Keywords
iterative method
predator-prey system
How to Cite
Abstract
In this paper we prove the existence and uniqueness of weak solutions for a kind of Lotka–Volterra system, by using successive linearization techniques. This approach has the advantage to treat two equations separately in each iteration step. Under suitable initial conditions, we construct an invariant region to show the global existence in time of solutions for the system. By means of Sobolev embeddings and regularity results, we find estimates for predator and prey populations in adequate norms. In order to demonstrate the convergence properties of the introduced method, several numerical examples are given.
Murray JD. Mathematical Biology. Springer-Verlag, Berlin, 1993.
Turchin P. Complex Population Dynamics. Princeton Univ. Press, Princeton, NJ. 2003.
Dunbar S. Traveling wave solutions of diffusive Lotka - Volterra equations, Journal of Mathematical Biology, 1: 11-32, 1983.
doi: 10.1007/BF00276112
Xu Z, Weng P. Traveling waves in a diffusive predator - prey model with general functional response, Electronic Journal of Differential Equations, 197: 1-13, 2012.
Tang De, Ma Li. Existence and uniqueness of a Lotka - Volterra reaction - diffusion model with advection term, Applied Mathematics Letters, 86: 83-88, 2018.
doi: 10.1016/j.aml.2018.06.015
Khan H, Li Y, Khan A, Khan A. Existence of solution for a fractional-order Lotka - Volterra reaction - diffusion model with Mittag - Leffler kernel, Mathematical Methods in the Applied Sciences, 42(9): 1-11, 2019.
doi: 10.1002/mma.5590
Pao CV. Dynamics of Lotka - Volterra competition reaction-diffusion systems with degenerate diffusion, Journal of Mathematical Analysis and Applications, 421(2): 1721-1742, 2015.
doi: 10.1016/j.jmaa.2014.07.070
Guo X, Wang J. Dynamics and pattern formations in diffusive predator - prey models with two prey - taxis, Mathematical Methods in the Applied Sciences, 42(12): 4197-4212, 2019.
doi: 10.1002/mma.5639
Roubicek, T. Nonlinear Partial Differential Equations with Applications. Birkhauser Verlag, Basel, Switzerland, 2005.
Evans LC. Partial differential equations, American Mathematical Society, Providence, RI, 2010.
Smoller J. Shock Waves and Reaction-Diffusion Equations. Springer-Verlag, Berlin, 1994.
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