Modelling Living Processes with Nonlinear Dynamics

1. Terminology

For clarity, the following terminologies are presented:

• Thermodynamics [31,32]:

  (a)

  System: A specific part (generally small) of the universe under consideration.

  (b)

  Surroundings (or Environment): All external elements that can interact with the system.

  (c)

  Universe: The total combination of the system and its surroundings.

• Biology [33]

  (a)

  Animate processes: Processes that emphasize mobility, behavior, and sensory responses.

  (b)

  Living processes: Processes that emphasize fundamental life functions and biochemical activities.

As our theory is more closely related to systems that reduce entropy, the term “living process” is more appropriate [34,35]

2. Logistic Map

The logistic map is $x_{n+1}=R{x}_n\left(1-x_n\right)$ where R is a parameter determining the nature of the map, regular or chaotic [36,37]. In our approach, living processes are characterized by a system that converges for a predetermined fixed value, denoted as C. For that reason, we express the logistic map as follows:

$$x_{n+1}=x_n{\displaystyle \frac{1-x_n}{1-C}}$$

(1)

We are interested in a model with at least two adjustable parameters to define diverse life and animate processes. A single parameter characterizes the logistic model and is, therefore, inappropriate. In the next section, we present a new, more adjustable model.

3. Nonlinear Recursive Model

In the system-environment interaction cycle, two processes co-occur through a regular interaction, thereby producing the map striving for a single value (type 1) and a process that drives the system into chaos (type 2). We associated inanimate systems with convergence toward a chaotic state, whereas living processes were characterized by stabilization at a constant value. The nonlinear recursive relation was designed such that the constant value could be predicted as far in advance as possible.

3.1. Regular process-living process

We formulate the following recursive relation describing a type 1-interaction:

$$x_{n+1}={\left[C^p-{\left(R{x}_n\left(C-x_n\right)\right)}^p\right]}^{{\displaystyle \frac{1}{p}}}$$

(2)

where x is the term life parameter, with $x=C$ indicating that the life parameter is in the best possible life status. Further, n is the time (e.g., to years), and p is another parameter that enables better control in adjusting the parameters to different biological phenomena, such as adjusting the lifetime characteristics of varying living species. In the simulations conducted in this study, we used the simplest model; that is, $p=1.$

If, during the iteration, $x=C$, the system stabilizes at that predefined value. Thus, the system is designed to reach the best life parameter value. The parameter R defines the map type, characterizing the process type. Numerical calculations, illustrated in Figure 1 and Figure 2, demonstrate that the final value, once achieved, remained as expected regardless of the initial condition.

3.2. System with both interactions

The second type of interaction involves the following modifications:

• We hypothesize a gradual increase in R. This adjustment facilitates two primary effects:

  (a)

  The system progresses toward chaotic behavior. Associating this increase in chaos with increasing entropy suggests that the increased R aligns with the second law of thermodynamics.

  (b)

  Beyond a certain threshold of R, the map becomes mathematically undefined (as observed in other maps, such as logistic maps, where for $R>4$, the map becomes mathematically undefined [36,37]). We interpret this critical condition as indicative of the process of death.

• The influence of the environment is modeled by introducing slight randomness in x and C.

A plot of x during the years is referred as to life plot, as shown in Figure 3 with the parameters $R_0=2.5$ and $R_{n+1}-R_n=0.002$. $C=1.5\pm 0.0075$ uniformly distributed, $p=1.2$ and $x_0=0.55$. We identify four distinct phases:

• Evolving section:
  For $n=0-8$, x approaches the constant value $c=1\pm 0.0075$.
• Sustainable zone:
  For $n=8-25$x is oscillating randomly around $c=1\pm 0.0075$.
• Aging:
  For $n=30-60$, x represents a process wherein any disorder increases.
• Death:
  For $n>70$ the system becomes mathematically undefined. If the progression of x toward a constant value signifies a living process, the absence of a definable evolution of x, whether decreasing, increasing, or stable, indicates that this living process has ceased to exist.

4. Example for Adjusting Parameters to Describe Life Process Type

In this study, we proposed a nonlinear map characterized by the parameters R and C. To each of them, we added a random component describing the influence of the environment. We considered it more appropriate to use the current formalism to describe living processes rather than combining all processes describing the animal itself. However, to simplify the example, we describe the life curve of a dog. In Figure 3, we demonstrated a life curve that fits the description of a human who has lived for approximately 70 years. The parameters to describe different animal species could be adjusted. For example, considering a specific dog with a life expectancy of 14.5 years, we adjusted the parameters as $R=2.5$ and $C=1$, as presented in Figure 4.

5. Summary

This study formulated a nonlinear recursion equation based on established characteristics of living processes; specifically, their capacity to reduce entropy locally while interacting with the environment, thus upholding the second law of thermodynamics. This equation facilitates the depiction of a living system that operates in compliance with physical laws as a spontaneous process. The proposed mathematical model offers several advantages for the scientific description of phenomena [38,39,40]. The benefits pertinent to our research are as follows:

• Conceptual Understanding: The application of nonlinear dynamics to the study of living processes yields insights that are often not discernible through qualitative analysis alone.
• Simulation: The proposed model facilitates the simulation of unobserved life forms. Although our current study neglects the bifurcation phenomenon common to nonlinear systems, the identification of living processes that exhibit this behavior could significantly contribute to the field and refine our model.
• Theoretical framework. The model provides a robust theoretical framework that assists in hypothesis generation, experimental design, and data interpretation. In our approach, various life processes were classified according to distinct nonlinear mappings. The evolution of a system within a specific category is determined by the numerical values of the mapping parameters. In the case of our mode, this may indicate characteristics such as the lifespan of a process.