^{1}

^{2}

^{*}

We reviewed a nonlinear dynamical model in 2
n
-variables which has conservative nonlinear interactions defined in terms of Noether’s theorem in dynamics. The 2-variable (
n
= 1) conservative nonlinear model with external perturbations produced a possible explanation for problems such as the 10-year cycles of Canadian Lynx and snowshoe hair, interactions of microbes, stability and conservation law of nonlinear interacting systems. In this paper, the atto-fox (10
^{-18}
-fox) problem on the LV nonlinear equation, properties of 4-variable conservative nonlinear interactions different from nonconservative nonlinear interactions are exam
ined and emphasized. Properties of the 4-variable (
n
= 2) conservative interaction model and a method to construct numerical solutions are discussed by employing the 2-variable solution. The periodic times of component variables and the net periodic time defined by superposition of component variables are discussed in order to study stability of the net 4-variable system. With symmetries and conservation laws, nonlinear analyses would be useful to study microscopic and macroscopic complex systems.

The concepts of stability, conservation law, symmetry are important in order to understand natural phenomena in physical, biological and engineering systems [

Though symmetry and conservation laws have important information on nonlinear systems [

Complex systems in natural sciences from microbiology to ecology, economy and environmental sciences are regarded as realizations of nonlinear dynamical interactions. Nonlinear phenomena are difficult to handle because of their complex interactions and structures, self-organizations, spontaneous emergence of order, which exhibit no simple laws or orders. However, complex systems constitute cells, organs, and living animals which make stable systems. There may be some conservation laws corresponding to the stability observed in micro- biological systems. This is a motive that we applied nonlinear analyses based on Noether’s theorem to dynamics, and interesting results for complex phenomena are discussed [

Our approach to nonlinear phenomena based on conservation laws is briefly reviewed for 2-variable model (

The conventional nonlinear approach is reviewed in Section 1.1, and the atto-fox problem of Lotka-Volterra (LV) type nonlinear equation is explained in Section 1.2. Then, a new approach with conservation law and stability is explained in Section 1.3. In Section 2, the time-dependent perturbations and restoration from the perturbation are discussed with applications to the 10-year cycle of Canadian lynx and snowshoe hair and food-web of microbes in Okanagan Lake. In Section 3, the conservative nonlinear 2-variable model is extended to the 4-variable conservative model and properties of the 4-variable calculations are discussed. Conclusion and remarks are in Section 4.

It is often explained that a model of interacting populations is written in general as,

where

The simple and important applications in the field of ecology are those of Malthus (1959) for a population analysis, A. Lotka (1925) [

The mathematical expressions of

The typical Lotka-Volterra (LV) equation used in applications is given by the following simple nonlinear equation:

where x and y are the population number of prey and predator respectively;

However, the nonlinear LV equation of the type (1.2) has an intrinsic problem known as atto-fox (10^{−18}-fox) problem [

interact with other animals through reasonably large integer numbers, however, the lower left of the

More specifically, the LV equation (2) is invariant with the scale change:

where a and b are arbitrary numbers. The equation for

Hence, the solution ^{−18}-fox) problem is proved. The LV equation of the type (1.2) has a serious potential problem when it is applied to real biological and ecological problems.

The problem of LV equations may be summarized as:

(a) the LV Equation (1.2) is too simple to apply to ecological and interactive systems.

(b) there are no restrictions on the nonlinear coefficients,

(c) density-independent external perturbations should be included in numerical simulations in order to include changes of climate, temperature, seasons and landscape and so forth. The external perturbations are independent of

Hence, it is necessary to find a new approach to resolve (a)-(c), which restricts values of nonlinear coef- ficients and numerically simulates real data with external perturbations.

It may seem natural to assume that a nonlinear system eventually dissipates energy (or an activation of the system) and arrives at an equilibrium state in the long run. However, the equilibrium state evolved as a dissipative process should be a maximum entropy state, and if the state needs to be activated, it needs some external inputs of energy. Hence, a nonlinear system demands certain external density-independent inputs or external perturbations which are not included in Lotka-Volterra and Kolmogorov type nonlinear Equations (1.1) and (1.2).

