Optimal Parametric Iteration Method for Solving Multispecies Lotka-Volterra Equations

Abstract

We apply an analytical method called the Optimal Parametric Iteration Method (OPIM) to multispecies Lotka-Volterra equations. By using initial values, accurate explicit analytic solutions have been derived. The method does not depend upon small parameters and provides us with a convenient way to optimally control the convergence of the approximate solutions. An excellent agreement has been demonstrated between the obtained solutions and the numerical ones. This new approach, which can be easily applied to other strongly nonlinear problems, is very effective and yields very accurate results.

1. Introduction

A substantial amount of research work has been invested in the study of nonlinear systems of differential equations. Systems of nonlinear differential equations arise in many scientific models such as biological systems and are used in various fields as engineering, chemistry, and ecology. In 1925, Lotka [1] developed the motion of an evolutionary system based on two fundamental changes, those involving matter between components of a system and those involving exchanges of energy [2]. Unlike being grounded in chemistry, Lotka believed that these ideas could be applied to any biological system. In 1926, Volterra [3] developed the well-known mathematical models of multispecies interaction. These models, the predator, prey and competition models are known today as Lotka-Volterra models. The environment of the species can be influenced by the effect of food availability, weather conditions, temperature, mating habits, contact with predators, and other resource or physical environmental quantities. Lotka-Volterra equations describe variations of population densities of few species that compete for the same resources. However, the ecological system is often affected by environmental changes and other human activities. In many practical situations, it is often the case that one of the species maybe suffers a significant loss or increase in density for some reason at some transitory time slots. These models, for instance, they can describe the competing fish species which are exploited by human activities, can also describe the dynamics of normal and tumour cells in a changing environment under the effects of the chemotherapy. These models are also applicable in case we are interested in the existence and stability of tumor-free solution and how treatment affects the interaction of tumour and normal cells. Although simplistic, these few models are still used as the foundation for mathematical models in biology [4]. These models can also describe the time history of a biological system and are used in various fields as engineering, chemistry, biology, or mathematics [5]. In fact the Lotka-Volterra model is one of the most popular ones to demonstrate a simple nonlinear control system. The accurate solutions of the Lotka-Volterra equations may become a difficult task either if the equations are stiff or when the number of species is large.

Nonlinear analytical techniques for solving nonlinear problems have been dominated by the perturbation methods, which have found wide applications in engineering [6]. But perturbation methods have their limitations: perturbation techniques are based on the existence of a small parameter. A majority of nonlinear problems have no small parameters at all. So, it is necessary to develop a kind of new nonlinear analytical methods which does not require small parameters at all.

There exist same alternative methods, such as the variational iteration method [7], the Adomian decomposition method [8], a modified Lindstedt-Poincare method [9], the homotopy analysis method [10], the homotopy perturbation method [11], and the optimal homotopy asymptotic method [12].

In recent years, a growing interest towards the application of iterative techniques in nonlinear problems has appeared in science and engineering. In 1987 Mickens [13] proposed an iterative scheme for nonlinear problems. In 2002 Lim and Wu [14], in 2006 Hu [15], and in 2008 Chen and Liu [16] proposed modified iteration procedures. Also He [17] proposed some iterative and asymptotic methods for nonlinear problems and later, Marinca and Herişanu [18] proposed in 2006 a new iteration method by combining Mickens’ and He’s iteration methods. Ramos [19] investigated other iterative techniques for nonlinear differential equations. These methods are valid not only for a small parameter, but also for very large parameters and have been used to solve nonlinear both conservative and nonconservative oscillators.

In this paper we propose a new approach to find analytic approximate solution of multispecies Lotka-Volterra models using a new iterative procedure, namely, the Optimal Parametric Iteration Method (OPIM). The efficiency of the proposed procedure is proved since an accurate solution is explicitly analytically achieved in an iterative manner after only one iteration. This new approach involves the presence of a finite number of initially unknown parameters, which are optimally determined, providing a rigorous way to control the convergence of the solutions [20]. Different methodologies have been applied to study Lotka-Volterra models [21–25]. Using this new approach, the obtained approximate solutions rapidly converge to the exact solution. Some alternative ways for mathematical biology problems are presented in [26–28].

