Inter-specific competition among trees in pythagorean fuzzy soft environment

A Pythagorean fuzzy set is very effective mathematical framework to represent parameter-wise imprecision which is the property of linguistic communication. A Pythagorean fuzzy soft graph is more potent than the intuitionistic fuzzy soft as well as the fuzzy soft graph as it depicts the interactions among the objects of a system using Pythagorean membership grades with respect to different parameters. This article addresses the content of competition graphs as well as economic competition graphs like k-competition graphs, m-step competition graphs and p-competition graphs in Pythagorean fuzzy soft environment. All these concepts are illustrated with examples and fascinating results. Furthermore, an application which describes the competition among distinct forest trees, that grow together in the mixed conifer forests of California, for plant resources is elaborated graphically. An algorithm is also designed for the construction of Pythagorean fuzzy soft competition graphs. It is worthwhile to express the competing and non-competing interactions in various networks with the help of Pythagorean fuzzy soft competition graphs wherein a variation in competition relative to different attributes is visible.


Introduction
A graph is an effective tool to depict the connectivity and relationship among the objects of a system. For all the straightforward as well as complex networks can be modeled in terms of graph. These networks include the telecommunication web in computer sciences, molecular structures of organic compounds in chemistry, electronic circuits of physics, road networks in transportation engineering, food webs in ecology and cellular dynamics in biology. Thus, all pairwise real-world relationships can be illustrated with the help of graphs. The remarkable work of Swiss mathematician Euler, in 1736, for the Königsberg problem of seven bridges is considered to be the formal commencement of graph theory. Afterwards, these mathematical structures were not only applied in other disciplines but also emerged as an independent subject. Hafiza Saba Nawaz sabanawaz707@gmail.com 1 Department of Mathematics, University of the Punjab, New Campus, Lahore 54590, Pakistan A graph in which a direction is associated with every edge is called directed graph. Cohen [7] considered the model of food web as a directed graph and stated that two species (vertices) are joined by edge in the corresponding competition graph (CG), if these species must have at least one prey in common and vice versa. These CGs were then applied in various systems to depict the competition among the constituents of that system. Various authors also studied different forms of CGs like p-CGs [14], tolerance CGs [4], m-step CGs [6] and competition hypergraphs [33]. Different examples of such systems are competition among sellers for buyers in imperfect markets, among the applicants for a post, among animals for food and shelter, among plants for resources and many more. Since all these systems have uncertainty in their data, therefore it is better to handle them with fuzzy set (FS) theory and its extensions.
The FS [36] was suggested by Zadeh in 1965 as an extension of crisp set to deal with uncertainty. It is a function μ that takes elements of universal set X as an input and maps on the members of unit closed interval, that is, μ : X → [0, 1]. The value assigned to the element x ∈ X in FS is called the membership grade μ(x) of that element x and illustrates the extent of belongingness of x in this set. So FS represents the evidence for x ∈ X , but it lacks the ability to represent the evidence against x ∈ X . To address this issue, Atanassov [3], in 1983, put forward the intuitionistic fuzzy set (IFS) by inserting a new component ν : X → [0, 1] that investigates the non-membership grade such that μ + ν ≤ 1. In this way, IFS provides more flexibility as compared to FS theory due its non-null hesitation part. Yager [35] introduced another non-standard fuzzy subset, the Pythagorean fuzzy set (PFS), by relating the negation operation with Pythagorean theorem. This set assigns the membership and non-membership values so that μ 2 + ν 2 ≤ 1. He also compared the IFS and PFS and declared that the set of intuitionistic membership values is smaller than that of Pythagorean membership grades. All these extensions of crisp sets dealt with many actual-world problems that have fuzziness and uncertainty.
Another form of uncertainty was indicated by Molodtsov [21] in 1999. He gave the idea of soft set (S f S) which is a parameterized collection of subsets of universe X . It is defined as a set-valued approximate mapping from a set of parameters (or attributes which are selected according to the members of X ) to the power set of X . This set was applied to various systems by different authors [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. In order to add the flavor of fuzziness to S f S, Maji et al. [16] put forward the fuzzy soft set and also presented its properties. This hybrid model contributed to the solution of many decision making problems highlighted by Roy and Maji [28]. Maji et al. [17] introduced the intuitionistic fuzzy soft set and Peng et al. [27] put forth Pythagorean fuzzy soft sets (PFS f Ss). These hybrid models have tremendous applications in various fields.  considered Pythagorean fuzzy sets with applications in multiple criteria decision-making.
Different graphs were studied in FS theory and its related extensions later on. Kaufmann [13] presented fuzzy graph and its operations were defined by Mordeson and Nair [22]. Paravathi and Karunambigai [26] put forward intuitionistic fuzzy graphs with suitable illustrations to its idea. For the implication of Pythagorean membership grades in graphs, Naz et al. [25] introduced the Pythagorean fuzzy graphs (PFGs). They also determined various properties of these graphs, Pythagorean fuzzy preference relations as well as applications to some decision-making problems modeled in Pythagorean fuzzy environment. Mordeson et al. [23] discussed many fuzzy graph theoretic problems regarding to human trafficking.
Soft graph which is based on the structure of S f S is, in fact, a parameterized family of graphs and was suggested by Thumbakara and George [34]. Akram and Nawaz [1] combined the concept of fuzzy graphs and soft graph to put forward a new notion of fuzzy soft graphs. For the involvement of Pythagorean membership grades to the idea of soft graph, Akram and Shahzadi [32] introduced the Pythagorean fuzzy soft graph (PFS f G) and discussed its regularity.
With the passage of time, the concept of CGs was also elaborated in different theories. Samanta and Pal [31] pro-posed fuzzy CGs and some of its generalizations. The m-step fuzzy CGs [30], intuitionistic fuzzy CGs [29], q-rung orthopair fuzzy CGs [12], q-rung picture fuzzy CGs [2] and fuzzy soft CGs [24] were suggested by different authors. All these research articles include various interesting applications of CGs in social, ecological as well as economic competing network models.
There exist various competing networks that exhibit variation with respect to distinct attributes. To show the competing entities of such systems, it is necessary to include the parametrization factor to Pythagorean fuzzy competition graphs (PFCGs). The motivation of this paper is the following: 1. The PFS f S is a hybrid model of PFS and S f S, and has the combined properties of both these ground sets.  tributed, they compete with one another for these resources in order to survive. The availability of these natural resources vary from season to season so the discussion of competition in PFS f environment permits to examine competition with respect to different parameters which prevent the information loss. Moreover, the assignment of Pythagorean membership grades to the constituents of this system provide us information about both the involvement and non-involvement of trees in competition. This competition is also presented graphically. The comparative analysis is given in the following section and the concluding section completes the article. The list of acronyms is provided in Table 1.

