Dynamical analysis of a Lotka Volterra commensalism model with additive Allee

: We propose and analyze a Lotka - Volterra commensal model with an additive Allee e ﬀ ect in this article. First, we study the existence and local stability of possible equilibria. Second, the conditions for the existence of saddle - node bifurcations and transcritical bifurcations are derived by using Sotomayor ’ s theorem. Third, we give su ﬃ cient conditions for the global stability of the boundary equilibrium and positive equilibrium. Finally, we use numerical simulations to verify the above theoretical results. This study shows that for the weak Allee e ﬀ ect case, the additive Allee e ﬀ ect has a negative e ﬀ ect on the ﬁ nal density of both species, with increasing Allee e ﬀ ect, the densities of both species are decreasing. For the strong Allee e ﬀ ect case, the additive Allee e ﬀ ect is one of the most important factors that leads to the extinction of the second species. The additive Allee e ﬀ ect leads to the complex dynamic behaviors of the system.


Introduction
Commensalism is a symbiotic relationship between two species in which one species benefits from another species, while the other species neither gains nor loses. In the past few decades, many scholars have done work on the dynamic behaviors of the commensalism model, and some essential progress has been obtained .
Sun and Wei [1] first time proposed and studied a two species commensalism symbiosis model: They investigated the local stability property of four equilibria, among which the boundary equilibria E 0, 0 1 ( ), E k , 0 2 1 ( ), and E k 0, 3 2 ( ) are unstable, and the unique positive equilibrium E k αk k , is always locally stable. However, they did not conduct a further study on the global stability of the positive equilibrium E k αk k , 4 1 2 2 ( ) + . Han and Chen [2] proposed the following commensalism model: They showed that the system admits a unique positive equilibrium, which is globally asymptotically stable. In addition, they added feedback control variables into the system (1.2) and found that the feedback control variable only changes the position of the positive equilibrium but still maintains its property of global stability.
When the populations have non-overlapping generations, the discrete-time models governed by difference equations are more appropriate than the continuous ones. Thus, Xie et al. [3] proposed the discrete commensal symbiosis model. Based on [3], Li et al. [4] proposed the discrete commensal symbiosis model with the Holling II functional response. They gave some sufficient conditions for the existence of positive periodic solution of the models they considered. Chen [5] and Yu et al. [6] studied the commensal symbiosis model with the Michaelis-Menten type harvesting. In On the other hand, in 1931, Allee [23] pointed out that when the population density is too low, individuals in the population will encounter difficulties in finding mates and resisting natural enemies, which will lead to a decrease in the birth rate and an increase in the death rate of the population. This phenomenon is called the Allee effect [24]. Since then, many scholars began to study the ecological model with the Allee effect. Bazykin [25] proposed a single model with multiple Allee effects for the first time as follows: where r represents the inherent per capita growth rate of the population and K represents the environmental carrying capacity. If m K 0 < < , it shows the strong Allee effect when the population is lower than the threshold, the population growth is negative, and the population is at risk of extinction; otherwise, the population can survive. While if m 0 ≤ , it shows the weak Allee effect; the population growth slows down but there is no risk of extinction. Furthermore, Dennis [28] proposed the model with the additive Allee effect for the first time as follows: Here, we denote the additive Allee effect by m x a + , m and a are both constants, and the additive Allee effect has the following properties: (1) If m a 0 < < , then system (1.4) has the weak Allee effect.
(2) If m a > , then system (1.4) has the strong Allee effect.
Merdan [29] proposed the following predator-prey model: Merdan showed that the system subject to an Allee effect takes a much longer time to reach its stable steady-state solution; also, the Allee effect reduces the population densities of both predator and prey at the steady state. However, the Allee effect has no destabilizing role.
We mention here that in nature, one of the typical commensal relationships between epiphyte and plants with epiphyte, as shown in Figure 1, the plant (host) generally speaking, is huge, need more space to grow, and its density is sparse; this certainly increases the chance of the Allee effect on the plant. Indeed, recently, Jiao et al. [30] have shown that in a coastal wetland, the plant population does exhibit the Allee effect. Hence, it is natural to propose and study the commensalism model with the Allee effect.
Recently, Wu et al. [7] added the Holling-type functional response and Merdan-type Allee effect (one could refer to [29] for more details) to the system (1.2), this leads to the following system: They showed that the unique positive equilibrium is globally stable and the Allee effect has no influence on the final density of the species, and that the stronger the Allee effect (u become large), the system takes a longer time to reach its steady-state solution. Later, Lin [8] considered adding the Merdan-type Allee effect in the first species of the system (1.2), and they studied the dynamic behaviors of the following system: where b i , a ii , i 1, 2 = , β and a 12 are positive constants, F x x β x ( ) = + represents the Allee effect of the first species. They observed that as the Allee effect increased, the final density of the species affected by Allee effect also increased. Moreover, the positive equilibrium of the system (1.6) is still globally stable.
Inspired by Wu et al. [7] and Lin [8], we consider replacing the Merdan-type Allee effect with additive Allee effect on the traditional Lotka-Volterra commensalism model, this leads to the following model: To the best of the authors' knowledge, this is the first time to propose and study the commensal model with the additive Allee effect. Our most important task is to find out the influence of the additive Allee effect on the system (1.7), especially on the y species. We also want to know if the system (1.7) has similar dynamic behaviors or any new properties compared with the systems considered in [2,7,8].
The rest of this article is arranged as follows: We investigate the existence of the equilibria in the next section and then study the local stability property of the equilibria in Section 3. In Section 4, we discuss the saddle-node bifurcations and transcritical bifurcations. In Section 5, we give sufficient conditions to ensure the global stability of the boundary equilibrium and the positive equilibrium, respectively. Finally, the article ends with some numeric simulations and a brief discussion.   Through the above analysis, we know that system (1.7) always has two boundary equilibria given by E 0, 0 0 ( ) and E , 0 r r b ( ) , and for the other possible equilibria, we have the following results:

