The Lynx and Hare Data of 200 Years as the Nonlinear Conserving Interaction Based on Noether’s Conservation Laws and Stability

We applied n-variable conserving nonlinear differential equations (n-CNDEs) to the population data of the 10-year cycles of Canadian lynx (1821-2016) and the snowshoe hare (1845-1921). Modeling external effects as perturbations to population dynamics, recovering and restorations from disintegrations (or extinctions), stability and survival strategies are discussed in terms of the conservation law inherent to dynamical interactions among species. The 2-variable conserving nonlinear interaction (2CNIs) is extended to 3, 4, ... n-variable conserving nonlinear interactions (n-CNIs) of species by adjusting minimum unknown parameters. The population cycle of species is a manifestation of conservation laws existing in complicated ecosystems, which is suggested from the CNDE analysis as a standard rhythm of interactions. The ecosystem is a consequence of the long history of nonlinear interactions and evolutions among life-beings and the natural environment, and the population dynamics of an ecosystem are observed as approximate CNIs. Physical analyses of the conserving quantity in nonlinear interactions would help us understand why and how they have developed. The standard rhythm found in nonlinear interactions should be considered as a manifestation of the survival strategy and the survival of the fittest to the balance of biological systems. The CNDEs and nonlinear differential equations with time-dependent coefficients would help find useful physical information on the survival of the fittest and symbiosis in an ecosystem.


Introduction
Ecology is a field of study on complex many-body interactions among organisms, living beings and natural environment. The conservation status of a group of living animals is measured in many countries, whether certain species still exist or how likely to become extinct in the near future. In reality, human activities from political, social and cultural conduct to agricultural, engineering proce-  [1] is a multilateral international treaty to protect endangered species, which has goals for conservation of biodiversity, environment and natural resources, including protection and management of living habitat.
In order to understand the importance of biodiversity and ecology, it is essential to collect specific population data and the natural habitat of endangered species.
Any population data would reveal interactions of species and natural environment, and nonlinear differential equations have been applied to population changes among 2 -4-variable factors of microbes, food chains and interacting species [2] [3] [4]. Though population changes can be reasonably expressed by nonlinear differential equations with data accumulations and computer simulations, it is imperative for us to extract physical reasons and dynamics of nonlinear interactions in ecological systems.
Nonlinear differential equations provide us with a possible account for complex change of populations by adjusting coupling constants, but they do not explain physical principles and reasons why such nonlinear interactions and coupling constants are physically chosen by nature, and in this sense, dynamical change of ecological system is not simply solved by computer simulations and big-data accumulations. Hence, we have proposed nonlinear conserving equations for n-body interactions [5] [6] [7] and applied them to population data of species and microbes in order to understand and extract useful information of The CNI generated a characteristic stable population cycle, and we termed it as the standard rhythm, which is a consequence of conservation laws maintained by interacting species. The standard rhythm may be related to the concept of the survival of the fittest and symbiosis among species, which should be investigated with big data in the various fields of ecology. The conserving nonlinear model suggests that biomechanical and microbiophysical systems show symbiosis among cells and organisms, characteristic stability and balance [34]