Preliminaries
Throughout this section, we will consider X as a non-empty universal set, P as the set of parameters regarding to all objects in the universe and Q as a subset of P: represent the membership and non-membership functions, respectively, such that 0 ≤ μ 2 is the Pythagorean fuzzy index of the element x to the set A.
Definition 2 [27] A PFS f S (A, Q) over X is defined by the Pythagorean fuzzy approximation function A : Q → P(X ), where P(X ) denotes the Pythagorean fuzzy power set over X containing all possible PFSs over X . Its set notation is is follows: for all x ∈ X , is a PFS relative to parameter q. Definition 3 [32] A PFS f G G on X is an ordered 3-tuple G = (A, B, Q) such that (i) Q denotes a non-empty set of attributes, That is, the membership and non-membership values of the Pythagorean fuzzy edge (x, y) in R(q) is given by Note that (Str(q; (x, y)) μ , Str(q; (x, y)) ν ) denotes the strength of Pythagorean fuzzy edge (x, y) in the PFG R(q) = (A(q), B(q)). i.e., following inequalities hold for all q: The edge (x, y) is called weak if above-mentioned conditions are not satisfied for some q.
In other words, a PFS f edge (x, y) is called independent strong in G = (A, B, Q) if Str(x, y) μ ≥ 0.5 and Str(x, y) ν ≤ 0.5, and weak if not. and Str(y, z) = (0.92, 0.47), therefore, the PFS f edges (x, z) and (y, z) are independent strong, whereas (x, y) is weak in the PFS f G G.
for each x, y ∈ X and q ∈ Q.
) is a Pythagorean fuzzy digraph, for all q ∈ Q.
That is, the membership and non-membership degrees of a Pythagorean fuzzy arc (x, y) in − → R (q) is given by for each q and x, y ∈ X , such that the condition 0 ≤

Definition 11
The PFS f out-neighborhood N + (x) of a ver- .
is an undirected PFS f G that has same PFS f vertex set as in − → G and for all q ∈ Q, it has a Pythagorean fuzzy edge between points x, y ∈ X in S(q) = (A(q), D(q)) if and only if N + (q)(x) ∩ N + (q)(y) is nonempty set. The membership as well as non-membership grade of edges in S(q) are computed as follows: We can calculate the edge membership and nonmembership values for the corresponding has same PFS f vertex set as in − → G and for all q ∈ Q, it has a Pythagorean fuzzy edge between points x, y ∈ X in is non-empty set. The membership as well as non-membership grade of edges in T(q) are computed as follows: Fig. 4.

