka, Elements of Physical Biology, Williams and Wilkins Company, Baltimore (1925).
  2. [2] M. Senthilkumaran, C. Gunasundari, Stability and Hopf bifurcation in a delayed predator-prey system with parental care for predators, J. Math. Comput. Sci., 7, No 3 (2017), 495-521.
  3. [3] M. Senthilkumaran, C. Gunasundari, Dynamics of delayed prey-predator model with parental care for predators, Intern. J. of Comput. and Appl. Math., 12, No 1 (2017), 3193-3203.
  4. [4] A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. B Biol. Sci., 237 (1952), 37-72.
  5. [5] S. R. Dunbar, Travelling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.
  6. [6] S. R. Dunbar, Traveling waves in diffusive predator-prey equations: periodic orbits and point-to-periodic heteroclinic orbits, SIAM J. Appl. Math., 46 (1986), 1057-1078.TRAVELLING WAVE SOLUTIONS FOR A DIFFUSIVE... 683
  7. [7] R. Gardner, Existence of traveling wave solutions of predator-prey systems via the connection index, SIAM J. Appl. Math., 44 (1984), 56-79.
  8. [8] X. Liang, X. Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.
  9. [9] J. H. Wu, X. F. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687.
  10. [10] S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: a heteroclinic connection in R4, Trans. Amer. Math. Soc., 286 (1984), 557-594.
  11. [11] W. Z. Huang, Traveling wave solutions for a class of predator-prey systems, J. Dynam. Differential Equations, 24 (2012), 633-644.
  12. [12] G. Lin, Invasion traveling wave solutions of a predator-prey system, Nonlinear Anal., 96 (2014), 47-58.
  13. [13] W. T. Li, G. Lin, S. G. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1273.
  14. [14] S. W. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314.
  15. [15] C. F. Wu, D. M. Xiao, Travelling wave solutions in a non-local and time-delayed reaction-diffusion model, IMA J. Appl. Math., 78 (2013), 1290-1317.
  16. [16] S. Fu and J. Tsai, Wave propagation in predator-prey systems, Nonlinearity, 28 (2015), 4389-4423.
  17. [17] T. Zhang, W. Wang and K. Wang, Minimal wave speed for a class of non-cooperative diffusion-reaction system, J. Differential Equations, 260 (2016), 2763-2791.
  18. [18] P. Hartman, Ordinary Differential Equations, 2nd Ed., Birkhauser, Boston (1982).
  19. [19] T. Zhang and Y. Jin, Traveling waves for a reaction-diffusion-advection predator-prey model, Nonlinear Anal. Real World Appl., 36 (2017), 203-232.684 N.B. Sharmila, C. Gunasundari
  20. [20] J. P. LaSalle, The Stability of Dynamical Systems, Society for Industrial and Applied Mathematics Philadelphia (1976).
  21. [21] A. Arapostathis, M. K. Ghosh and S. I. Marcus, Harnack’s inequality for cooperative weakly coupled elliptic systems, Comm. Partial Differential Equations, 24 (1999), 1555-1571.