2. Formulation and Solution Approach

In order to introduce the OPIM framework for the most general form of systems of nonlinear ODEs, we consider the following system:𝑥̇₁=𝑓₁𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑠𝑥̇₂=𝑓₂𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑠⋮𝑥̇_𝑠=𝑓_𝑠𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑠𝑥₁(0)=𝑎₁,𝑥₂(0)=𝑎₂,…,𝑥_𝑠(0)=𝑎_𝑠, (2.1) where dot denotes derivative with respect to time 𝑡, 𝑓_𝑖,(𝑖=1,2,…,𝑠) are nonlinear functions.

In the frame of OPIM we define the following iteration formulas:𝑥̇_1,𝑛+1=𝑓₁𝑥_1,𝑛,𝑥_2,𝑛,…,𝑥_𝑠,𝑛+𝑚₁𝑡,𝐶_𝑖𝑓_1,𝑥1𝑥_1,𝑛,𝑥_2,𝑛,𝑥_𝑠,𝑛+𝑛₁𝑡,𝐶_𝑖𝑓_1,𝑥2𝑥_1,𝑛,𝑥_2,𝑛,…,𝑥_𝑠,𝑛+⋯+𝑝₁𝑡,𝐶_𝑖𝑓_1,𝑥𝑠𝑥_1,𝑛,𝑥_2,𝑛,…,𝑥_𝑠,𝑛𝑥̇_2,𝑛+1=𝑓₂𝑥_1,𝑛,𝑥_2,𝑛,…,𝑥_𝑠,𝑛+𝑚₂𝑡,𝐶_𝑗𝑓_2,𝑥1𝑥_1,𝑛,𝑥_2,𝑛,𝑥_𝑠,𝑛+𝑛₂𝑡,𝐶_𝑗𝑓_2,𝑥2𝑥_1,𝑛,𝑥_2,𝑛,…,𝑥_𝑠,𝑛+⋯+𝑝₂𝑡,𝐶_𝑗𝑓_2,𝑥𝑠𝑥_1,𝑛,𝑥_2,𝑛,…,𝑥_𝑠,𝑛⋮𝑥̇_𝑠,𝑛+1=𝑓_𝑠𝑥_1,𝑛,𝑥_2,𝑛,…,𝑥_𝑠,𝑛+𝑚_𝑠𝑡,𝐶_𝑘𝑓_𝑠,𝑥1𝑥_1,𝑛,𝑥_2,𝑛,𝑥_𝑠,𝑛+𝑛_𝑠𝑡,𝐶_𝑖𝑓_𝑠,𝑥2𝑥_1,𝑛,𝑥_2,𝑛,…,𝑥_𝑠,𝑛+⋯+𝑝_𝑠𝑡,𝐶_𝑘𝑓_{𝑠,𝑥𝑠}𝑥_1,𝑛,𝑥_2,𝑛,…,𝑥_𝑠,𝑛, (2.2) where 𝑓_1,𝑥𝑖=𝜕𝑓₁/𝜕𝑥_𝑖, 𝑛=0,1,2… and 𝐶_𝑖,𝐶_𝑗⋯𝐶_𝑘 are unknown constants at this moment.

