Stability and domination exponentially in some graphs

Betül ATAY ATAKUL; Betül ATAY ATAKUL

doi:10.3934/math.2020325

AIMS Mathematics

2020, Volume 5, Issue 5: 5063-5075. doi: 10.3934/math.2020325

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Research article

Stability and domination exponentially in some graphs

Betül ATAY ATAKUL ^,

Department of Computer Education and Instructional Technology, Aǧri İbrahim Çeçen University, 04100, Agri, Turkey

For a graph

$G = (V, E)$ and the exponential dominating set

$S\subseteq V(G)$ of

$G$ such that

$\sum_{u \in S}(1/2)^ {\overline{d}(u, v)-1}\geq 1$ ,

$\forall v\in V(G)$ , where

$\overline{d}(u, v)$ is the length of a shortest path in

$\langle V(G)-(S-\{u\}) \rangle$ if such a path exists, and

$\infty$ otherwise, the minimum exponential domination number,

$\gamma_{e}(G)$ is the smallest cardinality of

$S$ . The minimum exponential domination number can be decreased or increased by removal of some vertices from

$G$ . In this paper, we continue to study on exponential domination number and stability of some graphs. We consider

$\gamma_{e}^{+}$ and

$\gamma_{e}^{-}$ stability of the lollipop graph

$L_{m, n}$ , the comet graph

$C_{m, n}$ , the sunflower graph

$SF_{n}$ , the helm graph

$H_{n}$ , the diamond-necklace graph

$N_{n}$ , the diamond-bracelet graph

$B_{n}$ and the diamond-chain graph

$L_{n}$ to give us an idea about the resistance of these graphs.

Keywords:

Citation: Betül ATAY ATAKUL. Stability and domination exponentially in some graphs[J]. AIMS Mathematics, 2020, 5(5): 5063-5075. doi: 10.3934/math.2020325

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Abstract

For a graph

$G = (V, E)$ and the exponential dominating set

$S\subseteq V(G)$ of

$G$ such that

$\sum_{u \in S}(1/2)^ {\overline{d}(u, v)-1}\geq 1$ ,

$\forall v\in V(G)$ , where

$\overline{d}(u, v)$ is the length of a shortest path in

$\langle V(G)-(S-\{u\}) \rangle$ if such a path exists, and

$\infty$ otherwise, the minimum exponential domination number,

$\gamma_{e}(G)$ is the smallest cardinality of

$S$ . The minimum exponential domination number can be decreased or increased by removal of some vertices from

$G$ . In this paper, we continue to study on exponential domination number and stability of some graphs. We consider

$\gamma_{e}^{+}$ and

$\gamma_{e}^{-}$ stability of the lollipop graph

$L_{m, n}$ , the comet graph

$C_{m, n}$ , the sunflower graph

$SF_{n}$ , the helm graph

$H_{n}$ , the diamond-necklace graph

$N_{n}$ , the diamond-bracelet graph

$B_{n}$ and the diamond-chain graph

$L_{n}$ to give us an idea about the resistance of these graphs.

1. Introduction

Biodiversity loss is a major global environmental issue ^[1]. The main reasons of the loss include habitat degradation, overexploitation and invasion, among other threats ^[2]. Now, invasion is considered as one of the major threats to global biodiversity and human livelihoods ^[3].

Recently, concerns about the potential invasion risks of alien species and the impacts on biodiversity have increased rapidly ^[4,5,6], because some non-native species can bring about serious problems for ecosystems, socio-economic and human health ^[7,8]. Some invasive species will compete for resources with native species ^[9], feed on native species, lead to the extinction of many native species ^[10,11], reduce local biodiversity ^[12,13], and result in ecosystem degradation ^[14]. Besides ecological impacts, economic costs are also high ^[15]. For example, the imported red fire ants invaded the United States destroyed local agriculture and environmental hygiene, causing huge economic losses ^[16]. The invasion of new pathogens also has a negative impact on human health ^[17,18]. The number of invasive species will increase if no measures are taken, and the negative impact will be serious. At present, ways like blocking incoming, mechanical processing, spraying drugs, and biological control are used to control invasion. This study will focus on the method of biological control.

