Analytical properties and exact solutions of the Lotka–Volterra competition system

Abstract

The Lotka-Volterra competition system with diffusion is considered. The Painlevé property of this system is investigated. Exact traveling wave solutions of the Lotka-Volterra competition system are found. Periodic solutions expressed in terms of the Weierstrass elliptic function are also given.

New Conditional Symmetries and Exact Solutions of the Diffusive Two-Component Lotka–Volterra System

“…It can be seen that Table 1 does not contain such a type of systems; therefore, the relevant exact solutions cannot be found. Moreover, we noted that all the exact solutions derived in other papers [9][10][11][12][13][14]17,18] are not applicable for description of the prey-predator interaction as well. Thus, the problem of constructing exact solutions for the DLV system (1) modeling the interaction between the prey and predator is a hot topic.…”

“…It should be pointed out that traveling wave solutions, which are widely studied for any nonlinear model and play an important role in qualitative analysis, usually cannot be used for solving the relevant models involving the zero flux boundary conditions in the bounded domains. Let us consider the traveling wave of the DLV system (32) with d 1 = d 2 = 1, which was firstly constructed in [10] (see also Section 3.2.3 in [31]) and much later rediscovered in [12] (see formulae (18) and (24) therein)…”

“…However, the number of the papers devoted to construction of exact solutions of the nonlinear system (1) is relatively small. Exact solutions in the form of traveling waves were constructed in [9][10][11][12]. In the case of the three-component DLV system, some traveling waves were found in [13,14].…”

Dynamic analysis on an almost periodic predator-prey system with impulsive effects and time delays

“…The investigation of the nonlinear dynamics about biomathematics model is one of the most active and important subjects in mathematics, ecology and biological world. They described many of the relationships between two or more species, such as competitive mechanism [21,30], predator-prey mechanism [26,47], mutualism [40] and so on. To investigate the predator-prey effects between two or more natural species, several biological models have been introduced and investigated by many ecologists, biologists and mathematicians [16,27,29,36,37,45].…”

On Solutions and Heteroclinic Orbits of Some Lotka-Volterra Systems

“…Although several methods for obtaining invariants for the system of nonlinear ordinary differential equations(ODEs) like LV system [10][11][12][13][14][15][16][17] are now available in the literature but getting their solutions in closed form (in terms of elementary known function) even for a simpler system is not an easy task. F. GonzalezGascon [11] developed time independent first integrals of LV systems based on the computation of generalized symmetry vectors of the vector field associated with the systems.…”