There are numerous examples of stable dynamical systems from microbiological systems such as DNA and RNA, amino acids and proteins, cells and organs, to macroscopic systems such as the 10-year cycle of Canadian lynx and snowshoe hair, wolves and caribous, creatures in marine coral reefs. The stability suggests that some conservation laws and symmetries are inherent from complex microbiological systems to macroscopic ecological systems. Hence, it could be possible to investigate conservation laws corresponding to a biological stability, which leads to a fundamental assumption for the model of conservative binary-coupled interactions [

The meaning of a conservation law in biological complex systems may be very different from conservation laws in physics described by way of a Lagrangian or a Hamiltonian. It is difficult to use direct mechanical concepts and analogies such as potential and kinetic energy, but it should be useful to employ the concept of conservation law and symmetry in terms of Noether’s theorem in order to investigate conservation laws corresponding to biodiversity. Conservation laws and stability in biological systems could be employed to protect, understand and sustain the biosphere.

By employing the 2-variable conservative nonlinear model [

The Lagrangian of 2-variable conservative nonlinear system is described with the following Lagrangian,

From (1.5), we get the following nonlinear differential equation,

The parameter

The conserved function, Y, of this system is given by,

The Psi-function, Y, may correspond to the Hamiltonian or the Lagrangian in a mechanical system, however, one should note that the Y-function does not have any velocity (time-derivative) terms, showing no individual mechanical energy terms. The terms of the Y-function look like potential energy terms in order to conserve the Y-function as a constant value, whereas the interpretation of the function depends on nonlinear systems.

The solutions

The solutions in the phase space,

In widely used nonlinear differential equations, the eight interaction terms on the right-hand sides of (1.6) and (1.7) require eight independent parameters (

the beginning in order to solve (1.6) and (1.7). This is helpful for numerical simulations of real data. The property of the 2n-variable conservative nonlinear equation is essentially different from properties of Lotka- Volterra and Kolmogorov equations.

We would like to answer some questions concerning difficulties on nonlinear differential equations in order to elucidate applicability of the 2n-variable conservative nonlinear differential equations.

(Question 1) There may be infinitely many periodic solutions in the nonlinear model that depend on the values of nonlinear coefficients,

(Answer) This is restated as that solutions should be transformed in any form of solutions in the phase space,

those of the atto-fox (10^{−18}-fox) solutions.

When initial starting values and nonlinear coefficients are inconsistent, one obtains no solutions and the Y-function diverges [

(Question 2) Selecting initial conditions of scarce prey and plentiful predators, unphysical solutions to nonlinear equations could occur. For example,

(Answer) This is a typical surmise derived from LV type nonlinear equations, but the assumption is not correct at all with the conservative nonlinear model. Because nonlinear coefficients are not entirely free parameters, it is not possible to find solutions by selecting any nonlinear coefficients and initial conditions of scarce prey and plentiful predators, and atto-fox type problems are restricted. A specific numerical example is shown in the paper [

Lotka-Volterra and Kolmogorov equations are not sufficient to study nonlinear systems because they do not include density-independent external perturbations, such as climate and seasonal changes, pesticides and herbicides, hunting, forest fires and storms, etc. These external factors are density-independent and they are not expressible as (1.1). Hence, it is necessary to simulate density-independent and time-dependent external factors, and they are introduced as piecewise continuous, constant perturbations in the nonlinear equations discussed in detail in the papers [

In order to investigate numerical responses of a system to external perturbations, piecewise continuous constants are introduced by using q-function such that

where

and coefficients

An execution of an external perturbation to (

The instant that an external perturbation is added to conservative nonlinear equations, the solution jumps to possible another conserved system. When the perturbation was switched off (see, Ep 1 in

We gradually changed the strength and duration time of a negative external perturbation. The system advanced to unstable states and disintegrated,

The strength and duration time of negative external perturbations which make the system unstable were evaluated numerically. The threshold of negative perturbation is related to the amplitude and periodic time of each component,

The threshold of strength and duration time of negative perturbations is important information to regulate nonlinear complex systems. The 2-variable conservative nonlinear model indicates that if we know the relation

between stability and rhythm of interacting systems and exert appropriate external perturbations at a correct time, it could be possible to save species from extinction.