Biological control is a method of using biological interactions to inhibit or eliminate harmful species, which include microbial control, parasitic natural enemy control and predatory natural enemy control ^[19]. The application of drugs left residues, while blocking incoming and mechanical processing are impractical on large scale biological invasion, but the introduction of natural enemies for invasive species can reproduce themselves and control pests for a long time, with long-lasting effects and relatively low costs.

Biological invasion happens quite often in recent years, and biological control was successful in some cases. Cases in point include: Locusts and planthoppers were controlled by releasing ducks in sweet potato and rice fields ^[20]; The introduction of Chrysoperla sinica Tjeder to control red spiders in fruit groves and agricultural pests in vegetable sheds ^[21]; Chouioia cunea Yang are stocked in garden to curb the spread of Hyphantria cunea by species interactions ^[22]. It should be noted that when applying biological control methods, in-depth researches on the introduced species need to be conducted to ensure that they do not become new harmful invasive species, making selection of control species a very important issue.

There are studies on biological control ^[23,24], but most ignore the spatial structure of populations. Pair approximation (PA) is used in this study to investigate the invasion control theory with the spatial structure of species in consideration. Compared with the mean-field approximation (MFA), PA, which is often applied to biological phenomena such as population dynamics and epidemic spread ^[25,26], and which has obtained good results by its theoretical insights, considers the relationship between adjacent individuals and constructs a set of closed autonomous differential equations to represent the transfer process between patches.

Many factors affect the results of species control, one of which is the timing of the control. Previous research has shown that once invasions are identified, even if they have not yet had an impact, preventive action should be taken. And if potential harms exist, they should be eliminated immediately ^[27]. A case in point is, only 17 days after the discovery of invasive algae in California coastal waters, measures were taken ^[28]. Previous studies also discover that species omnivore is one of the factors affecting species density ^[29,30,31]. Both predation preference and control starting time will be considered in this research.

Different dispersal patterns can alter the distribution of species in surrounding habitats, and previous studies have shown that dispersal patterns affect the density of species ^[25,32]. Whether the dispersal strategy of species will affect the control result is a question that have drawn much attention. Some studies have found that initial density may be an important determinant of competitiveness and may influence interactions between species ^{[33,34,35,36]}. Initial density of the species may also affect the control outcome. It is generally believed that when species invade new habitats due to human transportation or boat fishing, the invader density is small and scattered while the native species density is high. Few studies have explored the effects of initial species density and spatial distribution on control outcomes.

We establish relevant models to explore biological control in this study, considering the situation of one native species and one invasive species. First, the effect of invasive species on native species was studied by pair approximation. Then, control species were introduced at different time nodes to study the effects of time introduction, predation preference, and dispersion strategies of invasive species on the density of species, and to compare the effectiveness of biological control.

2. Model

2.1. Model description

We use the single-species occupation model in a discrete infinite lattice to describe a native species, assuming that the lattice is rotationally symmetric and the landscape is homogeneous, i.e., ρ_ij = ρ_ji, and all patches are suitable for species survival. In addition, von Neumann neighbors are used, and each patch has z = 4 neighbors. The second species will be introduced to the landscape as an invader, and the third species will be joined in the system as the control species. More details will be described in the next step on model building. Table 1 gives some symbols used to construct the model.

Table 1. The symbols and descriptions.