Existence of equilibria
has a boundary equilibrium E y 0, 1 1 ( ) and a positive equilibrium E x y , has no other equilibria.
) and two positive equilibria E x y , has a boundary equilibrium E y 0, 3 3 ( )and a positive equilibrium E x y , has no other equilibrium.

Local stability of equilibria
In this section, we investigate the local stability of the equilibria. The Jacobian matrix of system (1.7) is calculated as ( ) is a stable node for ae d = and a saddle-node for ae d ≠ . Proof < , then λ 0 2 > and hence E 0, 0 0 ( ) is an unstable node; if m ad = , then λ 0 2 = , in this case, the local stability property of E 0 is difficult to be judged directly from the characteristic root.
First, we expand system (1.7) in power series up to the third order around E 0, 0 0 ( ) and let τ r t d d = : x τ is a saddle, and then E 0, 0 0 ( ) is also a saddle.
is a saddle-node, and then is also a saddle-node.
(2) The Jacobian matrix of system (1.7) at E , 0 and then expand the new system in power series around the origin: Now, we apply the transformation: is also a saddle-node. This ends the proof of Theorem 3.1. □  In this case, E 3 is difficult to be judged directly from the characteristic root.
We first shift E y 0, 3 3 ( ) to the origin by the transformation x X y Y y , 3 3 3 = = + and then expand the new system in power series up to the third order around the origin: exists, it is a saddle-node.
Proof. The Jacobian matrix of system (1.7) at E x y , From the proof of Theorem 3.2 we know that (3) If E 3 * exist, for the eigenvalue λ 2 of J E 3 ( ) * , we have λ 0 2 = , we can easily obtain that E 3 * is a saddle-node (this proof is similar to Theorem 3.2 (3)).
This ends the proof of Theorem 3.3. □ We use Table 1 to sum up the above conclusions.

Bifurcation analysis
From Theorems 2.1 to 2.3, we conjecture that system (1.7) may have saddle-node bifurcations at E 3 and E 3 * , and transcritical bifurcations at the equilibria E 0 and E r , respectively. Indeed, we have the following results.

