Dynamics of a modified Leslie-Gower predator-prey model with double Allee e ff ects

: In this paper, we investigate the dynamic behavior of a modified Leslie-Gower predator-prey model with the Allee e ff ect on both prey and predator. It is shown that the model has at most two positive equilibria, where one is always a hyperbolic saddle and the other is a weak focus with multiplicity of at least three by concrete example. In addition, we analyze the bifurcations of the system, including saddle-node bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation. The results show that the model has a cusp of codimension three and undergoes a Bogdanov-Takens bifurcation of codimension two. The system undergoes a degenerate Hopf bifurcation and has two limit cycles (the inner one is stable and the outer one is unstable). These enrich the dynamics of the modified Leslie-Gower predator-prey model with the double Allee e ff ects.


Introduction
Leslie and Gower proposed a predator-prey model [1,2], as follows: where x and y denote the average population densities of the prey and predator at time t, respectively; K, r, s, m and n are all positive; K represents the environmental carrying capacity; r and s are the intrinsic growth rates of the prey and predator, respectively; m is the maximum per capita predation rate; n is a measure of the quality of the prey as food for the predator.In model (1.1), the environmental capacity of predators is directly proportional to the number of prey (i.e., nx), and y nx is called the Leslie-Gower term.
Korobeinikov [3] and Hsu and Huang [4] showed that system (1.1) has a globally asymptotically stable positive equilibrium under certain conditions.Lindstrom [5] studied the nonexistence and existence of limit cycles of system (1.1).
In system (1.1), predators are thought to feed on a single prey species.In the world, several predators pursue a wide range of prey.That is, the predator will seek alternative food sources if the predator's favorite food is not in sufficient supply.This type of predator is called a generalist predator, and it includes foxes, common buzzards, cats, etc. [6].In order to study the situation in which the predator is a generalist predator, Aziz-Alaoui and Okiye [7] replaced nx with nx + c and considered a predator-prey model with other food sources and a Holling type II functional response as follows: where y nx+c is called the modified Leslie-Gower term; c can be seen as other food sources for the predator.The authors [7] studied the boundedness of the solution of system (1.2) and proved that the interior positive equilibrium is globally asymptotically stable under certain parameter conditions by constructing a suitable Lyapunov function.Nindjin et al. [8] discussed the effect of time delay on the stability of the positive equilibrium.Xiang et al. [9] rigorously analyzed the high codimension bifurcation of system (1.2), such as the Hopf bifurcation of codimension 2 and degenerate Bogdanov-Takens bifurcation of codimension 3.Under the condition that the coefficients of system (1.2) are periodic, Zhu and Wang [10] proved the existence of positive periodic solutions and obtained some sufficient conditions for the global attractivity of positive periodic solutions.For more interesting results on the modified Leslie-Gower term, please refer to [11][12][13].
Allee [14] pointed out that clustering is conducive to the growth and survival of the population, but excessive sparsity and overcrowding can prevent the growth of the population; he also found that each population has its optimal population density.A population is endangered when its density falls below a certain threshold; that is, it has a minimum density to sustain the population.When the population density is too low, it will be difficult for individuals to find mates or resist 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, which leads to more complex dynamic behavior.Lots of biological phenomena can cause the Allee effect, such as mating difficulty, anti predator defense and genetic drift.One of the Allee effect's forms expression is the multiplicative Allee effect, which can be written as follows for a single species: where a is the Allee threshold.When 0 < a < K, that is, the population density is small, the per capita growth rate of the population is negative, which is called a strong Allee effect.Arancibia-Ibarra [15] proposed the following model with the multiplicative Allee effect and a generalist predator: 3) The author demonstrated the existence of separatrices that separate basins of attraction in the phase plane, which is associated with the oscillation, coexistence and extinction of predator-prey populations.
