Dynamic analysis of a modiﬁed algae and ﬁsh model with aggregation and Allee e ﬀ ect

: In the paper, under the stress of aggregation and reproduction mechanism of algae, we proposed a modiﬁed algae and ﬁsh model with aggregation and Allee e ﬀ ect, its main purpose was to further ascertain the dynamic relationship between algae and ﬁsh. Several critical conditions were investigated to guarantee the existence and stabilization of all possible equilibrium points, and ensure that the model could undergo transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation and B-T bifurcation. Numerical simulation results of related bifurcation dynamics were provided to verify the feasibility of theoretical derivation, and visually demonstrate the changing trend of the dynamic relationship. Our results generalized and improved some known results, and showed that the aggregation and Allee e ﬀ ect played a vital role in the dynamic relationship between algae and ﬁsh.


Introduction
Nowadays, great quantities of rivers and lakes around the world are more or less affected by algal bloom, which phenomenon has received more and more attention from aquatic ecologists and environmentalists [1,2]. Algae are the simplest vegetative organisms capable of photosynthesis, they have various kinds and are widely distributed. The outbreak of algal bloom will bring a lot of troubles to our life and environment. For examples, some algaes can secrete toxic substances, which can not be easily removed by conventional water treatment processes, then directly affect the quality and safety of water supply. A large number of algae populations will also consume too much oxygen, causing fish and other aquatic organisms to die due to the lackness of oxygen, which also further deteriorates water where X(t) and Y(t) represent the population densities of algae and fish respectively, the ecological significance of all parameters are detailed in the paper [26]. However, the aquatic ecological model (2.1) has two shortcomings. One is that the Allee effect is not considered in the growth function of algae. As far as we know, since the population size of microbial species is often "huge" and most microorganisms reproduce asexually, there are few researches on microorganisms with Allee effect. However, some scholars have found the existing evidences of Allee effect in microbial population experiments. For example, Smith et al. [27] engineered Allee effects in Escherichia coli, and Qi et al. [28] detected Allee effect in experimental population of Vibrio fischeri in a specific experimental environment. The causes of Allee effect are various and different for different species. Compared with algae monomer, algal aggregation has some obvious survival advantages [29][30][31]. There are gaps between algal aggregation, which will increase buoyancy then promote photosynthesis better [29]. Algal aggregation is usually composed of thousands of cells, which is large enough that can effectively prevent zooplankton from preying, and better avoid the invasion of viruses, bacteria and algae phagocytes [30]. Algal aggregation can secrete more microcystins than monomer, which reduces the risk of predation [31]. Thus algal aggregation can make algae to find a more suitable niche, then promote the growth and reproduction. The individual reproduction of algae monomer with low aggregation degree will be relatively reduced, which will result in Allee effect. The other is that the function r 2 Y(1 − X k ) of algae monomer affecting the abundance of fish is not reasonable, this is because that the function includes the influence of algal aggregation. Thus, relatively speaking, the function r 2 Y(1 − X−M K ) is more reasonable. Based on the above analysis, we will construct a modified aquatic ecological model, which can be described by the following differential equations: where N (0 < N < K) is Allee effect threshold, the prey population is doomed to extinction when the prey population density or size is below the threshold. It is worth mentioning that the term of feeding on the alternative prey R 2 Y 1 − X−M K in model (2.2) is more reasonable than it in model (2.1), this is because the aggregated part of algae M is always composed by thousands of units, which is too big for fish, hence fish can only turn to the monomer part of algae X − M for grazing.
There are too many parameters of model (2.2), which will bring inconvenience to our follow-up research, in order to reduce the number of parameters, replacing it with the following variables then model (2.2) can be rewritten as the following dimensionless form: where m = M A , k = K A , n = N A , β = α 1 β 1 R 1 , r = R 2 R 1 and d = D R 1 are positive constants and apparently 0 m k. It is worth noting that the density of prey X must greater than the aggregation parameter M, therefore x must satisfy m x and m must be euqal to zero when x = 0. To illustrate how Allee effect affects the per-capita growth rate of algae population, we numerically show the change trend of per-capita growth rate under different values of Allee effect control parameter n. It can be seen from Figure 1 that if the Allee effect is not considered, the per-capita growth rate of algae population decreases monotonically with the increase of x value, if the Allee effect is considered, the per-capita growth rate of algae population increases first and then decreases monotonically with the increase of x value, and if the algae population density is lower than an Allee threshold, the per-capita growth rate is negative, which implies that the algae population density can not support the large-scale reproduction of algae population, and will lead to the extinction of algae population. At the same time, it is also worth pointing out that the per-capita growth rate curve is concave, which can better describe the proliferation stage of algae population. Thus, considering Allee effect in the construction of algae ecological dynamics model is more in line with the actual ecological and environmental significance. In this paper, firstly, we analyze the relevant dynamic properties of model (2.3), and obtain several conditions of critical threshold to ensure that model (2.3) can undergo transcritical, saddle-node, Hopf and B-T bifurcation. Secondly, we carry out numerical experiments on the dynamic behaviors of model (2.3), which can confirm the effectiveness of theoretical derivation and visually display the dynamic behavior evolution process. Finally, we reveal the quality of dynamic relationship between algae and fish from the view of population dynamics evolution according to results of the numerical simulation.