The 2-Variable CNIs, Restoration from External Perturbations
Nonlinear equations are classified into non-conserving dissipative equations suitable for transient behaviors and phenomena short in time and conserving nonlinear equations suitable for life living-long and surviving against external destructive perturbations, which corresponds to irreversible dissipative phenomena and thermomechanical equilibrium [36]. Though life-involving phenomena are very different from those of physical materials and particles, viruses and cells, plants and animals are not against physical laws and principles; in other words, they are making good use of natural laws for their existence. The Lotka-Volterra types of nonlinear differential equations [37] [38] are primitive nonlinear equations and have been applied to study competitive and predator-pray interactions, mathematical models of disease spreading among species, pest-control, evolving ecosystem networks, population and epidemiology and so forth [39] [40] [41]. The Lotka-Volterra types of nonlinear differential equations are easy to use, but they have intrinsic blunder known as the atto-fox problem, which is prevented in the conserving nonlinear equations [5] [6] [7]. Since the Lynx-hare cycle is an approximate nonlinear conserving phenomena, it has a conserved quantity, Ψ-function which restricts unphysical atto-fox problem.
The 2n-variable conserving nonlinear differential equations give the following useful results.
1) The form of differential equations and coefficients of nonlinear interactions are strictly confined with initial conditions when a population of species is stable for a long period of time, designating a conservation law.
2) The conserved quantity Ψ-function produces a Lyapunov function usually employed to study solutions to nonlinear differential equations, which is helpful to study non-conserving dissipative interactions.
3) A complex interacting system can be approximately decomposed into an assembly of binary-coupled forms (BCF).
4) The binary coupled system with the conservation law indicates an addition law interpreted as the restoration or rehabilitation phenomena, such as a small damaged device replaced by a new one.
5) The conservation law is also useful to check accuracy of numerical solutions to conserving nonlinear differential equations.
The purpose of conserving or dissipative nonlinear analyses among species is to study stability and restoration, prosperity and degeneration mechanism of ecological systems, effects on natural or artificial environmental changes in dynamics of ecology. It should help superintend proper procedures and methods to sustain natural and ecological systems in the future based on data in the past.

A Brief Review of Noether's Theorem and Conserved Quantities
The necessary condition for extrema in Lagrangian formulation, Journal of Applied Mathematics and Physics This is the Euler-Lagrange equation which determines the equations of motion where s ranges over 1, , r , and ( ) o ε denotes the terms which go to zero faster than ε , This defines the conserved quantities of a system. Since the expressions Ψ defined in (2.5) are constant with the condition: , they are the Journal of Applied Mathematics and Physics first integrals of the differential equations of motion. In physical applications, the first integral (2.5) is interpreted as the energy of the system, whose governing equations are 0 k E = . In general, the function Ψ is constant with respect to time.
By employing the formalism, the conservation law and symmetry of nonlinear differential equations for coupled conserving systems are discussed.
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 directly use 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 dynamics of biodiversity.

Nonlinear Conserving 2-Variable Interactions
It may seem natural to assume that a nonlinear system eventually dissipates energy and arrives at an equilibrium state, and the equilibrium state would be considered as a maximum entropy state. If the state is to be activated, it needs some external inputs of energy, designating external inputs or perturbations.

Exponential Types of 2 Variable Coupled LV System
The Lyapunov function discussed in nonlinear problems can be obtained by Ψ-function derivation, which is shown in a simple example. The classical type of LV equation is defined as: The logarithm of (2.13) is similar to a well known Lyapunov function ( ) order to solve (2.7) and (2.8), which is helpful for numerical simulations of conserving nonlinear differential equations.

The 3-Variable Conserving Nonlinear Interactions
The conserving nonlinear interaction of 3-variable system will be produced by the Lagrangian of the following type: and the conserved quantity of (2.14) is given by x The Lagrangian (2.14) produces the following coupled nonlinear differential equations: The coupled nonlinear Equation (2.16) can be solved for 3 x , resulting in,

The Conservation Law and Properties of
( ) The Ψ-function may have similar physical meanings as Hamiltonian of a system of materials and particles. However, the Hamiltonian has definite meanings as the total energy of a system is a constant in time (the energy conservation law).
The energy is convertible to heat, light, electric energy, other kind of work, and has the dimension of force × displacement [42] [43]. The conserved quantity Ψ is constant in time, but it is constructed from variations of populations or density of populations of 2n species, not from the force, kinetic energy and potential energy. The meaning of a conservation law in biological complex systems may be very different from conservation laws in physics, whereas the Ψ-function could be the conserved quantity corresponding to biodiversity.
A nonlinear system demands certain density-independent or density-dependent external perturbations which are not included in Lotka-Volterra and Kolmogorov type nonlinear Equations (6.1) and (6.2). The 2-variable conserving nonlinear differential Equations (2.6)-(2.9) produce generalized nonlinear equations of Lotka-Volterra, Kolmogorov, and Lyapunov function type nonlinear equations, and so, the conserved Ψ-function is related to stability, control and properties of a nonlinear system. The concept of stability is studied in terms of an addition law of Ψ-function [5] [6] [7]. The Ψ-function could be applied to recovery or restoration phenomena caused on a system, which is important for microbiological and ecological systems. The 10-year cycle of Canadian lynx and snowshoe hare is studied as an example of the standard rhythm for recovery or restoration. The standard rhythm would be a consequence of symbiosis [18] [19], living together to survive in severe nature.

