Higher order solitary solutions to the meta-model of diffusively coupled Lotka–Volterra systems

A meta-model of diffusively coupled Lotka–Volterra systems used to model various biomedical phenomena is considered in this paper. Necessary and sufficient conditions for the existence of nth order solitary solutions are derived via a modified inverse balancing technique. It is shown that as the highest possible solitary solution order n is increased, the number of nonzero solution parameter values remains constant for solitary solutions of order n>3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n>3$\end{document}. Analytical and computational experiments are used to illustrate the obtained results.


Introduction
Even though solitons (also called solitary solutions) have been first discovered in the 19th century by John Scott Russell [32], later formalized by Korteweg and de Vries [14] and made famous in the mid-20th century by the Fermi-Pasta-Ulam computational experiment [12] and later works by Norman and Zabusky [41], they remain at the forefront of research to this day due to their unique physical properties.
In recent years, the emergence of powerful computer algebra software coupled with a marked rise in computing power has sparked a new interest in the subject. Analytical investigation is central to the construction of solitary solutions and the availability of aforementioned tools has greatly increased the number of studies in this area.
Classical methods used to construct solitary solutions to differential equations include the inverse scattering transform [2], the Bäcklund transform [31] and the Darboux transform [30] methods.
More recently developed techniques that make use of computer algebra software include the simplest equation method and its extensions [15,17], the equivalent (G /G)-extension and tanh-extension methods [16,40], and the homotopy analysis method [1]. A novel adaptation of the (G /G)-expansion technique is used to construct solitary wave solutions to the (2 + 1) dimensional Konopelchenko-Dubrovsky and Kadomtsev-Petviashvili equations in [5]. The (G /G)-expansion method is further adapted and applied to obtain solitary wave solutions to the (2 + 1)-dimensional time-fractional Schrödinger equation and the space-time nonlinear conformable fractional Bogoyavlenskii equations in [4]. A technique based on Lyapunov's second method is used to conduct an investigation of integrability of Volterra integro-differential systems in [37].
The determination of solitary solutions to differential equations is an important question in applied research. Some recent examples are given below. Soliton crystals have been observed and characterized in monolothic Kerr resonators, which offers a novel way to increase the efficiency of Kerr combs [9]. A generalized hydrodynamics theory based on soliton solutions is developed in [10] and is applied to the Lieb-Liniger model realized in cold-atom experiments. Solitons have been observed in the cell movement of a cellular slime mould in [13]. Biological population models have been shown to possess solitary solutions on multiple occasions, including [3,29].
The motivation for this study and its contributions to the theory of solitary solutions is given in the next section.

The diffusive and the multiplicative coupling
Mathematical modeling of interacting dynamical systems is a classical field of research. For example, a diffusive coupling between two (or more) dynamical systems is used to model the effect of synchronization. The paradigmatic model of two diffusively coupled dynamics is described by: where F is the vector field modeling the isolated chaotic dynamics; t is time; γ is the diffusive coupling parameter (usually set a positive constant). The systems are said to be completely synchronized when there is a set of initial conditions so that the systems eventually evolve identically in time and the divergence of trajectories of interacting systems is suppressed by the diffusive coupling [27]. However, dynamical systems can be coupled not only with terms representing the diffusive coupling. Another type of the coupling is the multiplicative coupling. A paradigmatic example of such type of coupling is the Lotka-Volterra model: where α, β, γ , δ are positive real constants. Complete synchronization is not possible in the Lotka-Volterra model (except for two trivial equilibriums when both competing species die out or coexist at fixed population levels). The classical Lotka-Volterra model (3)-(4) reduces to a system of linear uncoupled ordinary differential equations when the coupling constants β and δ vanish to zero. A more complex variant of the Lotka-Volterra model is the competitive Lotka-Volterra model which is based on the nonlinear logistic equation [20,38] instead, namely where r is inherent per-capita growth rate, and K is the carrying capacity. The competitive Lotka-Volterra model reads [6]: where the parameter β xy represents the effect species y has on the population of species x, and β yx represents the effect species x has on the population of species y. Both parameters β xy and β yx are usually set as positive constants due to harmful (competitive) interaction between species. Indeed, the dynamics of the competitive Lotka-Volterra model is more complex compared to the classical Lotka-Volterra model [7]. The competitive Lotka-Volterra model reduces to two uncoupled nonlinear logistic equations when the coupling constants β yx and β xy vanish. Note that the nonlinear logistic equation is a partial case of the paradigmatic Riccati equation with constant coefficients [28]: where a 0 , a 1 , a 2 ∈ R; a 2 = 0. The analogy between the Lotka-Volterra model (3)-(4) and the competitive Lotka-Volterra model (6)-(7) suggests the following model with the multiplicative coupling: where a 2x , b 2y = 0. The system (9)-(10) reduces to two uncoupled Riccati equations with constant coefficients when the coupling coefficients β xy and β yx vanish. It appears that such models are widely used to describe the interaction between healthy and cancer cells in phenomenological mathematical models of a single cancer tumor [19]. Such models comprising two Riccati-type equations coupled with multiplicative terms are used for the description of prostate cancer treatment with androgen deprivation therapy [42], cancer stem-cell-targeted immunotherapy [35], the maximization of viability time in general cancer therapy [8]. Elliptic and hyperbolic problems that stem from mechanical models also involve Riccati-type equations, as given in the following examples. A Riccati-type hyperbolic boundary value problem is considered in [18]. An elliptic problem related to membrane equilibrium equations is studied in [39].

