Complex systems may be mechanistically modelled by white-box modeling with using logical deterministic individual-based cellular automata. Mathematical models of complex systems are of three types: black-box (phenomenological), white-box (mechanistic, based on the first principles) and grey-box (mixtures of phenomenological and mechanistic models). Most basic ecological models are of black-box type, including Malthusian, Verhulst, Lotka–Volterra models. In black-box models, the individual-based (mechanistic) mechanisms of population dynamics remain hidden. Here we mechanistically model the S-shaped and double S-shaped population growth of vegetatively propagated rhizomatous lawn grasses. Using purely logical deterministic individual-based cellular automata we create a white-box model. From a general physical standpoint, the vegetative propagation of plants is an analogue of excitation propagation in excitable media. Using the Monte Carlo method, we investigate a role of different initial positioning of an individual in the habitat. We have investigated mechanisms of the single-species population growth limited by habitat size, intraspecific competition, regeneration time and fecundity of individuals in two types of boundary conditions and at two types of fecundity. Besides that, we have compared the S-shaped and J-shaped population growth. We consider this white-box modeling approach as a method of artificial intelligence which works as automatic hyper-logical inference from the first principles of the studied subject. This approach is perspective for direct mechanistic insights into nature of any complex systems.
A mechanistic approach corresponds to the classical ideal of science. Existing mathematical approaches to complex systems modeling are rather phenomenological than mechanistic. Ecologists investigate population dynamics phenomenologically, rather than mechanistically (
How to create an individual-based mechanistic model of population growth? First, we need to know how to mechanistically model a complex dynamic system. A complex dynamic system may be considered as consisting of interacting subsystems. Interactions between subsystems lead to the emergence of new properties, e.g., of a new pattern formation. Therefore we should define these subsystems and logically describe their interactions in order to create and investigate a mechanistic model. If we want to understand how a complex dynamic system works, we must understand cause–effect relations and part-whole relations in this system. The causes should be sufficient to understand their effects and the parts should be sufficient to understand the whole. There are three types of possible models for complex dynamic systems: black-box, grey-box and white-box models (
This is a schematic representation of a black-box model, a grey-box model and a white-box model with the level of their mechanistic understanding.
Black-box models are completely nonmechanistic. They are phenomenological and ignore a composition and internal structure of a complex system. We cannot investigate interactions of subsystems of such a non-transparent model. A white-box model of complex dynamic system has ‘transparent walls’ and directly shows underlying mechanisms. All events at micro-, meso- and macro-levels of a dynamic system are directly visible at all stages of its white-box model. Unfortunately, in most cases mathematical modelers prefer to use the heavy black-box mathematical methods, which cannot produce mechanistic models of complex dynamic systems in principle. Grey-box models are intermediate and combine black-box and white-box approaches. As a rule, this approach is used in ‘overloaded’ form, what makes it less transparent. Basic ecological models are of black-box type, e.g. Malthusian, Verhulst, Lotka–Volterra models. These models are not individual-based and cannot show features of local interactions of individuals of competing species. That is why they principally cannot provide a mechanistic insight into dynamics of ecosystems. Earlier, we demonstrated that the logical deterministic cellular automata approach allows to create the white-box models of ecosystems with interspecific competition between two, three and four grass species (
Creation of a white-box model of complex system is associated with the problem of the necessity of an a priori basic knowledge of the modeling subject. The deterministic logical cellular automata are necessary but not sufficient condition of a white-box model. The second necessary prerequisite of a white-box model is the presence of the physical ontology of the object under study. The white-box modeling represents an automatic hyper-logical inference from the first principles because it is completely based on the deterministic logic and axiomatic theory of the subject. The purpose of the white-box modeling is to derive from the basic axioms a more detailed, more concrete mechanistic knowledge about the dynamics of the object under study. We see no other way to obtain a specific and, at the same time, holistic mechanistic understanding of complex systems, apart from the white-box modeling. For providing a strong relevance of our model to the studied problem, we have specified the model’s rules (axioms) strictly in accordance with the subject under study. Each logical rule of the model has a correct ecological and physical interpretation. From an ecological point of view we model a vegetative propagation of rhizomatous lawn grasses. From a physical point of view we model propagation of excitation (autowaves, travelling waves, self-sustaining waves) in an excitable (active) medium. The presence of such physical interpretation makes our specific ecological model more general and more natural. The necessity to formulate an intrinsic axiomatic system of the subject before creating its white-box model distinguishes the cellular automata models of white-box type from cellular automata models based on arbitrary logical rules. If cellular automata rules have not been formulated from the first principles of the subject, then such a model may have a weak relevance to the real problem.