The Classification of Restorations and Disintegrations (Extinctions) in Terms of Ψ-Function
It is essential for nonlinear theoretical and mathematical analyses to have reliable predictions on an ecological system if it thrives or extincts over a given period of time. Let us suppose that a binary-coupled n-variable nonlinear system with ( ) 1 2 , , n x x Ψ and another 2-variable interacting system written by ( ) interact with each other, resulting in constructing a new, stable and conserving coupled system. The addition property of ( ) 1 2 , , x x Ψ function in the form: may be interpreted as an approximation to the restoration or rehabilitation phenomena known in a large system of biological systems, neural networks, cells of organs, or computer networks replacing a small broken device with a normal device. Journal of Applied Mathematics and Physics The recovery and degeneration in the current study are classified as: 1) Complete Recovery (CR), 2) Partial Recovery (PR), 3) Non-Recovery (NR), Degeneration or Extinction.

1) Complete Recovery (CR)
As shown in Figure 1, when a system maintains the form of ( ) 1 2 , , x x Ψ -function, the interacting system is completely stable. The addition property of Ψ-function means that a defect or negative perturbations will be recovered when a form of binary coupled form is maintained [5]; in other word, a form of CNI to maintain the constant Ψ-function is regained.

2) Partial Recovery (PR)
The partial recovery is not a complete recovery, but a transition to a reasonably stable, approximate state which maintains a conserving stable solution. The Ψ-function changes to a different constant value (see for example, Figure 2).

3) Non-Recovery (NR)
Negative external perturbations or defects of an interacting system eventually lead to the degeneration of an entire interacting system or the extinction, and the Ψ-function becomes negative and is not constant or diverges (see for example, Figure 3 and Figure 4).
There are characteristic combinations of variables, coupling constants to be conservative or dissipative until it reaches an ecological equilibrium. The extinction of species appears as population number zero or negative. The concept of extinction should be categorized as the acute extinction (AEX) and the chronic extinction (CEX). The concept of AEX is well explained with poisons that cause death, which can be specifically determined. On the other hand, the chronic extinction (CEX) is regenerating extinction caused by multi-factored problems, whose causes and effects are difficult to determine. The conserving coupled system and addition law of Ψ-function can help us understand a stable complex system and the concept of recoveries, which is an important result for the conservation of nature and sustainability.

Recovering, Degeneration and Extinction of Species from External Perturbations
The dynamical interactions of species are affected by climate changes, temporal temperature fluctuations and changes of natural environment by human activities. These factors are mathematically introduced as external perturbations and expressed by piecewise continuous constants, 1 c and 2 c , by using θ-functions such that represents a step function:  Table 1 (Condition 1). The solutions ( ) 1 2 , x x are deformed by the perturbation but, the system does not disintegrate and finds a new stable phase-space and a conserved relation. The perturbation ends at 1200 t = (Ep. 1), and the system recovers to, more or less, the original state ( ) 1 2 , x x . This is an example of the complete recovery (CR).  The conservation of nature and biodiversity on Earth by protecting species and their habitats from excessive rates of extinction and human activities are one of the essential purposes for theoretical and numerical nonlinear analyses [48].
In Figure 3, the response of a strong negative perturbation to prey after the peak of endogenous maximum is shown. The values of coefficients are listed in    The conservation law Ψ with a critical negative-perturbation. Ψ converges to zero after the critical negative-perturbation.
In the nonlinear interacting system, positive perturbations which will increase 1 x or 2 x do not always mean a positive effect on stability of the system. There is a limit to the value of a positive perturbation, because an increase of 1 x may lead to a decrease of 2 x in the stable system ( ) 1 2 , x x , which indicates that a Journal of Applied Mathematics and Physics system has internally allowed maximum and minimum populations, and so, the existence of critical point is important. The behaviors of ( ) are considered extinct. It should be noted that if a dynamical prey-predator system is active, the rhythm of maxima and minima of interacting species is clearly maintained, which is well known behavior in real prey-predator systems. However, if an external perturbation (exogenous interaction) exceeds a certain critical value of a competitive system, the rhythm of maxima and minima will disappear and then after a time, the system will diverge (disintegrate) as shown in