Symbols	Descriptions
b₁	The birth rate of native species 1
d_i	The mortality rate of species i
c₂₁	The local colonization rates of species 2
c^*₂₁	The global colonization rate of species 2
c^*₃₁	The predation preference of species 3 for species 1
c^*₃₂	The predation preference of species 3 for species 2
ρ_i	The density of i-type patch
ρ_ij	The probability of pair patches <ij>
q_i/j	The conditional probability of pair patches <ij>
T₁	Introduction time of species 3
Note: Species 1 is native species, species 2 is invasive species and species 3 is control species.

| Show Table

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2.2. Species invasion

Suppose the native species in the landscape can only disperse locally to the nearest empty patch. Species 2 invades the landscape, and it only reproduces by hunting native species 1, colonizing the patch where native species 1 existed by mixed dispersion mechanism, namely dispersing locally and globally. Global dispersion refers to sending offspring to random sites of all lattice, while local dispersion refers to sending offspring to random neighboring sites. If there are native species in the site patch, the invader has successful colonization, otherwise it will be wasted.

The possible states of each patch are 0, 1, and 2 (denoted empty patches, occupied by native species 1, and occupied by invasive species 2). Let ρ_i and ρ_ij represent the density of single patch i and pair patches <ij> . <ij> is a pair of adjacent patches, one of which is type i and the other is type j. Let q_i/j represents the conditional probability of arbitrarily selecting a j-type patch whose neighbor is i-type, and q_i/j = ρ_ij/ρ_j, i, j∈{0, 1, 2, 3}. Let c₂₁ indicates the colonization rate of species 2 by preying on species 1 in its neighbors, dispersing to the site where species 1 is located, and c^*₂₁ represents the colonization rate of species 2 by preying on any species 1 in the entire lattice. More symbols and explanations are in Table 1.

In order to construct a predation system by pair approximation, using the dynamic of ρ₁₂ in Eq (4) as an example, the two parts will cause the growth: (a1) Species 1 of the right site of pair sites <11> is occupied by species 2 through global and local predation, which can be denoted by ρ₁₁ (c^*₂₁ρ₂ + c₂₁q_2/11 (z-1)/z). (a2) The empty site of pair sites <02> is occupied by the offspring from its neighboring sites occupied by species 1, denoted by b₁ρ₀₂q_1/02 (z-1)/z. Three parts will reduce ρ₁₂: (b1) The death of pair sites <12> , denoted by -ρ₁₂(d₁ + d₂). (b2) Species 1 of pair sites <12> is killed by neighbor species 2, denoted by -ρ₁₂c₂₁(1/z+ q_2/12(z-1)/z). (b3) Species 1 of pair sites <12> is killed by arbitrary species 2 of all lattice, denoted by -ρ₁₂c^*₂₁ρ₂. The dynamic structure of other pair sites is the same as that of ρ₁₂ (see Appendix for details).

Depending on the relationship between single sites and pair sites, combined with the modeling methods of previous PA models ^[26,31], the corresponding equations can be obtained:

$\frac{d{\rho }_{1}}{dt} = {b}_{1}{\rho }_{1}(1-{q}_{1/1}-{q}_{2/1})-(d{}_{1}{\rho }_{1}+{c}_{21}^{*}{\rho }_{1}{\rho }_{2}+{c}_{21}{\rho }_{1}{q}_{2/1}),$

(1)

$\frac{d{\rho }_{2}}{dt} = {c}_{21}^{*}{\rho }_{2}{\rho }_{1}+{c}_{21}{\rho }_{2}{q}_{1/2}-{d}_{2}{\rho }_{2},$

(2)

$\frac{d{\rho }_{11}}{dt} = 2b{}_{1}{\rho }_{10}(\frac{1}{z}+\frac{z-1}{z}{q}_{1/01})-2{\rho }_{11}(d{}_{1}+{c}_{21}^{*}{\rho }_{2}+{c}_{21}\frac{z-1}{z}{q}_{2/11}),$

(3)