Let’s consider an example of the inadequacy of some ecological models in result of their incompleteness or incorrectness. There are many models of population dynamics that do not take into account what happens with individuals after their death. Dead individuals instantly disappear with roots, stubs, etc. “
Stephen Hubbell in his Unified Neutral Theory of Biodiversity (UNTB) in fact refuses a mechanistic understanding of interspecific competition: “
A vegetative propagation of rhizomatous lawn grasses is the biological prototype of our model (
A logical deterministic individual-based cellular automata model of an ecosystem with one species shows both population dynamics and pattern formation. The lattice consists of 25 × 25 sites. Individuals use the hexagonal neighborhood for propagation. The lattice is closed on the torus to avoid boundary effects. (A–C) Population dynamics of the species. S-shaped population growth curve (C). (D–F) Spatio-temporal patterns of the model are represented in numerical form of program implementation.
We have used logical deterministic individual-based cellular automata to model the S-shaped population growth mechanistically (
The presented cellular automata model is defined by the 4-tuple:
a cellular automata lattice, uniting a collection of sites;
a finite set of possible states for each lattice site;
a cellular automata neighborhood which consists of a site and its intrinsically defined neighbors;
a function of transitions between the states of a lattice site.
The best example of a white-box mechanism is a mechanical watch. Our model metaphorically resembles a mechanical watch in transparent case. A neighborhood logically binds dynamics of all cellular automata sites into one holistic complex dynamic system. There are three most known cellular-automata neighborhoods: von Neumann, Moore and hexagonal. The neighborhood may be of any type. Here we use the hexagonal and tripod neighborhoods which allow to model aggressive and moderate vegetative propagation of rhizomatous lawn grasses (
A cellular automata neighborhood models a vegetative propagation of plants and defines fecundity and spatial positioning of an individual’s offsprings. Positioning of offsprings is explained by how the cellular automata neighborhood is implemented successively for each lattice site. A central site of the neighborhood is defined by the array element with index (
Integration of reductionist and holistic approaches is one of the challenges for mathematical modeling. Our white-box model of single-species population dynamics opens up new possibilities to solve this challenge. This mechanistic model is hierarchically subdivided into micro-subsystems, meso-subsystems and the whole macro-system. A micro-level is modeled by lattice sites (cellular automata cells). A meso-level of local interactions of micro-objects is modeled by the cellular automata neighborhood. A macro-level is modeled by the entire cellular automata lattice. This is a ‘multy-level’ modelling as parallel logical operations performed on micro-level, meso-level and macro-level of the model. A unique feature of the cellular automata is the possibility to model part-whole relationships mechanistically. The relationships of the parts and the whole are modelled using the transition function (combination of the neigbourhood and rules of transition) between states of a lattice site. Parts are the lattice sites and the whole (ecosystem) is the lattice. On each iteration of evolution of the modeled macrosystem the states of its microsystems are changing simultaneously on the basis of logical ruless taking into account states of the neighbouring microsystems (neigbourhood’s sites). This allows to model how interactions of microsystems (parts) produce evolution of the macro-system (whole) which leads to emergence of its new properties (the ecosystem patterns). The white-box cellular automata model shows interactions of parts within the whole, i.e., ‘part-whole’ relations in the modeled complex system.
Directed graph of transitions between the states of a lattice site is represented in pictorial (A) and numerical forms (B). The graph represents a birth-death-regeneration process.
Here we show a description of the states of a lattice site (microecosystem) in the single species population growth model. Each site may be in one of the four states 0, 1, 2, 3 (
0—a free microhabitat which can be occupied by an individual of the species;
1—a microhabitat is occupied by a living individual of the species;
2—a regeneration state of a microhabitat after death of an individual of the species;
3—a site in this state represents an element of the boundary that cannot be occupied by an individual.
