ANALYSIS OF DYNAMIC EVOLUTION PROCESS OF THE N -PLAYER DIVISION OF LABOR GAME MODEL

. This paper investigates a three-strategy (cooperators, toxin producers, and cheaters) N - player division of labor game in bacterial populations. We construct the replicator equation to discuss the evolution of the frequency of the three strategies. Firstly, we prove that the interior equilibrium is always unstable, the three strategies cannot coexist. Secondly, according to Sotomayor’s theorem, the system undergoes transcritical bifurcation. Furthermore, the sensitivity of the two-dimensional evolutionary state diagrams to the third parameter (toxin rate, absorption rate, toxin quantity, etc) is analyzed. In summary, high toxicity rates, high levels of toxins, and low levels of competition tend to promote cooperation. All players choose to perform the task, and the cheater disappears. When the absorption rate of cooperators is high enough, only cooperators exist in the population over time. When the absorption rate of the cooperator is low, and the absorption rate of the toxin producer is greater than the threshold, the cooperator and the toxin producer coexist. All players perform the task. Finally, the triangle diagrams and three-dimensional diagrams are presented, which show the initial conditions of the three strategies also aﬀect the dynamic results. As the amount of toxin increases, the range of players who choose to perform tasks widens.

We often see a division of labor in which a group performs a task. In this case, there is a division of labor between players, which can lead to unequal distribution of work. So far, the division of labor game has attracted the attention of researchers [27,40,43,44]. Qin et al. [27] investigated the division of labor game with environmental feedback and obtained sufficient conditions for the stability of the equilibrium point. The stability of the equilibrium depends on the cost of each strategy, the benefits of cooperation, and the weighted benefits of players in the rich patch. Zhang et al. [43] used evolutionary game theory to analyze the strategy evolution of the self-organizing division of labor game. They obtained the conditions of promoting division of labor cooperation in the two-player division of labor game and multi-player division of labor game. Zhang et al. [44] discussed the evolutionary dynamics of the division of labor game on circular networks. Further, they explored the strategic interaction and network dynamics in the division of labor game based on self-organizing task allocation. Based on the actual situation, many researchers began to analyze N -player games [2,17,32,34,38]. The author [38] considered the N -player snowdrift game, and the delay of payoff led to Hopf bifurcation, and the critical value of time delay is calculated and analyzed in detail. This paper showed the general characteristics of cooperative dynamics of the N -player snowdrift game under different conditions.
In recent years, peer punishment has become an effective measure to promote the evolution of cooperation and restrain non-cooperative behavior and has attracted more and more attention from researchers [5,9,16,24,25,31,36,46]. Zhu et al. [46] combined peer punishment and pool punishment to analyze of the influence of different types of punishment on evolutionary dynamics. The authors found that the effect of different types of punishment on the evolution of cooperation was related to the number of punishers. They concluded that peer punishment promoted cooperation more than pool punishment. Fang et al. [9] considered the evolution of cooperation in spatial public goods games with third-party punishment. They considered four scenarios to determine what is most suitable for cooperation. In addition, the author also discussed the influence of defection tolerance and reward and punishment frequency and found that when the frequency of defection and reward was high, the evolution of cooperation was favorable.
Based on the above statements, we propose the N -player division of labor game system in bacterial populations. We use the evolutionary game theory to study how the toxin-producer strategy maintains cooperative behavior and makes the cheater strategy disappear. The system considers three types of strains: cooperators, toxin producers, and cheaters. Cooperators and toxin producers perform tasks for payoff, while cheaters perform no tasks. Cooperators provide the necessary goods to grow the strains, and toxin producers produce toxins harmful to cooperators and cheaters. We consider various factors in the system, to figure out under what conditions only cooperators exist in the population, or cooperators and toxin producers coexist, and they can resist the intrusion of cheaters.
