Extension of the Lotka-Volterra competition model

Year 2024, , 1401 - 1407, 15.10.2024

S.h. Rasouli

https://doi.org/10.15672/hujms.1315963

Abstract

In this paper, we introduce the ($p,q$)-Lotka-Volterra competition model which is extension of classical Lotka-Volterra competition model. The main purpose is to give some results on the existence and non-existence of positive solutions. Upper and lower solutions technique and comparison arguments plays a significant role in our main proof.

Keywords

Lotka-Volterra competition model, (p,q)-Laplacian, upper and lower solutions

References

• [1] J. Ali and R. Shivaji, Positive solutions for a class of p-Laplacian systems with multiple parameters, J. Math. Anal. Appl. 335, 1013-1019, 2007.
• [2] M.O. Alves, M.T.O. Pimenta and A. Suuárez, Lotka-Volterra models with fractional diffusion, Proc. Royal. Soc. Edin. 147A, 505-528, 2017.
• [3] R. Aris, Mathematical Modelling Techniques, Research Notes in Mathematics, Pitman, London, 1978.
• [4] G. Astrita and G. Marrucci, Principles of non-Newtonian fluid mechanics, McGraw- Hill, 1974.
• [5] L. Baldelli, Y. Brizi and R. Filippucci, Multiplicity results for (p, q)-Laplacian equations with critical exponent in $\mathbb{R}^{N}$ and negative energy, Calc. Var. 60 (8), 1-30, 2021.
• [6] V. Benci, D. Fortunato and L. Pisani, Soliton like solutions of a Lorentz invariant equation in dimension 3, Rev. Math. Phys. 10 (3), 315-344, 1998.
• [7] M. Bruschi and F. Calogero, Simple extensions of the Lotka-Volterra prey-predator model, The Mathematical Intelligencer 40, 16-19, 2018.
• [8] S. Carl, V.K. Le and D. Motreanu, Nonsmooth variational problems and their inequalities, Comparaison principles and applications, Springer, New York, 2007.
• [9] W. Cintra, M. Molina-Becerra and A. Suárez, The Lotka-Volterra models with nonlocal reaction terms, Communs. Pure. Appl. Anal. 21, 3865-3886, 2022.
• [10] C. Cosner and A.C. Lazer, Stable coexistence states in the Volterra-Lotka competition model with fiffusion, Siam. J. Appl. Math. 44, 1112-1132, 1984.
• [11] E.N. Dancer, On the existence and uniqueness of positive solutions for competing species models with diffusion, Trans. Am. Math. Soc. 326, 829859, 1991.
• [12] P.C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics 28, Springer Verlag, Berlin-New York, 1979.
• [13] J. López-Gómez and R. Pardo, Coexistence regions in Lotka-Volterra models with diffusion, Nonlinear Anal. 19, 1128, 1992.
• [14] R. Guefaifia, J. Zuo, S. Boulaaras and P. Agarwal, Existence and multiplicity of positive weak solutions for a new class of (p, q)-Laplacian systems, Miskolc Math. Notes 21, 861-872, 2020.
• [15] D.D. Hai and R. Shivaji, An existence result on positive solutions for a class of p- Laplacian systems, Nonl. Anal. 56, 1007-1010, 2004.
• [16] Sze-Bi Hsu and Xiao-Qiang Zhao, A Lotka-Volterra competition model with seasonal succession, J. Math. Biol. 64, 109-130, 2012.
• [17] S.A. Khafagy, Existence results for weighted (p, q)-Laplacian nonlinear system, Appl. Math. E-Notes 17, 242-250, 2017.
• [18] E.K. Lee, R. Shivaji and J. Ye, Positive solutions for infinite semipositone problems with falling zeros, Nonl. Anal. 72, 4475-4479, 2010.
• [19] L. Ma and S. Guo, Bifurcation and stability of a two-species diffusive lotka-volterra model, Commun. Pure. Appl. Annal. 19, 1205-1232, 2020.
• [20] A. Muhammadhaji, A. Halik and Hong-Li Li, Dynamics in a ratio-dependent Lotka- Volterra competitive-competitive-cooperative system with feedback controls and delays, Adv. Diffrence. Eqs. 230, 1-14, 2021.
• [21] E. Diz-Pita and M.V. Otero-Espinar, Predator-Prey Models: A Review of Some Recent Advances, Mathematics. 9, 1-34, 2021.
• [22] S.H .Rasouli, Existence of solutions for singular (p, q)-Kirchhoff type systems with multiple parameters, Elect. J. Diff. Eqs 69, 1-8, 2016.
• [23] S.H .Rasouli, Z. Halimi and Z.Mashhadban, A remark on the existence of positive weak solution for a class of (p, q)-Laplacian nonlinear system with sign-changing weight, Nonl. Anal. 73, 385-389, 2010.
• [24] M. Struwe, Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, Heidelberg, New York, 1996.
• [25] L. Wang and K. Li, On positive solutions of the Lotka-Volterra cooperating models with diffusion, Nonlinear Analysis. 53, 1115-1125, 2003.
• [26] Z. Zhu, R. Wu, L. Lai and X. Yu, The influence of fear effect to the Lotka-Volterra predator-prey system with predator has other food resource, Adv. Diffrence. Eqs. 237, 1-14, 2020.