The 4-Variable and (n + m)-Variable CNIs
The Nonlinearity exists in complicated interacting systems, such as the ecosystems of mammals [10] [11], microbes [12], systems of cells and organs [3]. These systems are highly evolved, interacting systems. We specifically discuss a practical technique for 4-variable and higher variable CNIs in the following discussions.

The 4-Variable Model for a Conserving Nonlinear Dynamical Interactions
The 4-variable CNI is derived from the 4-variable dynamical Lagrangian [7], and Journal of Applied Mathematics and Physics  x x  , , c c , represent strengths of external perturbations or effects, such as temperature changes, reductions of species by huntings or fishings, and so forth.
The free parameters are 28 α's and 4 ( 1 c to 4 c ) and resulting in 32 nonlinear coefficients to adjust. However, the number of adjustable coefficients can be decreased, since these parameters are restricted to positivity of variables, initial conditions and the conservation law.
Next, it is necessary for numerical analyses to roughly determine coupling constants of linear interactions, assuming coefficients of all nonlinear terms, 2 x , 3 1 , x x to be zero, and then, it is better to slowly turn on coefficients of nonlinear terms from 0 to 0 ε ± ( ε is a small number) to simulate data and adjust coefficients of linear terms, and numerical simulations are shown in the following section.
The numerical solution to the system of nonlinear differential equations from

The Superposition of CNI Solutions
The superposition means the total sum of interacting species, and in practice, it is tacitly assumed that the big data of several interacting species is too big to analyze, and the sum of population data would not be of any use for physical analyses. The superposition of solutions is mathematically exact only in linear differential equations to obtain the general solution. However, we applied it to  The emergence of a long-time pattern is a characteristic property in a CNI system. The superposition method may help analyze overall behavior and dynamics of big data, which is too complicated to comprehend by direct observations in short time.
The superposition of conserving nonlinear solutions seems to roughly display the net periodic time, n T , which is about ~100 n T in Figure 7 2) The 4-variable system becomes more stable than the 2-variable system against negative external perturbations.
The property would be a consequence of many coupling terms of the 4-variable CNIs, compared to those of the 2-variable CNIs. In other words, if an interacting system has many interacting terms with other species, the system may not be affected so much by a breakdown of external negative effects, because other interacting terms could alleviate the breakdown and defects on the system, indicating fundamental importance of biodiversity and mutual interactions.
In general, nonlinear systems have different classes of solutions, such as chaos and bifurcation [40] [57] [58]. The property is known that a small change of initial conditions or coupling constants in one state of nonlinear system can result in large differences in a later state. In other words, the underlying patterns and the deterministic laws of chaos and bifurcation are sensitive to initial conditions or coupling constants. The CNI state is similarly sensitive to initial conditions or coupling constants, but it maintains stability of the net system. The results of CNI are different from those discussed in dissipative, nonconserving nonlinear interactions in many literatures which discuss limit cycles and attractors. The conservation law and the stability of CNI system could be a key to understand dynamics of complex systems.