The edge membership and non-membership values of
In general if, in both cases, the sufficient condition holds relative to all attributes q, then the PFS f edge is said to be strong and vice versa.
give the membership and non-membership values of edge (x, y) in S(q) = (A(q), D(q)). As a consequence, we have and ν (q) (z) < 1 2 , respectively. Both these relations combine to give The strategy of the proof of (2) is similar to the above. .
Definition 14 For a non-negative real number k, the Pythagorean fuzzy soft k-competition graph The membership as well as non-membership grade of edges in S(q) are computed as follows: We can construct the corresponding A, D, Q), whose edge membership and non-membership degrees are computed as follows: is given in Fig. 5.

Definition 15
For a non-negative real number k, the Pythagorean fuzzy soft k-economic competition graph The membership as well as non-membership grade of edges in T(q) are computed as follows: We can construct the corresponding PFS f 0.5-economic CG E 0.5 ( − → G ) = (A, F, Q), whose edge membership and non-membership degrees are computed as is given in Fig. 6.
In general if, in both cases, the statement holds for all parameters q, then (x, y) is said to be independent strong PFS f edge.
= 1 2 which shows that the Pythagorean fuzzy edge (x, y) is independent strong in the Pythagorean fuzzy subgraph T(q) of E k ( − → G ). The proof to statement (1) is similar.
which has the same PFS f vertex set as in − → G and for all q ∈ Q, the Pythagorean fuzzy points for all x, y ∈ X and q ∈ Q.  Fig. 2.

Definition 17 An m-step PFS
The

Definition 18 An m-step Pythagorean fuzzy soft competition graph (m-step PFS
is an undirected PFS f G which has the same PFS f vertex set as in − → G and for all q ∈ Q, the Pythagorean fuzzy points x, y ∈ X are joined in S(q) = (A(q), D(q)) if and only if N + m (q)(x) ∩ N + m (q)(y) is nonempty set. The membership as well as non-membership grade of edges in S(q) are evaluated as follows: such that 0 ≤ μ 2 D(q) (x, y)+ν 2 D(q) (x, y) ≤ 1, for all x, y ∈ X and q ∈ Q. Fig. 8.

Example 9 The 2-step FS
The edge membership values can be calculated as

Definition 19 An m-step Pythagorean fuzzy soft economic competition graph (m-step PFS
is an undirected PFS f G which has the same PFS f vertex set as in − → G and for all q ∈ Q, the Pythagorean fuzzy points x, y ∈ X are joined in is non-empty set. The membership as well as non-membership grade of edges in T(q) are evaluated as follows: for all x, y ∈ X and q ∈ Q.

Example 10 The 2-step PFS
of Example 2, is presented in Fig. 9.
The edge membership values can be calculated as  , y), ∀x, y ∈ X . As S(q) has no edge and q is arbitrary, it is clear

Theorem 5 Let
is also void of edges. The proof for the result of E m ( − → G ) is similar to the above. x, y)for each x, y ∈ X and q ∈ Q.
To prove the statement (2) and

Definition 20 Let
be a PFS f digraph on X . Let s be a common Pythagorean fuzzy prey of m-step Pythagorean fuzzy out-neighborhoods of vertices s)) , respectively. We call the m-step Pythagorean fuzzy prey s ∈ X , a strong Pythagorean fuzzy prey in for all q, then s is termed as a strong PFS f prey.

Definition 21 Let
be a PFS f digraph on X . Let b be a common Pythagorean fuzzy buyer of mstep fuzzy in-neighborhoods of vertices x 1 , be the largest non -membership degree of edges in the Pythagorean fuzzy directed paths − → P m (q;(b,x 1 )) , , respectively. We call the m-step Pythagorean fuzzy buyer b ∈ X , a strong Pythagorean fuzzy buyer in The strength of a PFS f buyer b is denoted by where the mappings Str(q; b) μ : X → [0, 1] and Str(q; b) ν :

Example 11
Consider the PFS f digraph Fig. 2. Note that y is a common Pythagorean fuzzy prey of 2step Pythagorean fuzzy out-neighborhoods of vertices v and z in − → R (q 1 ) as well as − → R (q 2 ), and its strength is Str(y) = (0. 35, 0.71). Similarly, w is a common Pythagorean fuzzy buyer of 2-step Pythagorean fuzzy in-neighborhoods of vertices x and z in − → R (q 1 ) as well as − → R (q 2 ), and its strength is Str(w) = (0.47, 0.69).
be a PFS f digraph on a non-void set X .