The meta-model of coupled prey-predator systems
The mathematical meta-model of diffusively coupled Lotka-Volterra systems on heterogenous graphs in presented in [21]. When the number of systems is limited to two, the model of diffusively coupled predator-prey systems reads [21]: The system of diffusively coupled Lotka-Volterra models (11)-(14) reduces into a system of linear coupled differential equations when the multiplicative coupling constants λ 1 , μ 1 , λ 2 , μ 2 vanish. A natural extension of (11)- (14) is based on the expansion of the basic Lotka-Volterra model by the nonlinear terms (in accordance to (6)- (7)): System (15)-(18) does represent a meta-model of two diffusively coupled Riccati systems (each system comprises two Riccati equations coupled with multiplicative terms). System (15)-(18) splits into two uncoupled systems described by (9)-(10) when the diffusive coupling constants γ 1 and γ 2 vanish. Analogously, system (15)-(18) splits into four uncoupled Riccati equations (8) when both the diffusive and the multiplicative coupling constants vanish. In other words, the model described by (15)-(18) generalizes the competitive Lotka-Volterra model in the spatial domain.

The motivation of this paper
The existence of the first-order soliton-type solutions (kink solitons) to Riccati equation (8) is known for decades [28]. Necessary and sufficient conditions for the existence of second-order soliton-type solutions (dark/bright solitons) to system (9)-(10) has been recently reported in [23]. The existence of nth order soliton-type solutions to the metamodel of coupled Riccati equations (15)-(18) poses a serious challenge from the mathematical point of view. Providing two answers to the following questions -what is the maximal order n, and what are the necessary and sufficient conditions for the existence of solitons up to the nth order -is the main objective of this paper.

Definition of the solitary solution
Solitary solutions of the following form [22,33] are considered in this paper: where n ∈ N is the order of the solitary solution, t 0 , σ , η ∈ R, x k , t k ∈ C.
The following independent variable transformation is introduced: Using (20) on (19) simplifies the analytical expression of the solitary solution (19) as follows: where

Uncoupled Riccati equations
Consider the following Riccati equation with respect to x = x(t): where c 0 , c 1 , c 2 ∈ C.
Equation (23) can be transformed via the substitution (20), where η 2 = c 2 1 -4c 0 c 2 [28], as follows: The solution to (24) reads [28]: where s ∈ R is a free constant, ae 0 = st 0 and parameters σ , x 0 , t 0 satisfy the following identities: Thus the solution to (23) is This solution is known as the kink solitary solution [34]. It describes the transition of a system from one steady state to another via a monotonous trajectory. Example Suppose that the following Ricatti differential equation with respect to x = x(t) is given: The kink solitary solution to (30) is displayed in Fig. 1.