As a specific example, the 2-variable nonlinear model with perturbations is applied to analyses of the 10-year cycles of Canadian Lynx and snowshoe hair [

The standard rhythm is discussed in the paper [

The current conservative nonlinear model is applied to the data of food-web in Okanagan Lake [

The important property of the 2-variable nonlinear model with conservation law is that coupled systems can have certain recovering strength to external perturbations. As a predator needs a prey for its food, a prey needs a predator for the control and conservation of their own species. The conservation law and the standard rhythm of species are considered to be naturally constructed by species to live for a long time in nature. Therefore, the rhythm and stability of population cycle as a whole in an ecosystem would be interpreted as a manifestation of the survival of the fittest adapted to the balance in an ecosystem.

The stability and conservation law are constructed by species in mutual dependency or cooperation to survive for long-time periods in severe environmental conditions. The standard rhythm should be regarded as the result of strategy for species to live in nature. Whatever roles they have to play, the species that can fit and balance with other creatures can survive in nature. A strong predator cannot even survive if it ignores the law of the standard rhythm and conservation law of nature constructed by other members and the environment. The conservative BCF nonlinear model with perturbations enables one to examine a fundamental law of nature in a form of differential equations. We hope that this study will help understand both activities of animals and humans in natural life.

Based on the reviews and discussions, we extend the 2-variable conservative model to the 4-variable model for applications to more realistic interacting systems. Several examples can be found in microbiology and medical fields [

A morphogen is considered a substance governing the pattern of tissue development in the process of mor- phogenesis, or a signaling molecule that acts directly on cells to produce specific cellular responses depending on its concentration. When morphogens are identified, a dynamical method based on the 2-variable or 4-variable conservative nonlinear dynamical analyses could be possible in microbiology.

However, the 4-variable nonlinear differential equations have too many nonlinear parameters to be set at the beginning, and they are difficult to solve and apply to numerical simulations. Therefore, we discussed a strategy to construct solutions to 4-variable, 6-variable, … , conservative nonlinear differential equations. The construc- tion of 2n-variable solution is based on the solution of 2-variable solutions.

The 4-variable conservative nonlinear model for complex systems is described as:

where coupling constants,

The external perturbations are expressed as arbitrary, piecewise continuous constants,

The 4-variable conserved nonlinear equation is derived from Euler-Lagrange equation in Lagrangian dynamics. The Noether’s theorem in dynamics is employed to obtain the conservation law in complicated non- linear systems. Hence, coefficients of the right-hand side of (3.1)-(3.4) are restricted such that the system has conserved quantities.

In conventional 4-variable nonlinear differential equations of the type (3.1)-(3.4), nine terms in the right-hand side are supposed to be independent and adjusted freely. Hence, the number of nonlinear coefficients basically adds up to 40 parameters (

In order to make the 4-variable nonlinear equation useful for numerical simulations, we propose a strategy to construct a 4-variable solution by employing 2-variable solutions.

1) One should construct two sets of 2-variable numerical solutions independently by dividing interactions of 4 variables (

2) The 2-variable numerical simulations, (

3) The next step is to weakly couple the set of 2-variable solutions by using the 4-variable solution (3.1)-(3.4). The weak coupling of (

The coupling constants,

Following theoretical discussions of invariant variational method in dynamics, the Y-function for the 4- variable conserved nonlinear dynamical system is expressed as,

The physical quantities (

The 4-variables (

The 2-variable nonlinear solution

It is well known in linear differential equations that the net response (amplitude) y at a given place and time is

the sum of independent responses (

The superposition principle does not hold in nonlinear systems. However, we examined the net amplitude

defined by

total system.

In our conservative nonlinear interactions, we found that component solutions approximately produced a periodic shape of the net amplitude. The shape of the net amplitude is very complicated but oscillates at an approximate pitch like a resonance produced by a tuning fork on resonance box.

The approximate periodic times of each variable

Hence, conservative nonlinear interactions of component variables lead to a longer periodic time and a net stable system against external perturbations. In ecosystems, it could be interpreted so that living animals tend to have interactions with other animals as much as possible for their species to live long. The approximate periodic time of components and the net periodic time in nonlinear interactions may be one of important properties to understand stability of nonlinear interacting systems.