A special case of this general system is the Lotka-Volterra model with three species, which is described by the following system of nonlinear differential equation [29, 30]:,,̇,𝑥̇=𝑥1−𝑥−𝛼𝑦−𝛽𝑧𝑦̇=𝑦1−𝛽𝑥−𝑦−𝛼𝑧𝑧=𝑧1−𝛼𝑥−𝛽𝑦−𝑧 (2.3) where 𝛼 and 𝛽 are known parameters not necessary small. Initial conditions are𝑥(0)=𝑎,𝑦(0)=𝑏,𝑧(0)=𝑐. (2.4)

In this special case, it is clear that 𝑠=3 and𝑥₁=𝑥,𝑥₂=𝑦,𝑥₃=𝑧,𝑥₄=𝑥₅𝑓=⋯=0₁=𝑥1−𝑥−𝛼𝑦−𝛽𝑧,𝑓₂,𝑓=𝑦1−𝛽𝑥−𝑦−𝛼𝑧₃=𝑧1−𝛼𝑥−𝛽𝑦−𝑧,𝑓₄=𝑓₅=⋯=0. (2.5)

With the notations (2.5), the iteration formulas (2.2) can be written in the form𝑥̇_𝑛+1=𝑓𝑥_𝑛,𝑦_𝑛,𝑧_𝑛+𝑚₁𝑡,𝐶_𝑖𝑓_𝑥𝑥_𝑛,𝑦_𝑛,𝑧_𝑛+𝑛₁𝑡,𝐶_𝑖𝑓_𝑦𝑥_𝑛,𝑦_𝑛,𝑧_𝑛+𝑝₁𝑡,𝐶_𝑖𝑓_𝑧𝑥_𝑛,𝑦_𝑛,𝑧_𝑛𝑦̇_𝑛+1=𝑔𝑥_𝑛,𝑦_𝑛,𝑧_𝑛+𝑚₂𝑡,𝐶_𝑗𝑔_𝑥𝑥_𝑛,𝑦_𝑛,𝑧_𝑛+𝑛₂𝑡,𝐶_𝑗𝑔_𝑦𝑥_𝑛,𝑦_𝑛,𝑧_𝑛+𝑝₂𝑡,𝐶_𝑗𝑔_𝑧𝑥_𝑛,𝑦_𝑛,𝑧_𝑛𝑧̇_𝑛+1=ℎ𝑥_𝑛,𝑦_𝑛,𝑧_𝑛+𝑚₃𝑡,𝐶_𝑘ℎ_𝑥𝑥_𝑛,𝑦_𝑛,𝑧_𝑛+𝑛₃𝑡,𝐶_𝑘ℎ_𝑦𝑥_𝑛,𝑦_𝑛,𝑧_𝑛+𝑝₃𝑡,𝐶_𝑘ℎ_𝑧𝑥_𝑛,𝑦_𝑛,𝑧_𝑛, (2.6) where𝑓,𝑔,ℎ,𝑥,𝑦,𝑧=𝑥1−𝑥−𝛼𝑦−𝛽𝑧𝑥,𝑦,𝑧=𝑦1−𝛽𝑥−𝑦−𝛼𝑧𝑥,𝑦,𝑧=𝑧1−𝛼𝑥−𝛽𝑦−𝑧 (2.7) and 𝑓_𝑥=𝜕𝑓/𝜕𝑥, 𝑛=0,1,2…, and 𝐶_𝑖,𝐶_𝑗,𝐶_𝑘, 𝑖,𝑗,𝑘=1,2,… are unknown constants at this moment. There are many possibilities to choose the auxiliary functions 𝑚_𝑒(𝑡,𝐶_𝑖), 𝑛_𝑒(𝑡,𝐶_𝑗) and 𝑝_𝑒(𝑡,𝐶_𝑘), 𝑒=1,2,3. Basically, the shape of 𝑚_𝑒, 𝑛_𝑒, and 𝑝_𝑒 must follow the terms appearing in (2.6). Therefore, we try to choose the auxiliary functions so that in (2.6) the products 𝑚_𝑒𝐹_𝑥 are of the same shape with the function 𝐹_𝑥, where 𝐹∈{𝑓,𝑔,ℎ}. The constants 𝐶_𝑖,𝐶_𝑗,𝐶_𝑘 which appear in the auxiliary functions 𝑚_𝑒, 𝑛_𝑒, and 𝑝_𝑒 can be optimally determined via various methods, for example, putting the conditions that the residual functionals 𝐽₁, 𝐽₂, and 𝐽₃ given, respectively, by𝐽₁=^𝑏_𝑎𝑥̇−𝑓(𝑥,𝑦,𝑧)²𝐽𝑑𝑡,₂=^𝑏_𝑎𝑦̇−𝑔𝑥,𝑦,𝑧²𝐽𝑑𝑡,₃=^𝑏_𝑎𝑧̇−ℎ𝑥,𝑦,𝑧²𝑑𝑡 (2.8) be minimum, that is:𝜕𝐽_𝑖𝜕𝐶_𝑗=0,𝑖=1,2,3,𝑗=1,2,…,𝑠 (2.9) taking into account also the initial conditions. Other methods suitable to determine the constants 𝐶_𝑗 are the Galerkin method, the collocation method, and so on.