If a PFS f prey s of
− → G is independent strong then the strength of s satisfies Str(s) μ > 0.5 and Str(b) ν < 0.5.

If a PFS f buyer b of
− → G is independent strong then the strength of b satisfies Str(b) μ > 0.5 and Str(b) ν < 0.5.
Since q is arbitrary, 0.5 < min q Str(q; s) μ = Str(s) μ and 0.5 > max q Str(q; s) ν = Str(s) ν . The proof to statement (2) is similar. A, D, Q) which has the same PFS f vertex set as in − → G and for all q ∈ Q, the Pythagorean fuzzy points x, y ∈ X are joined in S(q) = (A(q), D(q)) if and only if for each q ∈ Q |supp(N + (q)(x) ∩ N + (q)(y))| ≥ p. The membership as well as non-membership grade of edges in

Definition 23
For a positive integer p, the p-economic competition Pythagorean fuzzy soft graph ( p-economic compe- which has the same PFS f vertex set as in − → G and for all q ∈ Q, the Pythagorean fuzzy points x, y ∈ X are joined in T(q) = (A(q), F(q)) if and only if for each q ∈ Q |supp(N − (q)(x) ∩ N − (q)(y))| ≥ p. The membership as well as non-membership grade of such that 0 ≤ μ 2 F(q) (x, y) + ν 2 F(q) (x, y) ≤ 1, for all x, y ∈ X and q ∈ Q, where n = |supp(N − (q)(x) ∩ N − (q)(y))|.

Example 13
Consider again the PFS f digraph − → G = (A, − → B , Q) of Example 12. In order to have a 2-economic competition PFS f G, we can compute the edge membership and nonmembership degrees as Fig. 12.

Competition among forest trees
Competition among organisms is a universal phenomenon that varies with respect to the factors involved in their development and growth. Its importance lies in the fact that severe competition among the species of a community can alter its structure. Competition inspires human beings to enhance their performance by putting more effort that brings satisfaction of doing things better. They usually compete for food, riches, prestige and mates. Animals and plants also struggle and compete with their own specie members as well as with the individuals of other species for their survival. Competition among the individuals of a specie is termed as intra-specific competition, whereas that within different species is called inter-specific competition.
Plants compete slowly which often went on for years as compared to animals. The resources that are essential for the survival and development of a plant are not necessarily available in sufficient quantity in the environment where it has to grow. There are various resources whose abundance or shortage influence the plants to compete for their survival. These are: (i) light (when one plant is shaded by nearby plants), (ii) water (when the dampness in soil is suboptimal), (iii) heat (when radiant energy is not available in cold areas), (iv) space (when the seeds sown are dense), (v) nutrients (when nutrients in soil are not ideally distributed), (vi) carbon dioxide (during vigorous photosynthesis in dense communities), and (vii) oxygen (as required by roots) [20]. Plants require nutrients throughout the year which include the macronutrients like nitrogen, potassium, phosphorus, calcium, sulfur and magnesium as well as micronutrients such as manganese, copper, iron, zinc etc., but their amounts needed are different for distinct plants. A plant can acquire nutrients from atmosphere, native soil nutrients, weathering of rocks and dropped litter of plants. Especially, nitrogen-fixing bacteria provide lots of nitrogen in soil. The amount of these resources in the environment of plants varies from season to season.
Consider the example of forest trees found in the mixed conifer forests of California. The climate of California changes to a great extent from burning hot to freezing cold. Some mixed conifer forests of California are found on the mountains of Sierra Nevada. In these mountains, the climate at the beginning of year is freezing cold which gradually changes to a pleasant weather during the early months of summer, hot during the summer season and then again turns to snowy winter at the end. During winter season in mixed conifer forests, the plants vigorously require water, sunlight and heat. Winter in these forests is so white that leaves and limbs of some trees cannot bear the snow load and thus fall while the other trees that have flexible branches shed heavy snow and thus survive. However, they require more space and light in summer because of regrowth of new limbs and leaves which steal the light as well as space of the relatively small or the new growing trees. Moreover, the plants that are shade-intolerant like Ponderosa pine which die in the constant absence of light require sunlight in all seasons. Similarly, white fir which is drought-intolerant need sufficient water in each season for its survival and development [15].
Consider a set {Ponderosa pine, Sugar pine, White fir, Incense cedar, Douglas-fir, Red fir, Lodgepole pine, Jaffrey pine} of forest trees and a set {Water, Nutrients, Sunlight, Heat, Space} of resources that are required for the development of the considered trees. Also, consider a set {q 1 , q 2 } of attributes where q 1 symbolizes winter season and q 2 symbolizes the summer season. A PFS f digraph that presents the forest trees and their interaction with resources is given in Fig. 13. With regards to the considered parameter (season), the membership as well as non-membership grade of (a) trees represent their growth and growth inhibition, (b) resources denote the resource availability and shortage, and (c) directed edges narrates the accessability and inaccessability of resource to tree, in that season. For example, in − → R (q 1 ), the directed edge from the Ponderosa pine to space has degree (0.82, 0.21) which shows that the space available to plant is 67.24%, not available is 4.41%, and the hesitation is 28.35%. Table 2 represents the PFS f out-neighborhoods of forest trees for growth resources.
The competition among trees for resources relative to winter (by S(q 1 )) and summer (by S(q 2 )) seasons is shown in Fig. 14 as a PFS f CG. The edge joining Ponderosa pine and Lodgepole pine has degree (0.4290, 0.0630) in S(q 1 ) which narrates that they are competing (for sunlight, heat as well as space) with strength 18.4% and are not involved in competition 0.4%. Table 3 gives the membership and nonmembership values of PFS f edges of PFS f CG. Note that the Pythagorean fuzzy subgraph S(q 1 ) is more dense than that of S(q 2 ). It is because of the fact that the resources are more depleted in winter as compared to summer, so those plants which are not tolerant to deficiency of a resource will not be able to survive and may pass by death.
The Algorithm 41 presents the procedure of the construction of PFS f CG for the PFS f digraph. It takes the PFS f digraph and for each of its parameter, computes the PFS f out-neighborhoods corresponding to each tree. The heights of intersecting vertex neighborhoods and the vertex degrees give rise to the strength of competing and non-competing interactions among the forest trees.