Results of mathematical analysis
Equilibrium points are the special solutions, which will exhibit rich properties of model (2.3), therefore, the existence and stability of all possible equilibrium points of model (2.3) will be discussed in this section, and we will also use the Poincare − Bendixson T heorem to confirm the existece of a limit cycle.

Existence and stability of equilibrium points
It is obvious that model (2.3) has at most five possible equilibrium points: trivial extinction equilibrium point E 0 (0, 0), two predator-free equilibrium points E 1 (k, 0) and E 2 (n, 0), two coexistence equilibrium points E * 1 (x 1 , y 1 ) and E * 2 (x 2 , y 2 ). The equilibrium points E 1 and E 2 exist unconditionally, and E 0 exists when m 1 from the mathematical perspective, while from the biological perspective it exists without condition. Where x 1 and x 2 are the roots of the equation: then we can obtain the concrete expressions of x 1 and x 2 as corresponding, the concrete expressions of y 1 and y 2 are The existence of coexistence equilibrium points is conditional, and the condition m x i k, i = 1, 2 must be satisfied, this is out of the consideration of biological significance. It is obvious that y i > 0 when x i > max {m, n}. The specific existence conditions of coexistence equilibrium points can be viewed in the appendix. Coexistence equilibrium points have a decisive influence on the dynamic behavior, for an example, the fish will finally die out and model (2.3) will not be persistent if there exists no coexistence equilibrium point.
The stability of equilibrium points can be obtained through the signs of the eigenvalues of Jacobian matrix, firstly we obtain the expression of Jacobian matrix about model (2.3) as then we have the following theorems about the stability of equilibrium points.
Theorem 3.1. E 0 (0, 0) exists when m 1 and it is stable when r < d, while E 0 is a saddle when r > d.
Proof. When x = 0, m must be equal to zero, therefore the Jacobian matrix of E 0 can be written as it is obvious that both the two eigenvalues of matrix J E 0 (0,0) , λ 1 = −1 and λ 2 = r − d, are negative when r < d, then E 0 is stable. However the two eigenvalues of J E 0 (0,0) have opposite signs when r > d, then E 0 is an unstable saddle.
Theorem 3.2. E 1 (k, 0) exists unconditionally, and E 1 is stable when d > β k−m 1+k−m + m k r, but E 1 is an unstable saddle when d < β k−m 1+k−m + m k r. Proof. The Jacobian matrix of E 1 can be written as the two characteristic roots of matrix J E 1 (k,0) are λ 1 = 1 − k n and λ 2 = β k−m 1+k−m + m k r − d, the former is negative since the Allee effect threshold N is smaller than the maximum environmental capacity K, then n < k is satisfied. Therefore, when d > β k−m 1+k−m + m k r, the two eigenvalues of J E 0 (0,0) are both negative, then E 0 is stable, but E 0 is an unstable saddle as d < β k−m 1+k−m + m k r. Theorem 3.3. E 2 allways exists as an unstable equilibrium point. When d < β n−m 1+n−m + r 1 − n−m k , E 2 is an unstable node or focus, otherwise E 2 is an unstable saddle if d > β n−m 1+n−m + r 1 − n−m k . Proof. The expression of the Jacobian matrix around the equilibrium point E 2 is given by: Obviously, J E 2 (n,0) has two characteristic roots λ 1 = 1 − n k and From the previous content that n < k then the root λ 1 is positive, which means E 2 is always unstable. Moreover, E 2 is an unstable node or focus when d < β n−m 1+n−m + r 1 − n−m k , and a saddle when d > β n−m 1+n−m + r 1 − n−m k . Next we will focus on the stability of the internal equilibrium points E * 1 and E * 2 , the Jacobian matrixs of them can be written as where i = 1, 2, then we have the following two theorems, which are under the conditions of their existence.
As for r > k k+1 d − β + √ ∆ , E * 1 is asymptotically stable in a small domain when T r J E * 1 < 0, while E * 1 is an unstable node or focus when T r J E * 1 > 0. Proof. It is not hard to get the following equation through calculation then for the stability of equilibrium point E * 1 , we have the following two situations (1) If 0 < r < k k+1 d − β + √ ∆ , then S 1 < 0, which means Det E * 1 < 0 i.e., the two eigenvalues of J E * 1 have opposite signs, therefore E * 1 is an unstable saddle. (2) If r > k k+1 d − β + √ ∆ , then S 1 > 0, which means Det E * 1 > 0. Under this condition, E * 1 is asymptotically stable in a small domain when T r J E * 1 < 0 i.e., J E * 1 has two negative eigenvalues or the real part of eigenvalues are negative, but E * 1 is an unstable focus or node when T r J E * 1 > 0 i.e., both the two eigenvalues of J E * 1 are positive or have positive real part.
Theorem 3.5. The equilibrium point E * 2 is an unstable saddle as long as it exists.
Proof. Similar to the above theorem, we can obtain the following equation It's obvious that S 2 < 0 due to the existing condition of E * 2 including r > d, which means Det E * 2 < 0, then J E * 2 has a positive eigenvalue and a negative eigenvalue i.e., E * 2 is an unstable saddle under the existing conditions.