On the other hand, the initial approximations 𝑥₀, 𝑦₀, and 𝑧₀ and also the auxiliary functions 𝑚_𝑒, 𝑛_𝑒, and 𝑝_𝑒 are not unique.

In short, the basic idea of the proposed procedure (OPIM) are the construction a new iteration scheme (2.6), the involvement of the auxiliary functions, and the convergence-control constants 𝐶₁,𝐶₂,… which lead to an excellent agreement of the approximate solutions with the exact ones.

3. Results and Discussions

In this section we present an example to show the efficiency of the method described in the previous section for solving (2.3).

Taking into account the initial conditions, we choose the initial approximations as functions of the form𝑥₀=𝑎𝑒^−𝑡,𝑦₀=𝑏𝑒^−𝑡,𝑧₀=𝑐𝑒^−𝑡. (3.1)

For 𝑛=0 into (2.6) we can construct the following iteration formulas:𝑥̇₁=𝑥₀−𝑥²₀−𝛼𝑥₀𝑦₀−𝛽𝑥₀𝑧₀+𝑚₁𝑡,𝐶_𝑖1−2𝑥₀+𝛼𝑦₀+𝛽𝑧₀+𝑛₁𝑡,𝐶_𝑖−𝛼𝑥₀+𝑝₁𝑡,𝐶_𝑖−𝛽𝑥₀,𝑦₁̇=𝑦₀−𝛽𝑥₀𝑦₀−𝑦²₀−𝛼𝑦₀𝑧₀+𝑚₂𝑡,𝐶_𝑗−𝛽𝑦₀+𝑛₂𝑡,𝐶_𝑗1−𝛽𝑥₀+2𝑦₀+𝛼𝑧₀+𝑝₂𝑡,𝐶_𝑖−𝛼𝑦₀,𝑧₁̇=𝑧₀−𝛼𝑥₀𝑧₀−𝛽𝑦₀𝑧₀−𝑧²₀+𝑚₃𝑡,𝐶_𝑘−𝛼𝑧₀+𝑛₃𝑡,𝐶_𝑘−𝛽𝑧₀+𝑝₃𝑡,𝐶_𝑘×1−𝛼𝑥₀+𝛽𝑦₀+2𝑧₀. (3.2)

For the auxiliary functions 𝑚_𝑖, 𝑛_𝑖, 𝑝_𝑖, 𝑖=1,2,3 we choose the expressions𝑚₁𝑡,𝐶_𝑖=𝐶₁𝑡+𝐶₂𝑒^−𝑡,𝑛₁𝑡,𝐶_𝑖=𝐶₃𝑡+𝐶₄,𝑝₁𝑡,𝐶_𝑖=𝐶₅𝑡+𝐶₆𝑒^−𝑡𝑚₂𝑡,𝐶_𝑗=𝐶₇𝑡+𝐶₈,𝑛₂𝑡,𝐶_𝑗=𝐶₉𝑡+𝐶₁₀𝑒^−𝑡,𝑝₂𝑡,𝐶_𝑗=𝐶₁₁𝑡+𝐶₁₂𝑒^−2𝑡𝑚₃𝑡,𝐶_𝑘=𝐶₁₃𝑡+𝐶₁₄𝑒^−𝑡,𝑛₃𝑡,𝐶_𝑘=𝐶₁₅𝑡+𝐶₁₆𝑒^−2𝑡,𝑝₃𝑡,𝐶_𝑘=𝐶₁₇𝑡+𝐶₁₈𝑒^−𝑡. (3.3)