$\frac{d{\rho }_{12}}{dt} = {\rho }_{11}\left({c}_{21}^{*}{\rho }_{2}+{c}_{21}\frac{z-1}{z}{q}_{\frac{2}{11}}\right)+{\rho }_{02}b{}_{1}\frac{z-1}{z}{q}_{\frac{1}{02}}-{\rho }_{12}[d{}_{1}+d{}_{2}+{c}_{21}^{*}{\rho }_{2}+{c}_{21}(\frac{1}{z}+\frac{z-1}{z}{q}_{2/12}\left)\right],$

(4)

$\frac{d{\rho }_{22}}{dt} = 2{\rho }_{12}\left[{c}_{21}\right(\frac{1}{z}+\frac{z-1}{z}{q}_{2/12})+{c}_{21}^{*}{\rho }_{2}]-2{\rho }_{22}d{}_{2}.$

(5)

Taking into account the population spatial distribution, according to lattice rotational symmetry, the pair approximation relationship (q_i/jk ≈ q_i/j, the spatial structure of two adjacent patches is approximate to that of three adjacent patches), the sum of the single sites is 1 (∑_iρ_i = 1), and the sum of the density of the pair sites is equal to the corresponding the density of the single site (ρ_i = ∑_jρ_ij). All other pair sites can be represented by these five quantities. Apparently Eqs (1)−(5) constitutes a closed autonomous system on which the invasion process is analyzed.

2.3. Biological control

Since invasive species feed on native species, if not controlled, the local ecosystem may be affected, so we adopted a predatory natural enemy control strategy and introduced species 3 as controller. Assuming that control species 3 globally hunt invasive and native species, with predation preferences. Let c^*₃₁ indicates the predation preference of species 3 for species 1, and c^*₃₂ represents the predation preference of species 3 for species 2. Since new species need to adapt to the local environment, we assume that native species have stronger reproductive capacity and have lower mortality rates than the other species.

According to the orthogonal neighborhood correlation algorithm given by Hiebeler, the density and spatial connectivity of control species should follow an inequality 2-1/ρ₃ <q_3/3 <1 ^[25], otherwise, species introduction is considered invalid. After introduction, the possible states of each site are 0, 1, 2, and 3 (indicating unoccupied empty site, occupied by native species 1, occupied by invasive species 2, and occupied by control species 3, respectively). Based on the previous study ^[26], we can infer that at the moment of introduction of species 3, the dynamics of single site and pair sites satisfy the following constraints (Appendix for details):

$\left\{\begin{array}{l}{\rho }_{i}^{+} = {\rho }_{i}\left(1-{\rho }_{3}\right), i = \mathrm{1, 2}\\ {\rho }_{3}^{+} = {\rho }_{3}.\end{array}\right. \left\{\begin{array}{l}{\rho }_{ij}^{+} = {\rho }_{ij}(1-2{\rho }_{3}+{\rho }_{3}{q}_{3/3}), i, j = \mathrm{1, 2}\\ {\rho }_{i3}^{+} = {\rho }_{i}{\rho }_{3}(1-{q}_{3/3}), \;\; \;\; \;\; \;\;i = \mathrm{1, 2}\\ {\rho }_{33}^{+} = {\rho }_{3}{q}_{3/3}.\end{array}\right.$

(6)

Based on above facts and assumptions, combined with the PA methods ^[26], a dynamic equation of nine variables is obtained, and the constraint (6) is substituted to finally obtain the system:

$\frac{d{\rho }_{1}}{dt} = {b}_{1}\left({\rho }_{1}-{\rho }_{11}-{\rho }_{12}-{\rho }_{13}\right)-{\rho }_{1}\left(d{}_{1}+{c}_{21}^{*}{\rho }_{2}+{c}_{31}^{*}{\rho }_{3}\right)-{c}_{21}{\rho }_{12},$

(7)

$\frac{d{\rho }_{2}}{dt} = {c}_{21}^{*}{\rho }_{1}{\rho }_{2}+{c}_{21}{\rho }_{12}-{c}_{32}^{*}{\rho }_{2}{\rho }_{3}-{d}_{2}{\rho }_{2},$