Existence of limit cycle
In this subsection, we assume that E * 1 is the unique internal equilibrium point, which is unstable, then as for model (2.3) we have the following main theorem about the existence of limit cycle. Proof. Consider the lines L 1 : x − m = 0, L 2 : x − k = 0, we have Therefore the orbit of model (2.3) will across the lines L 1 and L 2 from left and right, respectively. Moreover, we define the second line L 3 : x + y β − B = 0, then we have here B is a positive constant and large enough. According to the existence of E * 1 from appendix we have r < d, then r − r k (x − m) − d B is a quite small negative number for any specific x ∈ [m, k], , which means that model (2.3) has at least one limit cycle according to Poincare − Bendixson T heorem.  In order to make the results more visualized, we fix the parameters k = 5, n = 0.2, r = 0.15, d = 0.21, m = 0.4 and β = 0.25, then E * 1 is the unique equilibrium point of model (2.3), which is unstable. At the same time, the condition dL 3 dt < 0 is satisfied when B = 120, so it is obvious to find from Figure 2 that there exists one limit cycle at least. The condition dL 1 dt > 0 means that the density values of the two populations will tend to the right side of the line L 1 when the initial density values lie on the line, dL 2 dt < 0 and dL 3 dt < 0 denote similar meanings. Furthermore, the limit cycle describes such a phenomenon in biology that neither of algae and fish will be extinct, but reach a state of periodic oscillation and dynamic coexistence.

Local bifurcation analysis
In this section, the local bifurcation of model (2.3) will be discussed in detail. We not only consider the codimension one bifurcations, scuh as the transcritical, saddle-node and Hopf bifurcation but also explore codimension two bifurcation as B-T bifurcation. Here the transversality conditions for transcritical bifurcation and saddle-node bifurcation will be verified by the Sotomayor's theorem [32].

Transcritical bifurcation
Usually, the boundary equilibrium points are the main research object of transcritical bifurcation. In this subsection, the existence of transcritical bifurcation at the equilibrium point E 1 (k, 0) is studied. From Theorem 2, we know that E 1 is an unstable saddle when g (m) < 0, while E 1 is stable when letting m 1 m 2 be the two possible solutions of equition g (m) = 0, we have Thus, when m 1 or m 2 is in the set [0, k], the equilibrium point E 1 will translate its stability as the value of m passes through m 1 or m 2 . Furthermore, it should be attention that when m = m 1 or m = m 2 , Transcritical bifurcation takes place in model (2.3) at E 1 when m = m 2 with either of the following two cases is satisfied.
Proof. (a) m 1 ∈ (0, k), when case 1 or case 2 is satisfied. And E 1 shifts it's stability from stable to unstable with m passing through m 1 from left to right. When m = m 1 , the Jacobion matrix at E 1 can be expressed as we assume that V and W respectively are eigenvectors of J (E 1 ;m) and J T (E 1 ;m) with respect to eigenvalue zero, which means.
due to β 0 and n k, it's obvious that Therefore, by the S otomayor s theorem, transcritical bifurcation takes place in model (2.3) at E 1 when m = m 1 . (b) We omitted the process since it is similiar to the proof of (a).

Saddle-node bifurcation
The existing conditions of the two internal equilibrium points E * 1 and E * 2 are given in the appendix. When β is chosen as the bifurcation parameter, the collision of E * 1 and E * 2 may overlap as an equilibrium point E S N (x S N , y S N ) when ∆ = 0. With the change of the value of β, the value of ∆ will change, and when ∆ < 0, there is no internal equilibrium point E S N . The change of equilibrium point number is caused by the taking place of saddle-node bifurcation in model corresponding, we have Theorem 4.2. Saddle-node bifurcation takes place in model (2.3) when β = β S N under the conditions of (1) d > r, Proof. The equilibrium point E S N exists under the above two conditions according to the Appendix. The Jacobian matrix at E S N when β = β S N can be written as Letting the eigenvectors of the zero eigenvalues of J E S N and J T E S N are V and W respectively, where

then, we can get
Clearly, the transversality condition for the taking place of saddle-node bifurcation at E S N is satisfied when β = β S N . Therefore, it is obvious that the number of internal equilibrium point of model (2.3) changes from zero to two when the value of parameter β passes through β = β S N .