Alternatively, we can consider either other expressions for the initial approximations and the auxiliary function, follows as:𝑥₀=𝑎𝑒^−2𝑡,𝑦₀=𝑏,𝑧₀=𝑐𝑒^−𝑡,𝑚₁𝑡,𝐶^′_𝑖=𝐶^′₁𝑒^−2𝑡,𝑛₁𝑡,𝐶^′_𝑖=𝐶^′₂𝑡+𝐶^′₃𝑒^−𝑡,𝑝₁𝑡,𝐶^′_𝑖=𝐶^′₄𝑡+𝐶₅𝑒^−2𝑡𝑚₂𝑡,𝐶^′_𝑗=𝐶^′₆𝑡+𝐶^′₇𝑒^−𝑡,𝑛₂𝑡,𝐶^′_𝑗=𝐶^′₈𝑒^−2𝑡,𝑝₂𝑡,𝐶^′_𝑗=𝐶^′₉𝑡+𝐶^′₁₀𝑒^−𝑡𝑚₃𝑡,𝐶^′_𝑘=𝐶^′₁₁𝑒^−𝑡,𝑛₃𝑡,𝐶^′_𝑘=𝐶^′₁₂𝑡+𝐶^′₁₃,𝑝₃𝑡,𝐶^′_𝑘=𝐶^′₁₄𝑡+𝐶^′₁₅𝑒^−2𝑡, (3.4) and so on.

Now, substituting (3.1) and (3.3) into (3.2), we obtain the equations in 𝑥̇₁, 𝑦₁̇, and 𝑧₁̇, respectively,𝑥̇₁=−𝛼𝑎𝐶₃𝑡+𝑎−𝛼𝑎𝐶₄𝑒^−𝑡+𝐶₁−𝛽𝑎𝐶₅𝑡+𝐶₂−𝛽𝑎𝐶₆−𝑎²+𝛼𝑎𝑏+𝛽𝑎𝑐𝑒^−2𝑡−2𝑎+𝛼𝑏+𝛽𝑐𝐶₁𝑡+𝐶₂𝑒^−3𝑡𝑦₁̇=𝐶₉𝑡−𝛽𝑏𝐶₇𝑡+𝑏−𝛽𝑏𝐶₈+𝐶₁₀𝑒^−𝑡−𝛽𝑎+2𝑏+𝛼𝑐𝐶₉𝑡+𝛽𝑎𝑏+𝑏²+𝛼𝑏𝑐+𝛽𝑎+2𝑏+𝛼𝑐𝐶₁₀𝑒^−2𝑡−𝛼𝑏𝐶₁₁𝑡+𝛼𝑏𝐶₁₂𝑒^−3𝑡𝑧₁̇=𝐶₁₇𝑡+𝐶₁₈+𝑐𝑒^−𝑡−𝛼𝑎+𝛽𝑐+2𝑐𝐶₁₇+𝛼𝑐𝐶₁₃𝑡+𝛼𝑎𝑐+𝛽𝑏𝑐+𝑐²+𝛼𝑐𝐶₁₄+𝛼𝑎+𝛽𝑐+2𝑐𝐶₁₈𝑒^−2𝑡−𝛽𝑐𝐶₁₄𝑡+𝛽𝑐𝐶₁₆𝑒^−3𝑡. (3.5)