The remainder of this paper is organized as follows. The first section gives the replicator equation of the three-strategy N -player division of labor game. We figure out all equilibrium points of the system and discuss the sufficient conditions for the stability of equilibrium points. According to Sotomayor's theorem, we prove that the system undergoes a transcritical bifurcation under certain conditions in the third section. In the fourth section, we first provide the two-dimensional evolutionary state diagrams of different parameter combinations. Secondly, this section discusses the sensitivity of the two-dimensional evolutionary state diagrams to the third parameter. Finally, the influence of initial frequency and toxin amount on the area of attraction of the equilibrium point is explored by using triangle diagrams and three-dimensional diagrams. In the fifth and sixth sections, we summarize the conclusions and discuss them.

The N -player division of labor game model
First, we consider a division of labor game for infinitely large bacterial populations. There are three types of players: cooperators (X 1 ), toxin producers (X 2 ) and cheaters (X 3 ). Their frequencies are x, y, and z, respectively, and x + y + z = 1. At each time step, N individuals are randomly selected from infinitely large bacterial populations.
In this game, each cooperator contributes a fixed amount P , and all of them contribute to the game environment. The energy provided by the environment is E, and the public goods produced by the cooperators can promote the absorption of energy. When lacking public goods or energy, all three types of players fail to make benefits. Each toxin producer contributes a fixed amount T , and the toxin is harmful to both cooperators and cheaters. The toxicity rates for them are α 1 and α 3 respectively, where α 1 < α 3 . The toxin producer is resistant to the toxin but produces the toxin at its consumption rate of β. The total energy in the common environment is multiplied by the factor r (1 < r < N ) and divided equally among all N players. Because the cooperator and the toxin producer need to perform the task, they can not fully absorb the energy, their absorption rate is q 1 and q 2 respectively, while the cheater can fully absorb.
In a division of labor game with randomly formed bacterial populations, the probability that a focal player has N 1 cooperator, N 2 toxin producer, and N 3 cheater among (N − 1) co-players is Let π Xi (i = 1, 2, 3) represent the payoffs of the focal players as cooperators, toxin producers, and cheaters. Therefore, if the focal player is a cooperator, toxin producer, or cheater, then their payoff is Next, F Xi (i = 1, 2, 3) denote the expected payoffs of a cooperator, a toxin producer, and a cheater in the population, andF is the average expected payoffs of the whole population.
Accordingly, the payoff F X1 is followed by where we assume q1rP E N − P > 0. In the same way, the payoff F X2 and F X3 are followed by The average expected payoffF is Based on the above analysis and evolutionary game theory, we use the replicator equation to explore the evolutionary dynamics of three strategy frequencies in infinitely large bacterial populations. We can obtain the replicator equation as Accordingly, we can obtain the specific higher order replicator equation as (2.2) Since x + y + z = 1, i.e. z = 1 − x − y, the replicator equation (2.1) can be rewritten as In the same way, the specific higher order replicator equation (2.2) can be rewritten as

Existence and stability of equilibria
Based on the replicator equation (2.3), we further analyze the existence and stability of equilibria in Sections 3.1 and 3.2 respectively. Firstly, we make and define Secondly, we define
(II) We substitute x 2 = 0 and y 2 = 1 into the formula (3.2), and get the Jacobian matrix as .
is a locally asymptotically stable point, (III) If β < β * and A < B, the boundary equilibrium is a locally asymptotically stable point.
According to matrix (3.5), we can get its eigenvalues According to the formula (3.2), x 5 = x * 5 and y 5 = 0, we obtain the Jacobian matrix where According to matrix (3.6), we can get its eigenvalues Since q 1 < q * 1 , we can know that λ 1 < 0 and λ 2 < 0, so the boundary equilibrium is a locally asymptotically stable point. (III) Since z 6 = 0 and x 6 + y 6 = 1, the Jacobian matrix of the boundary equilibrium E 6 is We can obtain that the characteristic polynomial of the matrix (3.7) is is a locally asymptotically stable point.
Proof. Since 0 < x 7 , y 7 , z 7 < 1 and x 7 + y 7 + z 7 = 1, the Jacobian matrix of the interior equilibrium E 7 is We can obtain that the characteristic polynomial of the matrix (3.8) is From the above statement, we conclude that the interior equilibrium E 7 is unstable.
Since cooperators and toxin producers obtain payoff by performing tasks, cheaters do not perform any tasks. Cooperators, toxin producers, and cheaters do not co-exist in populations over time. Any population will not allow the behavior of not performing the task to get the payoff, so it is realistic that the interior equilibrium is not stable.