A method to Construct the 4-Variable and Higher-Variable Solutions by Employing 2-Variable Solutions
The conserving nonlinear n-variable interactions become complicated when the 1) Let us suppose that a 4-variable interacting system is constructed by a pair of 2-variable interaction systems; one system for the interacting species ( 1 2 , x x ) and another system for ( 3 4 , x x ), and the 2-variable interacting systems begin to interact.
3) The next step is to weakly couple the set of 2-variable CNI by observing coupling terms in the 4-variable solution (4.3)-(4.6). The weak coupling of ( 1 2 , x x ) and ( 3 4 , x x ) can be performed by adjusting slowly cross-coupling coefficients from 0 to 0 ε ± (for instance, ~0.01 ε ). This is a trial and error ap- , , , The reasonable values of nonlinear parameters would be readily searched in numerical simulations by the trial and error. Some numerical values of 2-variable conserving approximations are found in papers [5] [6] [7], and then, reasonable coupling constants can be found by the perturbation method explained above.
The strategy can be extended step-by-step to higher variable solutions, for example, a 5-variable CNI solution as a (2 + 3)-variable CNI solution, and a 6-variable CNI solution as a (2 + 2 + 2)-variable CNI solution and so forth.

The 2-Variable Conserving Approximation Applied to Nonlinear Interactions between Canadian Lynx and Snowshoe Hare
Lynx It is reported that there is no evidence to suggest that overall lynx numbers or distribution across Canada have declined significantly over the past two decades, although loss of habitat through increased urbanization and development and forestry is likely affecting lynx populations. Despite the dramatic decrease in harvest through the 1990 cyclic peak, there is evidence that lynx populations in most of the northern range were cycling normally. It is noticeable that field researches report that there is little evidence to conclude that the harvest during the 1980s had a long-term impact on contiguous northern lynx populations [8] [30]. It is necessary to have such a field research to understand ecological environment as well as impact of activity from human societies. The snowshoe hares were the dominant herbivore, and the changes in their numbers were correlated with those in numbers of arctic ground squirrel, spruce grouse, ptarmigan, lynx, coyote, great horned owl, goshawk, raven and hawk owl. The hare numbers were not correlated with numbers of red-backed vole [59] [60] [61] [62]. The lynx-hare population change is well examined by computer simulations of the 2-variable conserving CNI model with external perturbations as shown in Figure 8, showing that the CNI system keeps stable phase space, Ψ-function in 7.

