The integration of three-dimensional Lotka–Volterra systems

Abstract

The general solutions of many three-dimensional Lotka-Volterra systems, previously known to be at least partially integrable, are constructed with the aid of special functions. Examples include certain ABC and May-Leonard systems. The special functions used are elliptic and incomplete beta functions. In some cases, the solution is parametric, with the independent and dependent variables expressed as functions of a 'new time' variable. This auxiliary variable satisfies a nonlinear third-order differential equation of a generalized Schwarzian type, and results of Carton-LeBrun on the equations of this type that have the Painlevé property are exploited, so as to produce solutions in closed form. For several especially difficult Lotka-Volterra systems, the solutions are expressed in terms of Painlevé transcendents. An appendix on incomplete beta functions and closed-form expressions for their inverses is included.

An N-barrier maximum principle for elliptic systems arising from the study of traveling waves in reaction-diffusion systems

“…For the case of strong competition [S], (32) is proved, whereas for the case of weak competition [W], we obtain (39). Combining the two inequalities (32) and (39) yields the lower bound of q(x) given by (30). This completes the proof of (I).…”

“…First of all, we prove (I) for the case of strong competition [S]: σ1 c11 > σ2 c21 and σ2 c22 > σ1 c12 . Clearly, (30) in this case gives…”

“…As a suggestive example of species diversity, we investigate the situation where one exotic competing species (say, W ) invades the ecological system of two native 1504 CHIUN-CHUAN CHEN AND LI-CHANG HUNG species (say, U and V) that are competing in the absence of W . Then a problem related to competitive exclusion ( [2], [20], [21], [23], [31], [37]) or competitor-mediated coexistence ( [5], [26], [32]) arises, and a mathematical model for this situation can be provided by the diffusive Lotka-Volterra system of three competing species ( [1], [13], [15], [16], [17], [28], [30], [32], [33], [38], [41]) (1) where u(y, t), v(y, t) and w(y, t) denote the population densities of U , V and W at time t and position y. The parameters d i , σ i , c ii (i = 1, 2, 3), and c ij (i, j = 1, 2, 3 with i = j), which are all positive constants, stand for the diffusion rates, intrinsic growth rates, intra-specific competition rates, and inter-specific competition rates, respectively.…”

The balance simplex in non-competitive 2-species scaled Lotka–Volterra systems

“…For a Lotka-Volterra system with 2-species, the phase portraits are well known [7,16,19]. However, explicit solutions are rare, whether for actual solutions [20,28], or for invariant manifolds [6,31]. Here we obtain an explicit and analytic solution for the heteroclinic orbits that connect non-zero steady states in a scaled Lotka-Volterra system (see (1) below).…”

On Solutions and Heteroclinic Orbits of Some Lotka-Volterra Systems

“…It is known that when one of the coupling parameters , , equals unity and growth or decay rate are equal ( 1 = 2 = 3 ), the system (1) is completely integrable in the sense that a pair of first integrals exists [18]. Some general solution of this system of ODEs was published by Maier [20] but not as an explicit dependence on . Consequently, getting the analytic expression for heteroclinic orbit between two fixed points is not inevitable.…”