(8)

$\frac{d{\rho }_{3}}{dt} = {c}_{31}^{*}{\rho }_{1}{\rho }_{3}+{c}_{32}^{*}{\rho }_{2}{\rho }_{3}-{\rho }_{3}d{}_{3},$

(9)

$\begin{array}{c} \frac{d{\rho }_{11}}{dt} = 2b{}_{1}({\rho }_{1}-{\rho }_{11}-{\rho }_{12}-{\rho }_{13})(\frac{1}{z}+\frac{z-1}{z}\frac{{\rho }_{1}-{\rho }_{11}-{\rho }_{12}-{\rho }_{13}}{1-{\rho }_{1}-{\rho }_{2}-{\rho }_{3}})-2{\rho }_{11}(d{}_{1}+{c}_{21}^{*}{\rho }_{2}+{c}_{31}^{*}{\rho }_{3}+\\{c}_{21}\frac{z-1}{z}\frac{{\rho }_{12}}{{\rho }_{1}}), \end{array}$

(10)

$\begin{array}{c} \frac{d{\rho }_{12}}{dt} = {\rho }_{11}\left({c}_{21}^{*}{\rho }_{2}+{c}_{21}\frac{z-1}{z}\frac{{\rho }_{12}}{{\rho }_{1}}\right)+b{}_{1}\frac{z-1}{z}\left({\rho }_{2}-{\rho }_{12}-{\rho }_{22}-{\rho }_{23}\right)\frac{{\rho }_{1}-{\rho }_{11}-{\rho }_{12}-{\rho }_{13}}{1-{\rho }_{1}-{\rho }_{2}-{\rho }_{3}}-\\{\rho }_{12}\left[d{}_{1}+d{}_{2}+{c}_{32}^{*}{\rho }_{3}+{c}_{31}^{*}{\rho }_{3}+{c}_{21}^{*}{\rho }_{2}+{c}_{21}\left(\frac{1}{z}+\frac{z-1}{z}\frac{{\rho }_{12}}{{\rho }_{1}}\right)\right], \end{array}$

(11)

$\begin{array}{c} \frac{d{\rho }_{13}}{dt} = b{}_{1}\frac{z-1}{z}({\rho }_{3}-{\rho }_{13}-{\rho }_{23}-{\rho }_{33})\frac{{\rho }_{1}-{\rho }_{11}-{\rho }_{12}-{\rho }_{13}}{1-{\rho }_{1}-{\rho }_{2}-{\rho }_{3}}+{\rho }_{12}{c}_{32}^{*}{\rho }_{3}+{\rho }_{11}{c}_{31}^{*}{\rho }_{3}-{\rho }_{13}(d{}_{1}+d{}_{3}+\\{c}_{21}^{*}{\rho }_{2}+{c}_{31}^{*}{\rho }_{3}+{c}_{21}\frac{z-1}{z}\frac{{\rho }_{12}}{{\rho }_{1}}), \end{array}$

(12)

$\frac{d{\rho }_{22}}{dt} = 2{\rho }_{12}\left[{c}_{21}\left(\frac{1}{z}+\frac{z-1}{z}\frac{{\rho }_{12}}{{\rho }_{1}}\right)+{c}_{21}^{*}{\rho }_{2}\right]-2{\rho }_{22}(d{}_{2}+{c}_{32}^{*}{\rho }_{3}),$

(13)

$\frac{d{\rho }_{23}}{dt} = {\rho }_{13}\left({c}_{21}^{*}{\rho }_{2}+{c}_{21}\frac{z-1}{z}\frac{{\rho }_{12}}{{\rho }_{1}}\right)+{\rho }_{12}{c}_{31}^{*}{\rho }_{3}+{\rho }_{22}{c}_{32}^{*}{\rho }_{3}-{\rho }_{23}(d{}_{2}+d{}_{3}+{c}_{32}^{*}{\rho }_{3}),$

(14)

$\frac{d{\rho }_{33}}{dt} = 2{\rho }_{13}{c}_{31}^{*}{\rho }_{3}+2{\rho }_{32}{c}_{32}^{*}{\rho }_{3}-2{\rho }_{33}d{}_{3}.$

(15)

It can be seen that Eqs (7)−(15) constitute a closed system for the control process. Compared with the invasion process (1)−(5), (7)−(15) consider the effects of species 3 hunting species 1 and 2, as well as the intrinsic mortality of species 3.