The Population Regulation in Canadian Lynx and Snowshoe Hare in 200 Years
It is difficult to identify population regulation mechanism concerning the preypredator patterns of large mammals because large mammals' life span is relatively long compared with that of microbes. The prey-predator cycle for wolves and caribou takes some decades to observe; their interacting relation and behaviors were observed with modern technology (GPS-colored animals) [10]. However, the food web configuration between snowshoe hare and Canadian lynx is a well-known prey-predator-type phenomenon, and the ten-year cycle of Canadian lynx was examined from the data of Canadian lynx fur trade returns of the Northern Department of the Hudson's Bay Company [8]. The Canadian lynx and snowshoe hare have a synchronous ten-year cycle in population numbers [9] [10]. The fundamental mechanism for these cycles is affected by factors, such as nutrient, predation, and environmental factors and human social activities [11]. nonlinear conserving model suggests that species of a system consequently find a strategy or a mechanism to survive for long time periods. In other words, the cycle of population density is a manifestation of the strategy or mechanism to survive, suggested by the stability of phase space solutions of the system.
The data of Canadian lynx and snowshoe hare (1800-2018) is shown in Figure  9(a) and Figure 9(b). The lynx population data is clearly characterized in two periods between (1800-1920) and (1920-2018), which would be related to the modern world history of human societies, while the hare population is in a stable rhythm of the 10-year population cycles. The 2-variable nonlinear conserving model is employed to the lynx-hare data with external perturbations, shown in the Ψ-function in Figure 10. The Lynx data is reproduced in the 2-variable model with external perturbations as in Figure 9(c). The simulation of lynx data around 1800-1920 shows large and abrupt change in the Ψ-function, resulted from large external perturbations. Although the 10-year cycle of lynx population  has not been disappeared, the value of the Ψ-function has gradually decreased, readily observed by comparing 1821-1921 and 1960-2018 data.
It indicates that human activities affected the lynx population changes, which should be clearly observed, for instance, from the data of Canada lynx fur-trades return of the Hudson's Bay Company [31]. However, the value of Ψ-function becomes small but changes slowly during 1920-2018, which may indicate an outset of environmental conservation of natural species and resources. The number of lynx is very slowly decreasing, but it is clearly responding to the hare population cycle. In Figure 9(d), it is shown how the hare population can be affected by the lynx population cycle and suggests the hare population is not quantitatively affected by the lynx population cycle, which may express the hares' vitality and multiple dynamical links to other species.
The feeding and nutrient experiments reported in [11] are considered as external perturbations to the system. As shown in Figure 8, the perturbations cause certain effects on the system, but the system finds a rhythm to maintain the dynamics of species, which is related to the stability of nonlinear dynamical systems. The shapes of phase space ( ) 1 2 , x x are not so different from the original standard rhythm after several perturbations. Our numerical results agree with conclusions derived from feeding experiments and nutrient-addition experiments. Therefore, the properties of a system with a conservation law have a key to understand the unanswered question why these cycles exist.
The timing of perturbation leads to an essential difference to the feeding experiment of snowshoe hare: ... during the peak of the cycle in 1989 and 1990 (the feeding experiment) had no impact on reproductive output ... however, during the decline phase in 1991 and 1992, the predator exposure plus food treatment caused a dramatic increase in reproductive output ... [11]. The experiment can reproductive, offspring season.
The cycle of standard rhythm for Canadian lynx and snowshoe hare indicates that the stable dynamical system of lynx and hare functions persistently even if environmental nature is ruined by some degree. This is also compatible with the empirical fact that the ten-year cycle in snowshoe hare is resilient to a variety of natural disturbances from forest fires to short-term climatic fluctuations. But it has a limit as we have shown explicitly in Section 3, as recovery, disintegration and extinction phenomena. If a strong negative perturbation is applied persistently for a long period, the system would fall into danger of extinction. The important results of our simulation tell that before a system gets in danger of extinction, the standard rhythm of the system will disappear or tend to become ambiguous. Even if the population of lynx seems to be decreasing in Figure 9(d), the interacting hare species are stable as theoretically predicted in Figure 9(b), which indicates a possible recovery of the lynx species in terms of food and predation. Hence, the lynx population decrease in 1920-2018 should be concluded to be induced by other causes beside food and predation.
We cannot be sure at present what kinds of external perturbations caused the lynx population decrease, which is important to be investigated by local field researchers. A long-term (more than ten years) negative perturbation and a vast environmental change that humans could cause would definitely endanger the standard rhythm of snowshoe hare, lynx and related species. If we carefully observe the standard rhythm of specific ecological systems of species, we could help sustain the dynamics of ecological system and study specific solutions to preserve natural environment and sustainability.

Notes on Nonlinear Interactions, Lyapunov Function, Kolmogorov's Predator-Prey Model, Atto-Fox Problem
The CNI restricts the form of nonlinear interactions and reveals some characteristic properties related to nonlinear dynamics, which helps us understand biological mechanism by way of possible physical analyses. The applicability and physical meanings of nonlinear differential equations with time-dependent coefficients should be studied further for science from microscopic to macroscopic thermomechanical and biological systems.
As physical reasons of nonlinear differential equation have been studied in terms of conserving and nonconserving interactions, nonlinear differential equations with time-dependent coefficients may reveal new mechanism and improve understanding complex phenomena scientifically [36]. While nonlinear differential equations have been used for mathematical analyses and computer simulations to get coupling constants to produce big-data, one should know that nonlinear equations are just a mathematical and experimental tool for classification and systematics (taxonomy) of natural phenomena. Journal of Applied Mathematics and Physics There are unphysical and scientifically invalid problems, misunderstandings on nonlinear problems, such as the atto-fox problem [13] [14] [15] [16] [17].
Specifically, some researchers still insist that non-existing prey (negative number of prey) can be eaten by predators, and predators can increase or decrease by eating or interacting with negative prey, which comes from typical nonconserving nonlinear interactions, even with Lotka-Volterra and other general nonlinear equations. It is important to answer properties of conserving and nonconserving nonlinear dissipative problems in order to study nonlinear dynamics and some typical questions in this section.