= +
It is obvious that the matrix has a zero eigenvalue, named λ 1 . Let V and W represent the eigenvectors corresponding to the eigenvalue λ 1 for matrices J E3 and J E T 3 . By calculation, we can obtain: It is obvious that the matrix has a zero eigenvalue, named λ 1 . Let V and W represent the eigenvectors corresponding to the eigenvalue λ 1 for matrices J E 3 * and J E T 3 * . By calculation, we can obtain:   Proof. The Jacobian matrix at E 0 is It is obvious that the matrix has a zero eigenvalue, named λ 1 . Let V and W represent the eigenvectors corresponding to the eigenvalue λ 1 for matrices J E0 and J E T 0 . By calculation, we can obtain: Define   Proof. The Jacobian matrix at E r is It is obvious that the matrix has a zero eigenvalue, named λ 1 . Let V and W represent the eigenvectors corresponding to the eigenvalue λ 1 for matrices J Er and J E T r . By calculation, we can obtain: From (4.11)-(4.14), it follows that shown that E r is locally asymptotically stable if m ad > . In this section, we will provide some sufficient conditions for the global stability of E 1 * and E r . (1) m ad < ; (2) m ad = and ae d < .
Proof. From Table 1, we find that in addition to E 0 and E r , system (1.7) also has a boundary equilibrium E 1 and a positive equilibrium E 1 * when (1) or (2) holds. Under these conditions, E 0 , E r , and E 1 are all unstable, but E 1 * is locally asymptotically stable. Obviously, all x where F x r bx cxy, According to the Bendixson-Dulac discriminant [38], system (1.7) has no limit cycle in the first quadrant, so E 1 * is globally asymptotically stable.
This ends the proof of Theorem 5.1. □ Remark 5.1. Theorem 5.1 shows that for the weak Allee effect case, the stability of E 1 * is not affected, that is, systems (1.2) and (1.7) admit a positive equilibrium E 1 * , which is globally asymptotically stable. We also find that the values of x 1 * and y 1 depend on the value of a and m, which means that the additive Allee effect has an effect on the final density of the species. Proof. From Table 1, we find that the system (1.7) has two boundary equilibria E 1 and E r when system (1.7) satisfies one of conditions (1)- (4). Under these conditions, E 0 is always unstable and E r is locally asymptotically stable. Next, we will prove that E r is globally asymptotically stable. First, let us consider the system = . Therefore, V satisfies Lyapunov's asymptotic stability theorem [40], so y 0 = of the system (5.1) is globally asymptotically stable.
Noting that the second equation of the system (1.7) is only related to y, and independent of x. Therefore, under the assumption of Theorem 5.2, we can conclude that Hence, for any sufficiently small ε 0 > , there exists an integer T 0 > such that Then, it follows from the first equation of system (1.7): Applying Lemma 5.1 to the above inequality leads to From (5.2) and (5.3), we can conclude that Consequently, E r is globally asymptotically stable.
This ends the proof of Theorem 5.2. □

Numeric simulations
In this section, we use numerical simulations to verify the above theorem.
Example 6.1. We consider the following system: we obtain m ad = and ae d < ; then E 0 is unstable, E r is a saddle-node, E 1 is a saddle, E 1 * is a stable node (Figure 3(a)). For a d m 0.3, 0.3, 0.09 = = = , we obtain m ad = and ae d = ; then E 0 is a saddle, and E r is a stable node (Figure 3 , we obtain ad m m < < * and ae d < ; then E 0 , E 1 and E 2 * are saddle points, E r and E 1 * are stable nodes, and E 2 is an unstable node (Figure 4(a)). For a d m 0.2, 0.1, 1 = = = , we obtain ad m m < < * and ae d > , then E 0 is a saddle, E r is a stable node (Figure 4(b)). (4) For a d m 0.5, 1.5, 1, = = = we obtain m m = * and ae d < ; then E 0 is a saddle, E r is a stable node, and E 3 and E 3 * are saddle-nodes (Figure 4(c)). For a d m 1.5, 0.5, 1 = = = , we obtain m m = * and ae d > , then E 0 is a saddle, E r is a stable node (Figure 4(d)).  (Figure 4(e)).

Conclusion
In this article, we proposed and studied a commensalism model with the additive Allee effect. We study the dynamics behaviors under three conditions, i.e., m ad < , m ad = , and m ad > . For the case m ad < , system (1.7) has four equilibria, of which three boundary equilibria are always unstable, and the unique positive equilibrium E 1 * is globally asymptotically stable. Compared with system (1.2), the weak Allee effect in the system (1.7) has no influence on its stability but changes the position of the equilibria, when the Allee effect increases, the final density of x and y species are decreasing. For the case m ad = , if ae d < , then the situation is the same as m ad < ; if ae d = , system (1.7) has two boundary equilibria E 0 and E r , in which E 0 is unstable and E r is globally asymptotically stable, which means that the second species will be driven to extinction. If ae d > , system (1.7) has two boundary equilibria E 0 and E r , both of them are unstable.
For the case m ad > , we have two new findings. The first one is that system (1.7) has at least two boundary equilibria and at most six equilibria, this means that the additive Allee effect affects the number of equilibria and their stability. The other is that E r is always stable, and E r is globally asymptotically stable under some sufficient conditions, this shows that the additive Allee effect will cause the extinction of the second species.
In addition, from Theorems 4.1 to 4.4, we also proved that system (1.7) has saddle-node bifurcations at E 3 and E 3 * , respectively, and transcritical bifurcations at E 0 and E r under some suitable assumptions, respectively.
Through the above analysis, we can conclude that when the additive Allee effect is weak, both species x and y can survive, and the additive Allee effect only affects the position of the equilibria. However, when the additive Allee effect presents as a strong Allee effect, the dynamic behaviors of two species have changed, and the second species even faces the risk of possible extinction, which is quite different from the findings in [2,7,8]. Moreover, in some conditions, system (1.7) has saddle-node bifurcations and transcritical bifurcations, which are also not found in [2,7,8].
It seems that different types of Allee effect expression may make results in different dynamic behaviors, it seems interesting to investigate the commensalism model with additive Allee effect and functional response; we leave this for future investigation.