Through numerical simulations, they showed that system (1.3) undergoes Hopf bifurcation and Bogdanov-Takens bifurcation.Considering system (1.3) with Holling's type II functional response, Arancibia-Ibarra and Flores [16] investigated the different bifurcation and showed that the system exhibits the multi-stability phenomenon.Yin et al. [17] studied a predator-prey model with the Allee effect and prey refuge, and they showed that the system has two limit cycles and a Bogdanov-Takens bifurcation of codimension 3. Some scholars [18][19][20][21][22] have studied the impact of the other functional response functions and Allee effect on the dynamical behaviors of Leslie-Gower systems.
Most scholars have considered the impace of the Allee effect on prey.In fact, the Allee effect on predator populations can also affect the dynamic behavior of the system.For predators, when the population density is low, the success rate of cooperative hunting will decrease; the probability of finding a mate will also decrease, which will lead to a decrease in the population birth rate.Therefore, there is also the Allee effect in predator populations.Recently, the study of predator-prey models with the Allee effect on the predator has attracted the interest of a number of scholars.Fox example, Feng and Kang [23] investigated a predator-prey model with the Allee effect on both prey and predator; they showed that the double Allee effect greatly altered the survival of these two species.Alves and Hilker [24] studied the relationship between hunting cooperation and the Allee effect in the predator population, as well as its impact on the predator-prey system.
Therefore, inspired by [15,23], we consider a modified Leslie-Gower predator-prey system with double Allee effects on predator and prey, as follows: where all parameters are positive and 0 < a < K. Here, y y+b represents the Allee effect, and the per capita growth rate of the predator changes from s to sy y+b with the influence of the Allee effect [25].Obviously, as b increases, the Allee effect becomes stronger and the per capita growth rate of the predator is slower.
For simplicity, making a dimensionless transformation by using and dropping the bar, system (1.4) becomes where 0 < a < 1 and the other parameters are positive.From the biological background, assume that the following initial conditions: The rest of this paper is organized as follows.We respectively discuss the existence and stability of the equilibria in Sections 2 and 3.In Section 4, we investigate the existence of various bifurcations, such as saddle-node bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation.In Section 5, we give numerical simulations to show the influence of the Allee effect in the predator population on the dynamical behavior of the system.The paper ends with a brief conclusion.

Existence of equilibria
First, we show that all solutions of system (1.5) are positive and bounded if the initial condition is given by (x(0), y(0 Lemma 2.1.All solutions of system (1.5) are positive and bounded for the initial condition (x(0), y(0)) ∈ Ω + .
Since the initial values x(0) > 0 and y(0) > 0, we get that x(t) > 0 and y(t) > 0 for t > 0. Therefore, solutions of system (1.5) are positive.If x ≥ 1, from the prey equation of system (1.5), we obtain that ẋ < x (1 − x) (x − a) ≤ 0, which implies that lim sup t→∞ x(t) ≤ 1.According to lim sup t→∞ x(t) ≤ 1, we have that x(t) < 1 for large values of time t.In addition, from the predator equation of system (1.5), we have that ẏ < sy 1 − y 1+c for large values of time t.Then, we have lim sup t→∞ y(t) < 1 + c.Hence, solutions of system (1.5) are bounded.The proof is completed.
Next, we study the existence of equilibria of system (1.5).Obviously, system (1.5) has a trivial equilibrium E 0 (0, 0) and two predator-free equilibria E 1 (1, 0) and E 2 (a, 0).When c > p, system (1.5) has a prey-free equilibrium E 3 (0, c − p).Note that the positive equilibria of system (1.5) satisfy the following equations: From the first equation of (2.1), we can obtain the prey isocline of system (1.5): From the second equation of (2.1), we get the predator isocline of system (1.5): The prey isocline y 1 is a parabola passing through points E 1 , E 2 with vertex at a+1 2 , (a−1) 2 4m .The predator isocline y 2 is a monotonically increasing straight line passing through E 3 with a slope of 1. Also, we calculate that the slope of curve y 1 at E 2 is 1−a m .From (2.1), we obtain 2) The discriminant of Eq (2.2) is Obviously, the number of intersection points of curves y 1 and y 2 depends on the sign of ∆.If ∆ > 0, Eq (2.2) has two real roots x * 1 and x * 2 , which means that the number of intersection points of curves y 1 and y 2 is 2. If ∆ = 0, Eq (2.2) has a unique real root x * ; that is, the curve y 1 has only one intersection point with y 2 .