The Conserving and Nonconserving Nonlinear Equations
It is often explained that a model of interacting populations is written in general as, , d x xf x y t y yg x y t

Lyapunov Function Produced by Noether's Theorem
We showed that the conserved quantity, Ψ-function, can reproduce Lyapunov function of the classical Lotka-Volterra equations in the previous work [5] [6] [7]. The mathematical expressions and physical meanings of ( ) , f x y , ( ) , g x y Journal of Applied Mathematics and Physics and stability in differential Equations (6.1) have been investigated by many researchers in terms of limit cycles and atractors of Lyapunov functions [47]. There are two types of Lyapunov functions, which are strict Lyapunov (having negative definite time derivative along trajectories) and non-strict Lyapunov functions (negative semi-definite time derivatives along trajectories) [47]. The notion indicates how strongly a nonlinear system approaches its equilibrium state when t → ∞ . In other words, a nonlinear interacting system in the Lyapunov type corresponds to a dissipative system or a nonconservative system leading to an equilibrium after a long time.
The systems with Lyapunov function have limit cycles and attractors, which designates energy dissipations of the system. The systems with Ψ-functions are strictly conserving systems corresponding to limit cycles at given time. It is essential to understand that Kolmogorov or Lyapunov functions for systems of differential equations can be derived from Noether's theorem when a system has an equilibrium state corresponding to certain dynamical conservation laws. The strict and non-strict Lyapunov function could be expressed by nonlinear conserving differential equation corresponding to the global property of Lagrangian approach based on conservation laws. Though Noether's theorem relates stability and conserved quantity analogous to symmetries and conservation laws of energy and momentum in physics, it is not yet clear what physical meanings the conservation laws have in nonlinear biological and ecosystems. Conservation laws are helpful to understand a system of nonlinear differential equations in view of scientific or physical terms.

A Note on the Atto-Fox (10 −18 -Fox) Problem
It should be emphasized that an irrational problem known as atto-Fox (10 −18 -Fox) problem [16] [17] in a system of nonlinear differential equations should not occur in a realistic ecosystem, or a conserved system of differential equations, because the conservation law, initial conditions and the Ψ-function restrict admissible solutions for nonlinear interacting systems. The nonlinear ordinary differential equations of the 2n-CNI system with a realistic initial condition can have stable solutions, and the solutions consist a stable closed hyper-surface of ( 1 2 2 , , , n x x x ), as in Figure 6(c) and Figure 6(d), for example. The problem is restricted by properties of conservation laws, and dissipative non-conserved systems progress to a realistic equilibrium. One can directly observe in the CNI model that the Ψ-function cannot be constant or diverge when a physical solution does not exist with given coupling constants and initial conditions [5] [6] [7].
It is known that typical Lotka-Volterra (LV) equations used in applications produce the atto-Fox (10 −18 -Fox) problem. Let us prove it by the following LV type nonlinear equation: Journal of Applied Mathematics and Physics ical computations with given coefficients as shown in Figure 11 and Figure 12.
Living animals interact with other animals through reasonably large integer numbers as we know in a common sense. The lower left of Figure 12 shows that the small numbers such as ( ) ( ) ,~1,90 Hence, the solution in Figure 11 and Figure 12 can be equivalently constructed by the solution If we choose, for example,