Our strategy of constructing solutions of nonlinear differential equations is to find a class of solutions having approximate periodic times. This may be true, and we investigated the class of solutions because the amplitude, y, and net periodic time,

The ten-year cycles of Canadian lynx and hair, interactions of microbes and other stable cycles are considered as examples of the approximate periodicity of nonlinear interacting systems. Hence, a possible condition to sustain stability of complex symbiotic systems can be examined by applying a conservative nonlinear dif- ferential equation.

The remarkable properties derived from conservative nonlinear interactions are summarized as follows:

1) The net periodic oscillation and time can be approximately defined in the conservative nonlinear interactions, and the net periodic time is longer than the periodic time of component variables:

2) The 4-variable system becomes more stable than the 2-variable system against negative external per- turbations. Nonlinear interactions of diverse component variables seem to maintain stability of the net system against negative external perturbations.

These results are different from those discussed in dissipative, nonconservative nonlinear interactions in many literatures which discuss limit cycles and attractors. The conservation law and durability of a conservative nonlinear system against external perturbations could be a key to understand stability of complex systems. The pattern simulations combined with external perturbations are possible for conserved nonlinear interactions, and it may be helpful to understand characteristic periodicities and responses of complex interacting systems.

In physics and engineering applications, it is standard to analyze physical quantities caused by heat conduction, electromagnetic and mechanical problems by way of the superposition principle [

We reviewed characteristic properties for the 2-variable conservative nonlinear differential equation which generalizes Lotka-Volterra type prey-predator, competitive interactions, and explained that the atto-fox problem found in a simple Lotka-Volterra equation is not intrinsic to nonlinear differential equations. The 2-variable and 4-variable conservative nonlinear equations with external perturbations are useful to simulate real data num- erically.

The conservative nonlinear interaction with external perturbations numerically suggested the existence of the standard rhythm (the 10-year population cycle) of Canadian lynx and snowshoe hare. The analysis of the 2- variable conserved nonlinear equation indicates that nonlinear interactions could be a manifestation of strategy to live and thrive in nature [

It is naturally observed that many complex systems from microscopic to macroscopic world seem to maintain stable structure, recoveries and self-organizations [

The solution of the 2-variable model is applied to construct solutions of the 4-variable conservative nonlinear equations. It is difficult to find an appropriate numerical solution for the 4-variable nonlinear differential equation, which tends to make numerical analyses hard to carry out. However, we showed a strategy to find a solution by employing the 2-variable solutions, which helps construct solutions and accomplish numerical analyses ri- gorously by employing Y-function. This approach can be extended to the 6 and 8-variable conservative non- linear equations.

The conservative nonlinear model has not been actively applied yet, though it revealed certain properties of the 10-year cycles of lynx and hair, microbe interactions. We are searching for ecological data or medical applications as microscopic interacting systems, such as diabetes or other chronic diseases exerted by multiple interacting factors. Collaborations to analyze big data with the current model are anticipated.

The nonlinear differential equation approach is useful to understand physical phenomena. We hope that the 4-variable conservative nonlinear equation may find useful and practical applications from microscopic complex systems to macroscopic ecosystems.

Lisa Uechi,Hiroshi Uechi, (2016) Noether’s Conservation Laws and Stability in Nonlinear Conservative Interactions. Open Access Library Journal,03,1-18. doi: 10.4236/oalib.1102592

The numbering of nonlinear coefficients,

and the nonlinear Equations (3.1)-(3.4), are derived from the lagrangian (5.2) by way of Euler-Lagrange equation:

The numbering of nonlinear coefficients of parentheses,

The number of 4-variable nonlinear parameters is still large, but if 2-variable numerical solutions are used to construct the 4-variable solution, one is supposed to control only 8 parameters. The method of solution is ex- plained in the Section 3.1.

Submit or recommend next manuscript to OALib Journal and we will provide best service for you:

Publication frequency: Monthly

9 subject areas of science, technology and medicine

Fair and rigorous peer-review system

Fast publication process

Article promotion in various social networking sites (LinkedIn, Facebook, Twitter, etc.)

Maximum dissemination of your research work

Submit Your Paper Online: Click Here to Submit

Contact Us: service@oalib.com