Note that the positive equilibria are determined by the intersection of the curves y 1 and y 2 in the first quadrant.Based on the positional relationships of E 1 , E 2 , E 3 and the slope of curve y 1 at E 2 , we consider four cases: (1) When p − c < a, (1.a) if 1−a m > 1, the number of intersections of y 1 and y 2 may be 0, 1 or 2, depending on the sign of ∆.For example, y 1 and y 2 have two intersections (see Figure 1(a)) or only one intersection (see Figure 1 , the predator isocline is above the prey isocline in the first quadrant, as shown in Figure 1(c).So, there is no intersection of y 1 and y 2 , i.e., system (1.5) has no positive equilibrium.
( (2.b) if 0 < 1−a m ≤ 1, the predator isocline and the prey isocline have only one intersection at E 2 , as shown in Figure 1(e).Hence, there is no intersection in the first quadrant, which implies that system (1.5) has no positive equilibrium.
(3) When a < p − c < 1, there is one intersection of y 1 and y 2 in the first quadrant; see Figure 1(f).Hence, system (1.5) has a positive equilibrium.(4) When p − c ≥ 1, in the first quadrant, the prey isocline is to the left of the predator isocline, as shown in Figure 1(g) and (h).Therefore, system (1.5) has no positive equilibrium.Let Notice that p * * ≥ p * .Clearly, when c * 1 ≤ 0, or when c * 1 > 0 and c > c * 1 , we can get that p * > 0. By a simple calculation, we get that c Based on the above discussion, we derive the following theorem about the existence of the positive equilibria of system (1.5).
Theorem 2.1.The existence of the positive equilibria of system (1.5) are classified as follows. (

Stability of equilibria
Now we discuss the stability of system (1.5).The Jacobian matrix of any equilibrium of system (1.5) is The local stability of equilibria is determined by the eigenvalues of the Jacobian matrix (3.1) at each equilibrium.
Theorem 3.1.The boundary equilibrium E 3 (0, c − p) is always a stable node if c > p.
Obviously, J(E 3 ) has two eigenvalues λ 1 = −a − m(c − p) < 0 and λ 2 = − s(c−p) 2 c 2 < 0, which implies that E 3 is a stable node (see Figure 2(a)).Theorem 3.2.For equilibrium E 0 , we have the following conditions: (1) if p < c, E 0 is an attracting saddle node, including a hyperbolic sector in the upper half-plane; (2) if p = c, E 0 is a stable degenerate node; (3) if p > c, E 0 is an attracting saddle node, including a parabolic sector in the upper half-plane.
Proof.The Jacobian matrix of system (1.5) at E 0 (0, 0) is Obviously, J(E 0 ) has one zero eigenvalue, which means that E 0 is a degenerate equilibrium.Performing a Taylor expansion at the origin and applying dτ = −adt (still denoting τ as t), then system (1.5) becomes From the center manifold theorem, we assume that x = α 1 y 2 + β 1 y 3 + o(|y| 3 ).Substituting this into the first equation of system (3.3),we get that α 1 = 0 and β 1 = 0. Substituting x = o(|y| 3 ) into the second equation of system (3.3),we can obtain the reduced system, restricted to the center manifold: Clearly, s c 2 a > 0, which implies that E 0 is a stable degenerate node by Theorem 7.1 in [26] (see Figure 2(b)).The proof is completed.Theorem 3.3.For the equilibrium E 1 , we have the following conditions: (1) if p < 1 + c, E 1 is an attracting saddle node, with a hyperbolic sector in the upper half-plane; (2) if p = 1 + c, E 1 is a stable degenerate node; (3) if p > 1 + c, E 1 is an attracting saddle node, with a parabolic sector in the upper half-plane.