Answers to Some Questions on Nonlinear Equations
Some typical questions concerning difficulties on nonlinear differential equations are answered in order to elucidate applicability of the 2n-variable CNI equations.
• (Question 1) There may be many periodic solutions in the n-variable nonlinear model that depend on the values of nonlinear coefficients, 1 , α , and initial starting values, • (Answer) It is indeterminable from the beginning to know how many independent solutions a nonlinear differential equation can produce, or it could have none. This is restated such that solutions ( 1 2 , , , n x x x ) would be transformed in the phase space like Figure 12, resulting in the atto-fox problem or a physically meaningless problem, which may be correct for simple LV type x is positive and increasing, and 2 x may decrease or increase even if 1 x is negative. This means that non-existing prey can be eaten by predators, and predators can increase or decrease by eating or interacting with negative prey.
• (Answer) This is a typical conjecture derived from LV type nonlinear equations and the comment is not correct at all regarding the conserving 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 prohibited. A specific numerical example is shown in the paper [6] [7]. The solutions to n-variable CNI equations are different from those of conventional, nonconserving nonlinear equations.
It is important to stress that the atto-fox problem found in a simple Lotka-Volterra equation is not intrinsic to conserving nonlinear differential equations.
The n-variable CNI equations with external perturbations are useful to simulate real data numerically.

Conclusions on the Survival of the Fittest and Symbiosis from Conserving Nonlinear Interactions
We examined characteristic properties of several ecological systems based on CNIs which include generalized Lotka-Volterra type prey-predator, competitive interactions. The practical construction of solutions from 2-variable to n-variable CNIs is discussed in order to apply the model to more realistic biological phenomena and responses to external environmental changes. It is difficult to adjust values of coefficients for nonlinear differential equations to satisfy given initial conditions and real population ecological data. It is emphasized in the current CNI model that nonlinear differential equations are more than a convenient computer analysis of big data. When an appropriate set of nonlinear parameters in the conserving nonlinear model is found, it can be used to examine characteristic properties, such as stability and standard rhythm internally related to dynamics of a corresponding nonlinear system.
The important factors (nutrient, predation and social interactions) are needed for all species to survive in nature, but they are easily altered with environmental and natural conditions. An abnormal increase in population numbers of a species would endanger the survival of a species itself as well as other species, which could be observed by examining the population rhythm. Even if a mechanism of Journal of Applied Mathematics and Physics population increase is unknown, the CNI method would help study a cause in terms of the conservation law of the ecosystem, stability and recovering strength to external perturbations. It seems that as a predator needs a prey for its food, a prey needs a predator for the conservation of its own species. Nature simultaneously nurses and demolishes life because of the finite resources of food, energy and environment of the Earth. The conservation law and rhythm of species could be considered to have been constructed by species to survive in natural conditions for a long time with evolutionary strategies. Hence, the cycle (rhythm) of species would be interpreted as manifestation of the survival of the fittest to the balance of a biological system.
Conservation of forest-associated animals requires an understanding of habitat quality, along with the development of environmental spatio-temporal management strategies that are consistent with high-quality habitat [59]. The natural environment and habitat quality should be conserved for species, and so such conditions would manifest themselves as a dynamical rhythm of interaction and be related to conservation laws, whereas temporal and spatio-temporal conditions should be considered as external perturbations. The current conserving approximations with external perturbations help us understand cause and effect for complicated natural phenomena of ecosystems.
Nonlinear differential equations have bifurcation solutions depending on coupling strengths which cause a sudden qualitative and quantitative change into solutions, even when a small smooth change is made to parameter values (coefficients of given nonlinear differential equations) of nonlinear differential equations. In addition, nonlinear differential equations with time-dependent coefficients generate a different class of independent solutions compared to nonlinear differential equations with constant coefficients [36]. Nonlinear differential equations seem to have yet unexplored properties, which should be investigated further. The nonlinear differential equations with time-dependent coefficients would be related to the reaction-diffusion interactions of density-dependent growth and dispersal of viruses, density-dependent diffusion coefficient of cooperative phenomena and the evolutionary population dynamics [64]. The applications of CNI model to other dynamical fields will be investigated in the future.
We conclude that stability and conservation law are constructed by species in mutual dependence or cooperation to survive for long-time periods in severe nature. The standard rhythm should be regarded as the result of strategy for species to survive in nature. Therefore, the conserving nonlinear differential equations are suitable to reveal properties of complicated interacting ecological systems.
Whatever roles plants and animals have to play, species that can fit and balance 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 ecosystems constructed by other members and the environment. We hope that this study will help understand dynamical balance and activities for living animals Journal of Applied Mathematics and Physics and importance of natural environment for life.