Comparative analysis
The PFS f S proposed by Peng et al. [27] is very assistive in order to deal with the parameterized family of PFSs. The competition in problems that have parameterized Pythagorean uncertainty should be dealt with PFS f CGs; oth- x i s include resources c j , 1 ≤ j ≤ s and forest trees f k , 1 ≤ k ≤ r. Output: Competing and non-competing strength among forest trees f ku , 1 ≤ k u ≤ r − 1 and f kv , k u + 1 ≤ k v ≤ r.
procedure Strength of competing and non-competing interaction erwise, the competing and non-competing interactions could be inappropriate. We consider the PFG to present a comparison with the above discussed example of plant competition.   If we consider this system as to depict the annual interaction of forest trees and resources, the resulting PFCG would be incorrect. Its reason is that the resources that are responsible for the growth promotion and inhibition of plants do not remain same throughout the year in the surroundings of trees. Due to this reason, the competition among plants is not constant and varies from season to season. So, this system should be studied in the PFS f environment so that the forest management can take measures according to each season for the growth of plants. The analysis through PFS f CGs will also be helpful to understand that the resource for which most of the trees are competing is deficient and should be provided in sufficient amounts.

Conclusion
We often experience different types of competitions in our daily life that can be modeled as networks or graphs. The CGs of directed graphs suggested by Cohen [7] are effective to view the competing entities of system. The presence of uncertainty in actual-world problems intended the researchers to illustrate CGs in different theories. The PFS f theory is a good blend of PFS and S f S, which permits the assignment of Pythagorean membership values with respect to different parameters. The purpose of this article is to present competition as well as economic CGs in the PFS f environment. Additionally, we have discussed PFS f k-CGs which are, in fact, the PFS f CGs at level k, where k is a non-negative real number. The m-step PFS f CGs describe the competition among organisms for their m-step common preys. The edges in p-competition PFS f Gs highlight all those pairwise competitors that have at least p-common preys. We have explained all these variations of PFS f CGs through examples and also provided some interesting results. All these generalizations of PFS f CGs have significance at their own as they can represent different forms of competition in a parameter-dependent system. An example of competing forest trees for growth resources have also been discussed using PFS f CGs. The considered set of forest trees is common in the mixed conifer forests of California. The seasonal variations are responsible for changing concentrations of resources in the environment, so it is better to study such competitions in PFS f theory. An algorithm which calculates the strength of competing and non-competing interactions of the competing constituents of a system have also been given. The future research directions might address various topics. It is desirable to discuss (1) Pythagorean fuzzy soft competition hypergraphs, (2) Pythagorean fuzzy soft graph structure and (3) Interval-valued Pythagorean fuzzy competition graphs.

Conflict of interest
The authors declare that they have no conflict of interest regarding the publication of the research article.
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