Proof.The Jacobian matrix of system (1.5) at E 1 (1, 0) is which means that J(E 1 ) has one zero eigenvalue.So, E 1 is a degenerate equilibrium.Making the transformation (x 1 , y 1 ) = (x − 1, y) to move E 1 to the origin, system (1.5) becomes According to Theorem 7.1 in [26], if b 02 0, i.e., p 1 + c, E 1 is an attracting saddle node (see Figure 2(a), (b) and (c)).If p = 1 + c, system (3.5) is reduced to the following system: where Based on the center manifold theorem, let x2 ), and substitute it into the first equation of system (3.6);then, we have that 2 ) into the second equation of system (3.3),we obtain the reduced system restricted to the center manifold, as follows: E 1 is a stable degenerate node (see Figure 2(d)) since a < 1 (see [26]).The proof is completed.
Theorem 3.4.For the equilibrium E 2 , we have the following conclusions.
(1) When p > a + c, E 2 is a repelling saddle node, with a hyperbolic sector in the upper half-plane. ( (3) If p < a + c, E 2 is a repelling saddle node, with a parabolic sector in the upper half-plane.
Proof.The Jacobian matrix of system (1.5) at E 2 (a, 0) is The two eigenvalues of J(E 2 ) are λ 1 = (1 − a)a and λ 2 = 0; that is, E 2 is a degenerate equilibrium.By the transformation (x 1 , y 1 ) = (x − a, y), we have Next, making the transformation and applying dτ = (1 − a)adt (still rewriting τ as t), system (3.7)becomes the following system: where From Theorem 7.1 in [26], if d 02 0, i.e., p a + c, E 2 is a repelling saddle node (see Figure 2(a), (b) and (d)).Assume that p = a + c; system (3.8)becomes where By using the center manifold theorem and the first equation of system (3.9),we have Substituting (3.10) into the second equation of system (3.9),we obtain the reduced system, restricted to the center manifold, as follows: Hence, E 2 is a repelling saddle node since a 2 s(1−a) (a+c) 2 > 0 (see [26]).The proof is completed.(a) E 0 , E 1 are both attracting saddle nodes, E 2 is a repelling saddle node and E 3 is a stable node with a = 0.5, m = 0.8, s = 0.6, c = 1.2, p = 0.5.(b) E 0 is a stable node, E 1 is an attracting saddle node and E 2 is a repelling saddle node with a = 0.5, m = 0.8, s = 0.6, c = 0.5, p = 0.5.(c) E 0 , E 1 is an attracting saddle node and E 2 is an unstable node with a = 0.5, m = 0.8, s = 0.6, c = 0.5, p = 1.(d) E 0 is an attracting saddle node, E 1 is a stable node and E 2 is a repelling saddle node with a = 0.5, m = 0.8, s = 0.6, c = 0.5, p = 1.5.
Next, we study the stability of the positive equilibria of system (1.5).The Jacobian matrix of system (1.5) at positive equilibria is The determinant and trace of (3.12) are, respectively, By simple calculation, we get which means that E 4 is a saddle.Hence, we have the following theorem about the stability of positive equilibria E 4 and E 5 .

Define
Theorem 3.6.Assume that E 5 exists; then, E 5 is a focus or center if s = s > 0.
Proof.By simple calculation, we obtain and we get that T r(J(E 5 )) > 0, which means that E 5 is unstable.If s > s > 0, we get that T r(J(E 5 )) < 0, which implies that E 5 is stable.If s = s > 0, we get that T r(J(E 5 )) = 0, which implies that the eigenvalues of J(E 5 ) are a pair of conjugate complex roots and E 5 is a focus or center.The proof is completed. Define If the conditions of Theorem 2.1(1.b)hold, then s * > 0. When p = p * , by simple computation, we obtain where Assume that s s * , that is T r(J(E * )) 0. By applying the transformation where and the coefficients A i (i = 1, 2, 3, 4) are given in Appendix A. Using the conditions of Theorem 2.1(1.b),we get that ē20 > 0. From Theorem 7.1 in [26], E * is a saddle node.Considering the time variable, E * is an attracting saddle node if s > s * (see Figure 3(a)), and a repelling saddle node if 0 < s < s * (see Figure 3(b)).The proof is completed.When the conditions of Theorem 2.1(1.b)hold and s = s * , that is, T r(J(E * )) = 0, J(E * ) has two zero eigenvalues.Using the following two lemmas, E * is a cusp of codimension 2 or 3.

