On Distributed Predator-Prey System with Memories
DOI:
https://doi.org/10.5890/DNC.2022.09.003Abstract
In the present paper, we consider a class of reaction-diffusion systems based on the Lotka-Volterra differential equation model of a predator-prey interaction with the existence of memory terms. We show that every solution with initial values in $[0,l]$ and subject to homogeneous Neumann boundary conditions decays to a spatially homogeneous function of time.References
[1] Fisher, R.A. (1937), The wave of advance of advantageous genes, Ann. Eugenics, 7, 355-369.
[2] Kolmogorov, A., Petrovsky, I., and Piscounov, N. (1937), Etude de l'equation de la diffusion avec croissance de la quantite de matiere et son application a un pobleme biologique, Moscow Univ. Math. Bull., 1, 1-25.
[3] Dubois, D.M. (1975), A model of patchiness for pre-predator plankton population, Ecol. Modelling, 1, 67-80.
[4] Dunbar, S. (1981), Traveling wave solutions of diffusive Volterra-Lotka interactions equations. Univ. of Minnesota, Phd, thesis.
[5] Dunbar, S. (1983), Travelling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biolog, 17, 11-32.
[6] Dunbar, S. (1984), Travelling wave solutions of diffusive Lotka-Volterra equations. A geteroclinic connection in $R^4$, Transactions of American Math. Soci., 286, 557-594.
[7] Lotka, A.J. (1925), Elements of Physical Biology. Williams and Wilkins Company.
[8] Volterra, V. (1926), Fluctuations in the abundance of a species considered mathematically, Nature, 118, 558-560.
[9] Yamada, Y. (1988), On a certain class of semilinear Volterra diffusion equations, J. Math. Anal. Appl., 88, 443-457.
[10] Braik, A., Miloudi, Y., and Zennir, Kh. (2018), A finite-time blow-up result for a class of solutions with positive initial energy for coupled system of heat equations with memories, Math. Meth. Appl. Sci., 41(4), 1674-1682.
[11] Zennir, Kh. and Miyasita, T. (2020), Lifespan of solutions for a class of pseudo-parabolic equation with weak-memory, Alexandria Enginee. J., 59, 957-964.
[12] Zennir, Kh. and Feng, B. (2018), One spatial variable thermoelastic transmission problem in viscoelasticity located in the second part, Math. Meth. Appl. Sci., 41(16), 6895-6906.
[13] Bojadziev, G.N. (1981), Damped oscillating processes in biological and biochemical systems, Bulletin of Mathematical Biology, 42-5, 701-718.
[14] Volpert, A., Volpert, V., and Volpert, V. (1994), Travelling Wave Solutions of Parabolic Systems, Providence, AMS.
[15] Conway, E., Hoff, D., and Smoller, J.A. (1978), Large time behavior of systems of nonlinear diffusion equations, SIAM J. Appl. Math., 35, 1-16.
[16] Al Noufaey, K.S., Marchant, T.R., and Edwards, M.P. (2015), The diffusive Lotka-Volterra predator-preysystem with delay, Mathematical Biosciences, 270, 30-40.
[17] Pao, C.V. (2003), Global asymptotic stability of Lotka--Volterra 3-species reaction--diffusion systems with time delaysy, J. Math. Anal. Appl., 281, 186-204.
[18] Gourley, S.A. and Britton, N.F. (1996), A predator-prey reaction-diffusion system with nonlocal effects, J. Math. Biol., 34, 297-333.
[19] Chen, S., Zhang, J., and Young, T. (2003), Existence of positive periodic solution for nonautonomous predator-prey system with diffusion and time delay, J. Comput. Appl. Math., 159, 375-386.
[20] Yan, X.P. and Chu, Y.D. (2006), Stability and bifurcation analysis for a delayed Lotka-Volterra predator-prey system, J. Comput. Appl. Math., 196, 198-210.
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