A free microhabitat is the intrinsic part of environmental resources per one individual and it contains all necessary resources for an individual’s life. A microhabitat is modeled by a lattice site.
The cause–effect relations are logical rules of transitions between the states of a lattice site (
0→0, a microhabitat remains free if there is no one living individual in its neighborhood;
0→1, a microhabitat will be occupied by an individual of the species if there is at least one individual in its neighborhood;
1→2, after death of an individual of the species its microhabitat goes into the regeneration state;
2→0, after the regeneration state a microhabitat will be free if there is no one living individual in its neighborhood;
2→1, after the regeneration state a microhabitat will be occupied by an individual of the species if there is at least one individual in its neighborhood;
3→3, a site remains in this state, which defines a boundary site.
These logical statements are realized for all micro-levels (sites) with their meso-levels (neighborhoods) and thus for the whole macro-level (lattice) of the complex system on each time iteration. We consider implementation of this algorithm as hyper-logical operations or automatic hyper-logical inference from the first principles of the studied subject.
According to
Different initial conditions may lead to formation of different spatio-temporal patterns and as a result they may lead to different dynamics of the system. Using the Monte Carlo method, we have investigated the influence of different initial conditions on population dynamics of one species. We have investigated two different boundary conditions, two different cellular automata neighborhoods and four different lattice sizes (
Investigation of the influence of boundary conditions, initial positioning of an individual and lattice sizes on single-species population dynamics. (A–D) The lattice is closed on the torus to avoid boundary effects. (E–H) The lattice has a boundary. Every Monte Carlo simulation consisted of 100 repeated experiments with different initial positioning of an individual on the lattice.
Investigation of the influence of boundary conditions, initial positioning of an individual and lattice sizes on single-species population dynamics. (A–D) The lattice is closed on the torus to avoid boundary effects. (E–H) The lattice has a boundary. Every Monte Carlo simulation consisted of 100 repeated experiments with different initial positioning of an individual on the lattice.
Periodic fluctuations in numbers of individuals are observed at the plateau phase in most of the experiments. With increasing of the lattice size, these periodic fluctuations in population size become less visible. The periodic fluctuations on the plateau phase are absent when the lattice consists of 3 × 3 sites in the case of the tripod neighborhood (
We show four
In more detail individual-based mechanisms of the double S-shaped population growth curve are presented in
The lattice size which is available for occupation consisted of 50 × 50 sites in all four cases. (A) S-shaped curve with short phase of decelerating growth. Cellular automata neighborhood is hexagonal and the lattice is closed on the torus (
In
We have investigated the S-shaped population growth which is limited by following factors: finite size of the habitat (limited resources), habitats’ size, type of boundary conditions of habitat, intraspecific competition, lifetime of individuals, regeneration time of microhabitats, fecundity of individuals (
Propagation of individuals occurs in the absence of intraspecific competition and any restrictions on the resources. A species colonizes an infinite ecosystem under ideal conditions. (A) The number of offsprings per individual equals three. (B) The number of offsprings per individual equals six. (C) Geometric population growth in the first case (A). (D) Geometric population growth in the second case (B).