Lemma 3.2 ( [27]
).The system given by ẋ = y, ẏ = x 2 + a 30 x 4 + y(a 21 x 2 + a 31 x 3 ) + y 2 (a 12 x + a 22 x 2 ) + o(|x, y| 4 ) by some nonsingular transformations in the neighborhood of (0, 0), where G = a 31 − a 30 a 21 . Define Theorem 3.8.Assume that p = p * and s = s * .E * is a cusp of codimension 2 if one of the following conditions holds: (1) 0 < a < 7 − 4 √ 3, and either Proof.If s = s * , then J(E * ) has two zero eigenvalues and system (3.13) can be written as follows: where Taking the transformation where According to Lemma 3.1, system (3.15) is equivalent to where If the conditions of Theorem 2.1(1.b)hold, we have that By simple calculation, the discriminant of Ã5 is 3, we obtain the following: g( a+1 Assume that 0 < a < 7 − 4 √ 3 holds.According to the conditions of Theorem 2.1(1.b),E 0 if one of the following conditions holds: (1) 0 which implies that E * is a cusp of codimension 2 [28] (see Figure 4(a)).The proof is completed.
From the proof of Theorem 3.8, if 0 < a < 7 − 4 Hence, the following theorem shows that E * is a cusp of codimension of at least 4.
Theorem 3.9.Assume that 0 < a < 7 − 4 where . where the coefficients of system (3.17Through the following two transformations:  Note that e 20 = − m(−m+1+a) 2 4 < 0. Making the following transformation where Then, the sign of G is determined by B 0 .In fact, from [28], E * is a cusp of codimension 3 if B 0 0 (see Figure 4(b)).E * is a cusp of codimension of at least 4 if B 0 = 0. Remark 3.1.Because B 0 is a complicated polynomial with respect to a and m, it is difficult to discuss whether B 0 is zero.Next, by numerical simulation, we find that the value of B 0 can be zero or nonzero.Let a = 0.01; then, 1+a 2 = 0.505, M 1 = 0.949947725 and B 0 = −12m 5 + 32.32m 4 − 41.0241m 3 + 33.737434m 2 − 14.99700015m + 2.0195960202.

Bifurcation
In this section, we will discuss some bifurcation phenomena that occur in system (1.5), such as saddle-node bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation.

Saddle-node bifurcation
From Theorem 2.1, when p < p * , p = p * and p * < p < p * * , system (1.5) has 0, 1 and 2 positive equilibria, respectively.Therefore, selecting the Allee threshold p = p S N = p * as the bifurcation parameter, and by using Sotomayor's theorem in [28], we verify that system (1.5) undergoes a saddlenode bifurcation around the positive equilibrium E * .Proof.The Jacobian matrix of the positive equilibrium E * can be expressed as Then, Det(J(E * ; p S N )) = 0 and T r(J(E * ; p S N )) 0. So, J(E * ; p S N ) has a zero eigenvalue.Let V and W be the eigenvectors corresponding to the zero eigenvalues of the matrix J(E * ; p S N ) and J(E * ; p S N ) T , respectively.By computation, we obtain Hence, Thus, we have According to Sotomayor's theorem [26], system (1.5) undergoes a saddle-node bifurcation at E * .The proof is completed.
Clearly, when Det(J( Ē5 )) > 0, the stability of the equilibrium Ē5 is determined by the trace T r(J( Ē5 )).Hence, we have the following theorem.