System of Riccati equations coupled via multiplicative terms
Let us consider the following system of Riccati equations coupled via multiplicative terms: It is shown in [26] that equations (31)-(32) can be uncoupled under the assumption that solutions x, y are in an inverse relationship: In this case, the solutions have the same kink solitary solution form as described in the previous subsection. Furthermore, it has been shown in [23] that if the condition (33) does not hold, the system (31)-(32) admits the following dark/bright solitary solutions: where σ , γ , η, t 0 are constants and parameters t 1 , t 2 , x 1 , x 2 , y 1 , y 2 are functions of initial conditions posed for system (31)-(32) at the point t = t 0 .
It is proven in [23] that the above solution holds if and only if the solution parameters satisfy the relations: = - The system parameters a k , b k , k = 0, . . . , 3 must satisfy the following conditions: Furthermore, the parameter η is Example Suppose that the following system of Riccati equations with respect to x = x(t) and y = y(t) is given: The dark/bright solitary solutions to (40)-(41) are illustrated in Fig. 2.
It will be demonstrated in this paper that the system (15)-(18) also has both kink and dark/bright solitary solutions, as well as admits higher-order solitary solutions.

Inverse balancing technique
The inverse balancing technique [24] is used in order to obtain necessary and sufficient conditions for the existence of solitary solutions to a system of differential equations, as well as to determine the possible order of such solutions. The main idea of this technique is to insert the solitary solution as an anzatz into the considered model, which yields a system of linear equations from which the model parameters can be determined in terms of solitary solution parameters. The inverse balancing technique is applied to the system (15)- (18) in Sects. 4 and 5.
Note that a direct balancing approach consisting of inserting the solutions of the form (19) into the system (15)- (18) would result in a high-order nonlinear system of algebraic equations with respect to the solution parameters. Direct construction of a solution to this system would not be possible. Due to this, the inverse balancing technique is applied for the analysis of the system (15)-(18).
Note that systems (81) and (84) are identical. When (46) are kink solitary solutions (n = 1), systems (81) and (84) have infinitely many solutions of the following form: where l = 1, 2. Note that in this case solutions L kl and N kl (k = 1, 2, 3) are not necessarily equal. However, when n ≥ 2, systems (81) and (84) can only have a single unique solution: if and only if the following conditions with respect to solitary solution (46) parameters hold true: where l = 1, 2.
The system (91) yields the expressions for K 1l , K 2l : and the following conditions: where l, r = 1, 2, l = r. Note that applying (80) and (82) to (46) yields the following expression of the solitary solution: x l (t) = x l ( t) = σ 1l X l T = σ 1l ( t) n + p (n-1)l ae n-1 ( t) n-1 + · · · + p 0l ae 0 ( t) n + ae n-1 ( t) n-1 + · · · + ae 0 , ( t) n + h (n-1)l ae n-1 ( t) n-1 + · · · + h 0l ae 0 ( t) n + ae n-1 ( t) n-1 + · · · + ae 0 , for l = 1, 2. Algebraically solving the system of necessary and sufficient conditions defined in Lemma 2 for various values of n yields the conclusion, summarized in the lemma below. The auxiliary parameters A kl , L ml , N ml , and K ml form an essential link between the parameters of the system (15)- (18) and the solitary solution (46). If it is possible to determine the auxiliary parameters from the solitary solution, the system parameters can be computed via (77) and (78). Conversely, if the system parameters are known they can be used to determine auxiliary parameters, which in turn yield the solitary solution parameters via (69)-(76).

Analytical computation of auxiliary parameters from solitary solution parameters
In this subsection, the presented derivations are illustrated: starting from the given thirdorder solitary solutions (n = 3), the auxiliary parameters are first derived. From the auxiliary parameters, the coefficients of the system (15)- (18) are then computed.
The following example demonstrates these steps in detail, closely following the derivations presented in Sects. 4 and 5. See Sect. 6.2 for a more application-oriented example, where solitary solutions to a given system of diffusively coupled Lotka-Volterra equations are constructed.