According to the above description, we summarize in Table 1. Based on the above analysis, there is the following multiple stability theorem: Theorem 3.4. If q 1 ∈ (a, b), then system (2.2) has exactly two steady states: E 5 and E 6 .

Bifurcation analysis
In this section, using Sotomayor's theorem [30], we prove that the system undergoes a transcritical bifurcation.
Theorem 4.1. When the parameter q 1 crosses the transcritical threshold q 1 = q * 1 = 1 rE + N −1 N , the system undergoes a transcritical bifurcation at E 1 .

Equilibria
Existence condition Stability condition Instability condition Proof. According to the Jacobian matrix J E1 of point E 1 , when < 0, and the other eigenvalue is zero. The zero eigenvalues of the matrices J E1 and J T E1 correspond to eigenvectors are According to (2.4), we have When the parameter q 1 crosses the transcritical threshold q 1 = q * 1 , the system undergoes a transcritical bifurcation at E 1 . Proof. According to the Jacobian matrix J E2 of point E 2 , when β = β * = α 1 (N − 1) + P − q 1 rP E N T , the eigenvalue of matrix βT − α 3 T (N − 1) < 0, and the other eigenvalue is zero.
The zero eigenvalues of the matrices J E2 and J T E2 correspond to eigenvectors arẽ According to (2.4), we have When the parameter β crosses the transcritical threshold β = β * , the system undergoes a transcritical bifurcation at E 2 .
According to the above theorem, we can summarize in Table 2.

Analysis of dynamic evolution process
In this section, we mainly investigate the influence of various parameters with different values on the steadystate of the system. Different parameters represent different biological implications, and we look at the effects of each parameter on dynamics and strategies.
First, we investigate the effect of toxicity rates α 1 and α 3 on the steady-state of cheater strategy frequency. Since the sum of the three strategy frequencies is 1 (x + y + z = 1), we can obtain the variation of the frequency of players performing tasks (cooperators and toxin producers) according to the variation of cheater strategy frequency. In particular, Figure 2a-c shows how cheater frequency evolves with different α 1 and α 3 at (N = 6, T = 1), (N = 6, T = 2), and (N = 5, T = 1). The results show a clear boundary of the disappearance of cheaters as α 1 and α 3 change. There is a threshold for cheaters, and when the toxicity rate α 3 exceeds this threshold, cheaters disappear, only the players who perform the task exist in the game. Looking at Figure 2 from a different perspective, we fix the toxicity rate α 1 . When the toxicity rate α 3 value is low, the toxin is less toxic to cheaters, there are more cheaters in the population, and correspondingly, there are fewer players performing tasks. In addition, comparing (a) and (b), we find that the boundary line moves to the left. Namely, the area of the player performing the task becomes wider as the value of T increases. Comparing (a) and (c), we find that the boundary line moves to the right (note the change in the x-coordinate here), and as the number of players decreases, a higher toxicity rate α 3 is required to make cheaters disappearance. But when the toxicity rate α 3 is below the threshold, the frequency of cheaters decreases.