Theorem 4.2.Suppose that then, for system (4.3),Ē5 is (1) a stable hyperbolic focus or node if s * ≤ 0 or s > s * > 0; (2) an unstable hyperbolic focus or node if 0 < s < s * ; (3) a center or fine focus if s = s * > 0. Now, if Theorem 4.2(3) holds, system (4.3) may undergo Hopf bifurcation at Ē5 .First, we check the transversality condition for the occurrence of Hopf bifurcation.By calculation, we get d ds T r(J( Ē5 )) Next, we calculate the first Lyapunov coefficient that can determine the stability of limit cycles around Ē5 .In biology, if two species coexist in the form of periodic oscillations, the system will have a limit cycle.Making the transformations X = x − 1 and Y = y − 1, we have where Letting D = a 10 b 01 − a 01 b 10 , obviously, Taking the following transformation system (4.4) can be expressed as follows where and the coefficients of H 1 (X 1 , Y 1 ) and H 2 (X 1 , Y 1 ) are given in Appendix C. The first-order Lyapunov number in [26] at Ē5 is given by the following formula where the coefficient C 6 is given in Appendix D. Clearly, the sign of l 1 is determined by C 6 .Hence, we have the following theorem.(1) System (4.3) undergoes a subcritical Hopf bifurcation and an unstable limit cycle around Ē5 when C 6 < 0 (see Figure 5 Because the first-order Lyapunov number l 1 is too complicated, we give an example to show that system (1.5) undergoes a degenerate Hopf bifurcation of codimension 3. Letting a = 1 2 and c = 1, system (4.3)becomes where k > 3 2 and p > 2k−3 k−1 .Through a series of transformations and methods described in [26], the first and two Lyapunov numbers are obtained as follows: 6 , and the coefficients C 7 and C 8 are given in Appendix E. , p = 6 10 + 1 10 , s = 289 500 + 1 1000 , system (4.3)underdoes a degenerate Hopf bifurcation and multiple two-limit cycles (the inner one is stable and the outer one is unstable) around Ē5 .(f) Amplified phase portrait of (e).
Therefore, system (4.5) can undergo Hopf bifurcation of codimension 2 and 3.According to the above analysis, we can summarize the following remark.
(1) Ē5 is a weak focus of order 1 if L 1 0.
(3) Ē5 is a weak focus of order of at least 3 if L 1 = L 2 = 0.

Bogdanov-Takens bifurcation
From Thoerem 3.8, the unique positive equilibrium E * is a cusp of codimension 2 under some suitable conditions.In this section, we choose some suitable parameters as bifurcation parameters to show that system (1.5) undergoes a Bogdognov-Taken bifurcation of codimension 2.
Theorem 4.4.Assume that the conditions of Theorem 3.8 hold.Choosing p and c as two bifurcation parameters, system (1.5) undergoes a Bogdanov-Takens bifurcation of codimension 2 around E * .
Proof.Letting p = p * and s = s * , and substituting p = p * + λ 1 and c = c + λ 2 into system (1.5), we get the following system: ẋ where (λ 1 , λ 2 ) is in a small neighborhood of (0, 0).Making the transformations X = x − x * and Y = y − (x * + c − p), system (4.6)becomes where . Taking the following transformations Using the transformations where When λ 1 and λ 2 are sufficiently small, we have then, system (4.9)becomes (still denoting τ as t) where Besides, taking the transformations X 4 = X 3 − ẽ10 2 and Y 4 = Y 3 , system (4.10)becomes where Note that c c * 2 .When λ 1 and λ 2 are sufficiently small, Finally, letting and rewriting τ as t, system (4.11)becomes where We express µ 1 and µ 2 in terms of λ 1 and λ 2 , as follows: where and Note that the transversality condition for the existence of Bogdanov-Takens bifurcation, i.e., holds.According to the result in [28], system (1.5) undergoes a Bogdanov-Takens bifurcation of codimension 2 when (λ 1 , λ 2 ) is in a small neighborhood of (0, 0).The proof is completed.
S N, H and HL respectively denote the saddle-node bifurcation curve, Hopf bifurcation curve and homoclinic bifurcation curve of system (1.5) around E * .