The solutions of (3.5) with the initial conditions𝑥₁(0)=𝑎,𝑦₁(0)=𝑏,𝑧₁(0)=𝑐 (3.6) can be written in the form𝑥₁(𝑡)=𝑎+𝛼𝑎𝐶₃𝑡𝑒^−𝑡+𝑎−𝛼𝑎𝐶₃−𝛼𝑎𝐶₄1−𝑒^−𝑡−12𝐶₁−𝛽𝑎𝐶₅𝑡𝑒^−2𝑡+14𝐶₁−𝛽𝑎𝐶₅+2𝐶₂−𝛽𝑎𝐶₆−2𝑎²+𝛼𝑎𝑏+𝛽𝑎𝑐1−𝑒^−2𝑡+132𝑎+𝛼𝑏+𝛽𝑐×𝐶₁𝑡𝑒^−3𝑡−192𝑎+𝛼𝑏+𝛽𝑐𝐶₁+3𝐶₂1−𝑒^−3𝑡𝑦₁(𝑡)=𝑏+𝑏−𝛽𝑏𝐶₈+𝐶₁₀1−𝑒^−𝑡+𝛽𝑏𝐶₇−𝐶₉𝑡𝑒^−𝑡−14𝛽𝑎+2𝑏+𝛼𝑐𝐶₉+2𝛽𝑎𝑏+𝑏²+𝛼𝑏𝑐+2𝛽𝑎+2𝑏+𝛼𝑐𝐶₁₀1−𝑒^−2𝑡+12𝛽𝑎+2𝑏+𝛼𝑐𝐶₉𝑡𝑒^−2𝑡−19𝛼𝑏𝐶₁₁+3𝐶₁₂1−𝑒^−3𝑡+13𝛼𝑏𝐶₁₁𝑡𝑒^−3𝑡𝑧₁(𝑡)=𝑐+𝐶₁₇+𝐶₁₈+𝑐1−𝑒^−𝑡−𝐶₁₇𝑡𝑒^−𝑡−142𝛼𝑎𝑐+𝛽𝑏𝑐+𝑐²+2𝛼𝑐𝐶₁₄+2𝛼𝑎+𝛽𝑐+2𝑐𝐶₁₈+𝛼𝑐𝐶₁+𝛼𝑎+𝛽𝑐+2𝑐𝐶₁₇×1−𝑒^−2𝑡+12𝛼𝑐𝐶₁₃+𝛼𝑎+𝛽𝑐+2𝑐𝐶₁₇𝑡𝑒^−2𝑡+13𝛽𝑐𝐶₁₅𝑡𝑒^−3𝑡−14𝛽𝑐𝐶₁₅+3𝐶₁₆×1−𝑒^−3𝑡. (3.7) By substituting (3.7) into (2.3) it results in the residuals:𝑅₁𝑡,𝐶_𝑖=𝑥̇₁−𝑥₁1−𝑥₁−𝛼𝑦₁−𝛽𝑧₁,𝑅₂𝑡,𝐶_𝑖=𝑦₁̇−𝑦₁1−𝛽𝑥₁−𝑦₁−𝛼𝑧₁,𝑅₃𝑡,𝐶_𝑖=𝑧₁̇−𝑧₁1−𝛼𝑥₁−𝛽𝑦₁−𝑧₁. (3.8)

Making collocation in the arbitrary points 𝑡_𝑖,𝑖=1,2,…,6𝑅₁𝑡₁,𝐶_𝑖=𝑅₁𝑡₂,𝐶_𝑖=⋯=𝑅₁𝑡₆,𝐶_𝑖𝑅=0₂𝑡₁,𝐶_𝑖=𝑅₂𝑡₂,𝐶_𝑖=⋯=𝑅₂𝑡₆,𝐶_𝑖𝑅=0₃𝑡₁,𝐶_𝑖=𝑅₃𝑡₂,𝐶_𝑖=⋯=𝑅₃𝑡₆,𝐶_𝑖=0, (3.9) we obtain the optimal values of the constants 𝐶₁,𝐶₂,…,𝐶₁₈ and therefore the solution (3.7) in the first approximation is well determined.