Furthermore, we carefully observe the changes on the steady-state of the three strategy frequencies with the toxicity rate α 3 in Figure 3. The red, green and blue lines show the frequency of cooperators, toxin producers and cheaters, respectively. The four figures show a threshold α 3 that divides the system into two stable states. Most participants adopt the cheater strategy when the toxicity rate is below the threshold. When the toxicity rate is above the threshold, the cheater disappears, and the player changes strategy and becomes either a cooperator or a toxin producer. In other words, the toxin effect on cheaters is low, and the player uses the cheater strategy to gain payoffs by not performing the task. Once the toxin becomes more harmful to the cheater, the participants change their strategy and choose to perform tasks. So a higher toxicity rate α 3 can encourage the emergence of cooperators. In addition, by comparing (a) and (b), with the increase of toxicity rate α 1 , we can observe that the threshold α 3 becomes smaller, and the frequency of cooperation decreases, which is understandable. Compared with (b) and (c), the threshold decreases when the amount of toxin increases. At the threshold, the frequency of cooperation increases, and the high toxin levels can promote cooperation. Then, by comparing (b) and (d), the threshold increases, and the frequency of cooperators increases as the number of participants decreases, so fewer participants can promote cooperation. Next, we investigate the effect of the cooperator's absorption rate q 1 and the toxin producer's absorption rate q 2 on the steady-state frequency of the cheater's strategy. Since the sum of the three strategy frequencies is 1 (x + y + z = 1), we can obtain the variation of the frequency of players performing tasks (cooperators and toxin producers) according to the variation of cheater strategy frequency. Particularly, Figure 4(a)-(c) shows how cheater frequency evolves with different q 1 and q 2 at (N = 6, T = 1), (N = 6, T = 2), and (N = 5, T = 1). As q 1 and q 2 change, we can see that each figure is divided into three areas. At the top of each figure, the color is dark blue, and the frequency of cheaters is zero. In other words, when the cooperator's absorption rate q 1 is high enough, cheaters disappear from the population, and the cooperator takes over. Comparing (a), (b) and (c), we find that the dark blue area widens with the decrease of players, so fewer players can promote the occurrence of cooperation. Now, let us move on to the lower part of the figure. Each figure has a boundary line from bottom left to top right. By comparing (a) and (b), we find that the slope of the boundary decreases with the increase of T value, that is, the area of cheaters disappearance increases, so the number of players performing the task increases. As the amount of toxin increases, the number of players who perform tasks increases. By comparing (a) and (c), when the number of participants decreases and q 1 is fixed, the threshold of cheaters disappearance increases. In this case, the toxin producer needs a high toxin rate to resist the presence of cheaters.
Then, we explore the influence of absorption rates q 1 and q 2 on the stable state of the system in Figures 5  and 6, respectively. The red, green and blue lines show the frequency of cooperators, toxin producers and cheaters, respectively. In Figure 5, we can observe the threshold q 1 that divides the system into two distinct behavior stages. For q 1 below the threshold, the frequency of cooperators increases and the frequency of toxin producers decreases as the absorption rate of cooperators increases. For q 1 above the threshold, the toxin producer disappears, and cooperators and cheaters coexist in the system. But as q 1 gets bigger, all players are the cooperators in the game. All players become cooperators when the absorption rate of the cooperator increases to a certain value, and other parameters remain constant. The increased absorption rate of the cooperator promotes the occurrence of cooperation. In addition, compared with (a) and (b), the threshold q 1 increases with the increase of toxin amount, and the frequency of cooperators near the threshold values increases. Toxins can promote cooperation. By comparing (a) and (c), as the number of participants decreases, the threshold value of the absorption rate q 1 of the cooperator decreases, and the frequency of the cooperator near the threshold also decreases. But when the population is full of cooperators, the value of q 1 decreases. A small number of participants facilitates cooperation.
Similarly, we analyze Figure 6. The three figures show that the threshold value q 2 divides the system into two stages of behavior. When the absorption rate q 2 is below the threshold, cooperators and cheaters coexist in the system, and the absorption rate q 2 of the toxin producer does not affect the steady-state frequency. In this case, the absorption rate of the toxin producer is so low that the player does not want to produce the  . Schematic diagram of influence of toxicity rate α 3 on the steady-state of three strategy frequencies. A toxicity rate threshold divides the system into two stable states. When the toxicity rate exceeds the threshold, the cheater disappears, and the cooperator and toxin producer coexist. The red, green and blue lines indicate the frequency of cooperators, toxin producers and cheaters, respectively, where r = 4, q 1 = 0.55, q 2 = 0.8, β = 0.2, P = 1, E = 3.   Figure 6. The figures show the effect of the absorption rate q 2 of the toxin producer on the stable state of the three strategy frequencies. The toxin producer does not exist when the absorption rate q 2 is below the threshold. As the absorption rate increases, the cheaters disappear, cooperators and toxin producers coexist, and the frequency of toxin producers increases. The red, green and blue lines indicate the frequency of cooperators, toxin producers and cheaters, respectively, where r = 4, α 1 = 0.025, α 3 = 0.27, β = 0.2, P = 1, E = 3.
toxin. When the absorption rate q 2 is above the threshold, cheaters disappear, the frequency of toxin producers increases, and the frequency of cooperators decreases. Compared with (a) and (b), when the amount of toxin increases, the threshold value of toxin producer absorption rate decreases. High toxin levels reduce the scope of cheaters. Comparing (a) and (c), the number of participants decreases, and the threshold value for toxin producer absorption rate increases, but the frequency of cooperators increases throughout the range of variation.