Numerical simulations
We discuss the influence of the Allee effect in the predator population on the dynamical behavior of system (1.5).Letting a = 0.45, m = 0.14, c = 3 and s = 1, Figure 7 shows the bifurcation diagram in the (p, y)-plane of system (1.5).We find that there exist four Allee thresholds: p = p * ≈ 3.14982, p = p H ≈ 3.17156, p = p * * = 3.45 and p = 1 + c ≜ p * = 4.According to Theorem 2.1 and the bifurcation diagram, if the Allee effect parameter satisfies that p < p * or p ≥ p * , system (1.5) has no positive equilibrium.If the Allee effect parameter satisfies that p = p * , system (1.5) has a unique positive equilibrium E * .If p * < p < p * * , system (1.5) has two positive equilibria E 4 and E 5 , where E 4 is always a saddle and E 5 is unstable if p * < p < p H , and stable if p H < p < p * * .If the Allee effect parameter satisfies that p * * ≤ p < p * , system (1.5) has a unique positive equilibria E 5 , which is stable.Also, we give the two-parameter bifurcation diagram of system (1.5) in the (a, p)-plane, as shown in Figure 8.
We selected p as the control parameter and plotted the phase portraits of system (1.5) at different values (see Figure 9).If p = 0, that is, the predator population exists without the Allee effect, system (1.5) has no positive equilibrium and the boundary equilibrium E 3 is globally asymptotically stable, which means that the predator can survive and the prey will tend to extinction (see Figure 9(a)).When p = 3.1 (p < p * ), system (1.5) has no positive equilibrium and the origin is globally asymptotically stable.That is, the predator will become extinct with the influence of the Allee effect on the predator population, which means that both predator and prey will tend to extinction (see Figure 9(b)).When p = 3.165 (p * < p < p H ), system (1.5) has two positive equilibria, where E 4 is a saddle and E 5 is an unstable focus; see Figure 9(c).In this case, the predator and prey will still become extinct.When p = 3.1725 (p H < p < p * * ), E 4 is still a saddle but E 5 becomes a stable focus; also, an unstable limit cycle appears around E 5 ; see Figure 6(d).Hence, the unstable limit cycle acts as a separatrix between the attraction of the origin and E 5 .When p = 3.17642 (p H < p < p * * ), there exists an unstable homoclinic loop in system (1.5); see Figure 9(e).When p = 3.1767 (p H < p < p * * ), the homoclinic loop disappears and system (1.5) has a hyperbolic saddle E 4 and a hyperbolic stable focus E 5 ; see Figure 9(f).That is, the two stable manifolds of saddle E 4 act as a separatrix between the attraction of the origin and E 5 .When p = 3.45 (p * * ≤ p < p * ), system (1.5) has a unique positive stable equilibrium E 5 and a degenerate saddle E 2 ; see Figure 9(g).Obviously, the stable manifold of degenerate saddle E 2 is taken as a separatrix between the attraction of the origin and E 5 .When p = 4 (p ≥ p * ), system (1.5) has no positive equilibrium and a repelling saddle node E 2 ; see Figure 9(h).From Figure 9(h), the stable manifold of E 2 acts as a separatrix between the attraction of the origin and E 1 .
As shown in Figure 9(a), if the predator has no Allee effect, the prey will tend to extinction, but the predator can survive because they have alternative food.When the Allee constant p increases, the alternative food source does not guarantee the survival of the predator.Then, both the predator and prey will become extinct; see Figure 9(b)-(c).However, when the Allee effect on the predator is strong (i.e., the Allee effect constant p is large), there is a bistable phenomenon (see Figure 9(d)-(g)).The unstable limit cycle or the stable manifold of E 2 acts as a separatrix between the origin and E 5 .That is, the prey and predator may be able to coexist.Finally, under the condition that the Allee effect on the predator is strong enough, Figure 9(h) shows that the predator will become extinct, whereas the prey may become extinct or survive, depending on the initial value.On the whole, when the Allee effect in the predator population is strong enough, the predator will become extinct, whereas the prey will survive or become extinct, depending on the initial value.Hence, in contrast to the dynamic behavior of the predator without the Allee effect, a strong Allee effect can lead to the extinction of the predator and the increase of the survival rate of the prey.