To assess the effect of intraspecific competition and regeneration of microhabitats on population growth, we have compared our model of the S-shaped (
Time (Number of iteration and generation) | 0 | 1 | 2 | 3 | 4 | 5 |
Number of individuals in the S-shaped population growth model ( |
1 | 6 | 13 | 24 | 37 | 54 |
Number of individuals in the J-shaped population growth model ( |
1 | 6 | 36 | 216 | 1,296 | 7,776 |
Time (Number of iteration and generation) | 0 | 1 | 2 | 3 | 4 | 5 |
Number of individuals in the S-shaped population growth model ( |
1 | 3 | 6 | 10 | 15 | 21 |
Number of individuals in the J-shaped population growth model ( |
1 | 3 | 9 | 27 | 81 | 243 |
The basic ecological model, which has been presented in this paper, can easily be expanded by the introduction of additional states, different neighborhoods, nested and adjoint lattices (
We have presented and investigated a mechanistic model of dynamics of single species plant population. This model is based on pure logical deterministic individual-based cellular automata. It has a physical and ecological ontology. Here the physical ontology is the ontology of the active medium and ecological ontology represents an ecosytem with one vegetatively propagated plant species. We investigated deterministic individual-based mechanisms underlying the S-shaped and double S-shaped population growth of vegetatively propagated plants. Imitating modeling of vegetatively propagated rhizomatous lawn grasses was not our main goal. The main goal was demonstration of possibilities of the white-box modeling on example of population growth. The white-box model was made on the basis of physical axioms of excitation propagation in excitable medium. These basic physical axioms of the model have a universal character that, in principle, allows transferring the obtained results to other subject areas. An additional important result is itself demonstration of the white-box modeling of complex systems using logical cellular automata. We consider the details of the “white-box modeling” methodology as the main results of our work. We would like to make this perspective approach more widely used in the practice of mathematical modeling of complex systems. And we have tried to supplement the discussion about “the value of white boxes” by considering specific ways of implementation this model approach. Our study directly introduces the white-box approach into ecological modeling. The white-box approach opens up new perspectives in modeling by implementing a multy-level mechanistic modeling of complex systems. Our deterministic logical cellular automata model works as a system of artificial intelligence. Cellular automata are known as the method of artificial intelligence. But there is a problem how to use this method for investigation of complex systems. We show how logical deterministic cellular automata may be used for mathematical white-box modeling of complex systems on example of ecosystem with one species. Parallelism of the logical operations of cellular automata in total volume of the modeled macrosystem allows to speak that the model hyper-logically provides automatic deductive inference. The term ‘deductive’ is used here because all logical operations are based on axioms. We consider that the main difficulty of this white-box modeling is to create an adequate axiomatic system based on an intrinsic physical ontology of the complex system under study. The main feature of the approach is the use of cellular automata as a way of linking semantics (ontology) and logic of the subject area. Our white-box model of an ecosystem with one species combines reductionist and holistic approaches to modeling of complex systems. We consider the white-box modeling by logical deterministic cellular automata as a perspective way for investigation of any complex systems.
Every Monte Carlo simulation consisted of 100 repeated experiments with different initial positioning of an individual on the lattice. Here are shown five repeated experiments. The lattice is uniform, homogeneous and limited. It consists of 23 × 23 sites available for occupation by individuals. The lattice is closed on the torus and the neighborhood is hexagonal.
Every Monte Carlo simulation consisted of 100 repeated experiments with different initial positioning of an individual on the lattice. Here are shown five repeated experiments. The lattice is uniform, homogeneous and limited. It consists of 23 × 23 sites available for occupation by individuals. The lattice has a boundary and the neighborhood is hexagonal.
Every Monte Carlo simulation consisted of 100 repeated experiments with different initial positioning of an individual on the lattice. Here are shown five repeated experiments. The lattice is uniform, homogeneous and limited. It consists of 23 × 23 sites available for occupation by individuals. The lattice is closed on the torus and the neighborhood is tripod.
Every Monte Carlo simulation consisted of 100 repeated experiments with different initial positioning of an individual on the lattice. Here are shown five repeated experiments. The lattice is uniform, homogeneous and limited. It consists of 23 × 23 sites available for occupation by individuals. The lattice has a boundary and the neighborhood is tripod.
The lattice size which is available for occupation consists of 50 × 50 sites. Cellular automata neighborhood is hexagonal and the lattice is closed on the torus.
The lattice size which is available for occupation consists of 50x50 sites. Cellular automata neighborhood is hexagonal and the lattice has a boundary.
The lattice size which is available for occupation consists of 50x50 sites. Cellular automata neighborhood is tripod and the lattice is closed on the torus.
The lattice size which is available for occupation consists of 50x50 sites. Cellular automata neighborhood is tripod and the lattice has a boundary.
We thank Kylla M. Benes for helpful suggestions and edits of the earlier version of the manuscript. We would like to thank the Academic Editor Markus Dahlem and the anonymous reviewers for fruitful comments.
The authors declare there are no competing interests.