In Figure 7, we mainly observe the frequency evolution of the three strategies with different β and T values. We find that each figure is divided into two main areas. From Figure 7(a), (b) and (c), in the left and the top areas of each figure, we can conclude that there are no cooperators and toxin producers but only cheaters in the population. Next, we look at the rest of each figure. Figure 7a is observed from bottom left to top right, the frequency of cooperators increases with the increase of β and T , namely, the amount of toxin that can promote cooperation. In Figure 7b, it is understandable that the frequency of toxin producers decreases as the amount of toxin T and its consumption rate β increase. In Figure 7c, the cheater disappears in this case. In short, if we increase the amount of toxins and control the rate at which toxin producers consume themselves, then all players choose to perform the task in the population.
Next, in Figure 8, we analyze how parameter β affects the steady-state frequency of the system. Looking at the two figures, we see that the threshold value divides the system into two stages of behavior. When β is When β is below the threshold, the frequency of cooperators increases as β increases, whereas toxin producers do the opposite. The red, green and blue lines indicate the frequency of cooperators, toxin producers and cheaters, respectively, where r = 4, α 1 = 0.03, α 3 = 0.27, q 1 = 0.55, q 2 = 0.72, P = 1, E = 3, N = 6. below the threshold, there are no cheaters in the system, cooperators and toxin producers coexist, which we want. When β is greater than the threshold value, toxin producers consume too much to produce toxin, so toxin producers disappear, and the player changes strategy. And a smaller percentage of players adopt the cooperative strategy, which we do not want to accept. Compared with (a) and (b), with the increase of the toxin amount, the threshold value of β increases, and the scope of the existence of only cooperators and toxin producers expands, and at the threshold value, the frequency of cooperators in (b) is greater than that in (a). Namely, high toxin levels promote cooperation.
In Figure 8, it is clear that when the toxin level is below the threshold, fewer players adopt cooperative strategies, most players adopt cheater strategies, and no toxin producers exist. Once the toxin level exceeds the threshold, cheaters disappear, and the frequency of cooperative strategies increases. So we know that toxin production can promote cooperation.
According to the above descriptions, we obtain the influence of each parameter on the steady-state of the three strategy frequencies in the two-dimensional evolutionary state figures. And the sensitivity of the system to parameters also has different performance. The specific description is as follows: high toxicity rate α 3 , fewer participants and higher toxin amount are favorable for the existence of cooperator strategies, and conversely, fewer players adopt cooperator strategies. If the cooperator absorption rate is high enough, or the cooperator absorption rate is less than the toxin producer absorption rate and q 2 is greater than a threshold, then the cheater disappears and all players perform the task. We can find corresponding points in the triangle for any initial frequency distribution. Figure 10 shows how different initial frequencies evolve according to the replicator equation. In Figure 10, note that the initial points used in each figure are the same. Figure 10 Figure 10 show that the system has no stable interior equilibrium. And we can see similarities between the three diagrams (a)-(c): different initial frequencies are either stable on the X 1 and X 3 axis or stable on the X 1 and X 2 axis. The red line shows that the initial point evolves to a stable equilibrium point between the X 1 and X 2 axis, where there are no cheaters. And the blue line shows that the initial point evolves to a stable equilibrium point between the X 1 and X 3 axis. Looking at Figure 10, with the increase of the T value, the area where the initial point is stable to the equilibrium point between axis X 1 and X 2 increases; conversely, the area where the initial point is stable to the equilibrium point between axis X 1 and X 3 decreases. And with the increase of the T value, the equilibrium point between X 1 and X 2 axis moves to the left, and the proportion of cooperators increases in the population. In Figure 10(d)-(f), we find more initial points stable to the no-cheater equilibrium as T increases. To sum up, with other parameters holding constant, increasing the amount of toxin encourage the player to perform tasks and increases the percentage of cooperators. In Figure 10, we find that the system is bistable, which is consistent with Theorem 3.4.