Conclusions
In this manuscript, a modified Leslie-Gower predator-prey model with Allee effect on both prey and predator is proposed.We showed that the boundary equilibrium E 3 (0, c − p) is a stable node, while E 0 , E 1 and E 2 are unstable if c > p.Hence, E 3 is globally asymptotically stable if system (1.5) has no positive equilibrium.That is, a weak Allee effect on the predator is conducive to the survival of the predator.However, if p > 1 + c, that is, the Allee effect in the predator population is strong, system (1.5) has no positive equilibrium by Theorem 2.1.Then, the predator and prey do not coexist, which means that a strong Allee effect on the predator is detrimental to the survival of both predator and prey.Moreover, the other three boundary equilibria E 0 , E 1 and E 2 are non-hyperbolic (see Figure 2).We proved that the unique positive equilibrium E * is a saddle node or a cusp of codimension 3 (see Figures 3 and 4).Further, because the expression of B 0 is complicated, we showed that E * is a cusp of codimension of at least 4 by concrete example (see Remark 3.1).We showed that system (1.5) undergoes saddle-node bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation.In more detail, system (1.5) can undergo a degenerate Hopf bifurcation for some suitable parameter values and result in two limit cycles (the inner one is stable and the outer one is unstable; see Figure 5).Biologically, this indicates the bistable phenomenon, where the predator and prey will oscillate periodically or become extinct, depending on the initial values.That is, the predator will coexist and oscillate periodically if the initial values lie within the unstable limit cycle.However, the predator and prey will tend to extinction if the initial values lie outside of the unstable limit cycle.
In addition, we give an example to illustrate that system (1.5) has a weak focus of order of at least 3 and can undergo a degenerate Hopf bifurcation of codimension 3.Moreover, we proved that, within system (1.5), there is a Bogdanov-Takens bifurcation of codimension 2; we also presented its bifurcation curves.
In the absence of an Allee effect on the predator, that is, p = 0, system (1.5) reduces to system (1.3).Arancibia-Ibarra [15] proved the existence of separatrices in the phase plane separating basins of attraction.They showed that system (1.3) has at most two positive equilibria, where the smaller positive equilibrium is always a saddle, whereas the larger positive equilibrium can be either an attractor or a repeller surrounded by a limit cycle.They showed that system (1.3) undergoes Hopf bifurcation and Bogdanov-Takens bifurcation without rigorous mathematical proof.
Incorporating the Allee effect on the predator into system (1.3), we investigated the stability and bifurcation of system (1.5) by using the Allee effect as a threshold condition.When the Allee effect on the predator is weak (i.e., p < c), E 3 is globally asymptotically stable if system (1.5) has no positive equilibrium (see Figure 9(a)).As the Allee effect constant on predator increases, E 3 disappears, which means that the origin is globally asymptotically stable (see Figure 9(b)-(c)).However, in [15], the predator always can always survive due to alternative food.Hence, we showed that, even if the predator has the alternative food source, as long as the Allee effect on the predator is strong enough, both prey and predator will become extinct, which is different from [15].When the Allee effect on the predator is sufficiently strong, the predator will tend to extinction and the extinction and existence of the prey depend on the initial value.Therefore, a strong Allee effect on the predator is beneficial to the survival of the prey, but detrimental to the survival of the predator.Unlike [15], we give rigorous mathematical proof to prove that E * is a cusp of codimension 3, and that system (1.5) undergoes degenerate Hopf bifurcation and Bogdanov-Takens bifurcation of codimension 2. We also showed that system (1.5) has a cusp of codimension of at least 4 and can undergo a degenerate Hopf bifurcation of codimension 3 by concrete examples.Therefore, compared with system (1.3), the Allee effect on the predator greatly affects the dynamical behavior of the system, resulting in more complex dynamical behavior.This enriches the dynamics of the modified Leslie-Gower predator-prey model with the double Allee effect.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
a) if 1−a m > 1, y 1 and y 2 have only one intersection in the first quadrant; see Figure1(d).Hence, system (1.5) has only one positive equilibrium;