In the case when 𝑎=0.2, 𝑏=0.3, 𝑐=0.5, 𝛼=0.1, 𝛽=0.1, 𝑡₁=0.3, 𝑡₂=0.7, 𝑡₃=1, 𝑡₄=2, 𝑡₅=3, 𝑡₆=5, from (3.9) we obtain𝐶₁=1.669841466,𝐶₂=−0.452081708,𝐶₃=0.899181851,𝐶₄𝐶=2.361528973,₅=−1.940901217,𝐶₆=0.582517076,𝐶₇=1033.008542,𝐶₈𝐶=−540.7564632,₉=30.90993616,𝐶₁₀=−14.59841284,𝐶₁₁=29.8078082,𝐶₁₂𝐶=94.01456893,₁₄=53.21346396,𝐶₁₅=−22.21139178,𝐶₁₆=−35.8538265,𝐶₁₇𝐶=−0.345181562,₁₈=0.993490199. (3.10)

By substituting (3.10) into (3.7) we obtain after only one iteration, the following expression for the first-order approximate solution of Lotka-Volterra equations in three species (2.3):𝑥₁(𝑡)=0.8281811−(0.452081708𝑡+1.669841466)𝑒^−𝑡+(2.361528973𝑡−0.899181851)𝑒^−2𝑡+(0.582517076𝑡+1.940902217)𝑒^−3𝑡,𝑦₁(𝑡)=0.83333001+(0.08032009𝑡−1.924281055)𝑒^−𝑡+(1.715501457𝑡+0.342446025)𝑒^−2𝑡+(0.298078082𝑡+1.04850505)𝑒^−3𝑡,𝑧₁(𝑡)=0.836102311+(0.345181562𝑡−1.148308637)𝑒^−𝑡+(0.001606241𝑡+1.532769033)𝑒^−2𝑡−(0.370189863𝑡+0.720960396)𝑒^−3𝑡. (3.11)(3.12)(3.13)

It is easy to verify the accuracy of the obtained solution if we graphically compare the analytical solution with the numerical one. Figures 1, 2, and 3 show the comparison between the present solutions and the numerical integration results obtained by a fourth-order Runge-Kutta method.

Figure 1 

Comparison between the approximate solution (3.11) and numerical results obtained for (2.3) in case 𝑎=0.2,𝑏=0.3,𝑐=0.5,𝛼=0.1,𝛽=0.1: — numerical solution; - - - approximate solution.

Figure 2 

Comparison between the approximate solution (3.12) and numerical results obtained for (2.3) in case 𝑎=0.2, 𝑏=0.3, 𝑐=0.5, 𝛼=0.1, 𝛽=0.1: — numerical solution; - - - approximate solution.

Figure 3 

Comparison between the approximate solution (3.13) and numerical results obtained for (2.3) in case 𝑎=0.2, 𝑏=0.3, 𝑐=0.5, 𝛼=0.1, 𝛽=0.1: — numerical solution; - - - approximate solution.

It can be seen from Figures 1–3 that the solution obtained by OPIM is in very good agreement with numerical integration results.

4. Conclusions

In this work the Optimal Parametric Iteration Method (OPIM) is employed to propose new analytic solutions for some nonlinear differential equations. The validity of the method is illustrated on the Lotka-Volterra model with three species. Our procedure is independent of the presence of small parameters. Our construction of iterations is different from other known iteration techniques. In short, the basic new ideas of our method are related to the construction of a new iteration scheme which involves some auxiliary functions 𝑚_𝑒, 𝑛_𝑒, and 𝑝_𝑒, 𝑒=1,2,3 whose parameters 𝐶₁,𝐶₂,… (viz. the convergence-control constants) lead to an excellent agreement of the approximate solutions with the exact ones. Our procedure is very effective, explicit, and accurate for nonlinear approximations, rapidly converging to the exact solution after only one iteration. This procedure provides a simple but rigorous way to control and adjust the convergence of the solutions by optimally determining the parameters 𝐶₁,𝐶₂,…. Our method gives analytic solutions valid globally in time unlike other known methods, for instance. Adomian decomposition method, which unfortunately does not guarantee analytic solutions valid globally in time as proved by Rèpaci [31].