Conclusions
This paper considers the N -player division of labor game dynamics in bacterial populations. We give the reasonable payoffs of cooperator, toxin producer and cheater, respectively, and establish replicator equations to explore the evolution of the three strategy frequencies. We find all equilibrium points of the division of labor game system and determine the existence conditions of equilibrium points. By analysis and proof, sufficient The black dots indicate a stable equilibrium point. The red line shows the tendency of the initial point to stabilize to the equilibrium points on the X 1 and X 2 axis, and the blue line shows the tendency of the initial point to stabilize to the equilibrium points on the X 1 and X 3 axis. The vertices of the triangle represent all the cooperators (x = 1), all the toxin producers (y = 1), and all the cheaters (z = 1).
conditions for the stability of the equilibrium point are obtained. The interior equilibrium is always unstable, so the three strategies cannot coexist. According to the hypothesis, if the cooperator does not exist under the initial conditions in the system, then the player disappears. In other words, the population occurs the tragedy of the commons because no cooperators provide toxin producers and cheaters with the necessary goods for growth. With the change of time, when the system reaches a stable state, the system eventually has three results: (1) cooperator exists alone, (2) cooperator and toxin producer coexist, (3) cooperator and cheater coexist. (1) and (2) are the desired outcomes, where players in the population choose to perform the task and then obtain the payoff. The third section presents the transcritical bifurcation analysis. For equilibrium points E 1 and E 2 , we choose different bifurcation parameters. According to Sotomayor's theorem, the system undergoes transcritical bifurcation under different bifurcation parameter conditions. In the fourth section, we investigate the influence of each parameter on the steady-state frequency of three strategies in the division of labor game system. The two-dimensional evolutionary state diagrams with different parameter combinations are shown, such as Figures 2, 4, and 7. We also analyze the sensitivity of the twodimensional evolutionary state diagrams to the third parameter. In summary, high toxicity rates (for cheaters), high levels of toxins, and low levels of competition tend to promote cooperation. All players choose to perform the task, and the cheater disappears. Conversely, low toxicity rates (for cheaters), low levels of toxins, and high levels of competition inhibit cooperation. Since the toxin is less harmful to cheaters, players do not choose to perform tasks to gain a payoff. In this case, most players in the population choose the cheater strategy, and the toxin producer does not exist. When the absorption rate of cooperators is high enough, only cooperators exist in the population over time. When the absorption rate of the cooperator is low and the absorption rate of the toxin producer is greater than the threshold, the cooperator and the toxin producer coexist, that is, all players perform the task. In Figure 10, the triangle diagrams and the three-dimensional diagrams are presented to discuss the influence of the initial values of the three strategies on the evolution of strategy frequency. With the same initial value, the range of players who choose to perform tasks widens as the amount of toxin increases. We can also find that the stable point where cooperators and toxin producers coexist shifts to the left. Namely, the frequency of cooperators increases, and the production of toxins promotes cooperation.
In the process of proving the stability of each equilibrium point, we find that the interior equilibrium point is unstable. That is to say, cooperators, toxin producers, and cheaters do not co-exist, which is a realistic scenario. Cooperators and toxin producers obtain a payoff by performing tasks, while cheaters get a payoff by not performing any tasks. The player resists presence of the cheater, thus they make the cheater disappear. In real life, players are not allowed to hitchhike.

Discussion
Compared with reference [18], we use the idea of evolutionary game theory. The three strains are regarded as three strategies of the player, and the replicator equation is given to discuss the evolution of the three strategy frequencies. Although the N -player division of labor game dynamics are constructed in bacterial populations, the framework can be applied to many other populations. Therefore, our research has far-reaching significance and influence. This paper considers the dynamics of a division of labor game in a homogeneous environment. We do not distinguish environmental heterogeneity, nor do we consider the effect of stochastic noise in the system. Many researchers have explored the effects of environmental feedback [11,14] and stochastic noise [15,26,45] in dynamic systems. In the future, we will investigate the dynamics of the division of labor game with environmental feedback and stochastic noise.