2.4. Simulation process

Due to the complexity of the model, we use MATLAB to calculate the above differential equations. And when the population density is less than 0.00001, this species is considered to be extinct. We set parameters based on mass simulations. Firstly, the four dispersion strategies of invasive species 2 were set as follows: global dispersion (c^*₂₁ = 0.5, c₂₁ = 0), partial global dispersion (c^*₂₁ = 0.4, c₂₁ = 0.1), partial local dispersion (c^*₂₁ = 0.1, c₂₁ = 0.4), and local dispersion (c^*₂₁ = 0, c₂₁ = 0.5) (see Appendix Figures B1 and B2). Based on the assumption of reproduction rates and mortality rates let b₁ = 0.7, d₁ = 0.1, d₂ = 0.15, and d₃ = 0.2 (other parameter space situations see Appendix Figures B7−B10). In order to prevent invasive species from overly affecting native species, control should not start too late. So we set the introduction time T₁ = 10, 20, 40 respectively. In order to explore the influence of the predation preference of control species, we suppose that control species prefer to prey on invasive species (c^*₃₁ < c^*₃₂), so we fixed the predation preferences of species 1 and 2 as c^*₃₁ = 0.3 and c^*₃₂ = 0.6, respectively (see Appendix Figures B3−B6 for other situations). Based on the above parameters, we study the effects of the initial density and initial spatial structure of introduced species on results. These parameters are biologically significant. And there are other available parameter values, so we changed multiple parameter values in the simulation and determined that they would not change our general results and conclusions (see Appendix Figures B1−B6).

3. Results

In order to visually observe the effect of biological control, we simulated the procedure on a lattice of 100 × 100 and drew landscapes of the invasion process of species 2 and the introduction process of species 3 by cellular automata (Figure 1). Set the initial densities of species 1 and 2 to 0.8 and 0.5. When species 2 invades into a landscape where only species 1 exists, if left uncontrolled, we find that the density of species 1 decreases to approximately 0.1, and species 2 occupies most of the habitat (Figure 1(Ⅰ)). After that, we introduce species 3, set the initial density and initial aggregation of species 3 to ρ₃ = 0.2 and q_3/3 = 0.8 (Figure 1(Ⅱ)), then the graph (Figure 1(Ⅲ)) will be formed instantaneously, at which time three species coexist. After the three species reached equilibrium (Ⅳ), compare with Figure 1(Ⅲ), the number of species 2 after control became significantly smaller, and the number of species 1 increased significantly, indicating that biological control was effective.

Figure 1. Landscapes for species invasion and biological control. Green represents native species 1, red indicates invasive species 2, blue is control species 3, and white is empty sites. (Ⅰ): Landscape after invasion of species 2. (Ⅱ): Distribution of control species 3, setting ρ₃ = 0.2, q_3/3 = 0.8. (Ⅲ): Instantaneous landscape of introduction of species 3. (Ⅳ): landscape after stabilization.

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AIMS Mathematics

Stability and domination exponentially in some graphs

Related Papers:

Abstract

1. Introduction

2. Model

2.1. Model description

2.2. Species invasion

2.3. Biological control

2.4. Simulation process

3. Results

4. Discussion

Acknowledgments

Conflict of interest

Appendix

A. The dynamic structure of pair sites

B. Simulation of multiple parameter values

References

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