Hopf bifurcation
From the analysis in the previous content, it's easy to conclude that the equilibrium point E * 1 has different stability under different restrictions of parameters, which may caused by Hopf bifurcation. In order to figure out how algal aggregation and Allee effect influence the dynamic behavior of model (2.3), m and n are chosen as the control parameter of Hopf bifurcation respectively, then we have the following two Theorems. Proof. As for matrix J E * 1 , the characteristic equation of it can be written as λ 2 − T r J E * 1 λ + Det J E * 1 = 0, then a Hopf bifurcation takes place when m = m H p such that Theorem 3.4. Therefore we only need to certify the transersality condition (3) to guarantee the changes of stability of E * 1 through Hopf bifurcation.
The condition (3) is satisfied through our numerical simulation, then Hopf bifurcation takes place in model (2.3) at m = m H p . To find out the stability of the limit cycle brought by Hopf bifurcation, the first Lyapunov number l 1 at the equilibrium point E * 1 is going to be computed following. Firstly, translating the equilibrium point E * 1 to the origin by using the transformation x = x m + x 1 , y = y m + y 1 , then model (2.3) can be rewritten as , where a 10 , a 01 , b 10 , b 01 are the components of the Jacobian matrix at E * 1 , we have a 10 + b 01 = 0 and Det = a 10 b 01 − a 01 b 10 > 0. The coefficients a i j and b i j are determined by The limit cycle around equilibrium point E * 1 caused by Hopf bifurcation is unstable if l 1 > 0, otherwise the limit cycle is stable if l 1 < 0. For the expression of the first Lyapunov number l 1 is too cumbersome to tell the sign, the accuracy of this Theorem will be verified in the Section 5. Proof. This Theorem is similar to the above one, we only need to prove the third condition Through calculation we can obtain then we complete the proof.

Bogdanov-Takens bifurcation
Usually several main parameters will affect model (2.3) collectively, therefore it is significant to focus on the dynamic behavior caused by the combined parameters. We study a bifurcation of codimension two with parameters m and β in this section, we pay main attention to the B-T bifurcation, which is caused by the simultaneous occurrence of saddle-node bifurcation and Hopf bifurcation. Substituting m = m BT + ξ 1 , β = β BT + ξ 2 into model (2.3), where ξ 1 and ξ 2 stand for two small perturbations. Then we can obtain the following model Through introducing new variables u = x − x 1 and v = y − y 1 , the equilibrium point E * 1 is translated to the origin, we have du dt = a 00 (ξ) + a 10 (ξ) u + a 01 (ξ) v + a 20 (ξ) u 2 + a 11 (ξ) uv + a 02 (ξ) v 2 , and P 3 (u, v, ξ) is power series in (u, v) with terms u i v j satisfying i + j 4, whose coefficients are depend on ξ 1 and ξ 2 smoothly.
Then, in a small domain of the origin (0, 0), we take the following C ∞ change of coordinates:  and P 4 (n 1 , n 2 , ξ) is power series in (n 1 , n 2 ) with terms n i 1 n j 2 satisfying i + j 4, whose coefficients are depend on ξ 1 and ξ 2 smoothly.

Results of simulation analysis
In real life, ecological control of cyanobacteria bloom is one of the most effective methods, especially, controlling algae bloom by fish. Although fish can effectively graze algae, algae can prevent grazing through aggregation. Thus, to better understand the dynamic variation of fish and algae, some numerical simulation work needs to be implemented. Since model (2.3) contains too many parameters, we fix some of the parameters as follows for the seek of convenient: k = 5, n = 0.2, r = 0.15, d = 0.21. On the premise of the fixed parameters previously, we fix the parameter m = 1.5 and let parameter β vary within a certain range as Figure 3(a). It is can be find from Figure 3(a) obviously that model (2.3) has abundant dynamic properties. When β = β S N = 0.1748528137 and β = β TC = 0.2121428571, there takes place a saddle-node bifurcation and transcritical bifurcation in model (2.3) respectively. Firstly the internal equilibrium points E * 1 (x 1 , y 1 ) and E * 2 (x 2 , y 2 ) are not exist when β < β S N . In the meantime, E 0 and E 2 are unstable, but E 1 is a stable node. With the value of β increasing and passing through β S N , there appear two additional internal equilibrium points E * 1 (x 1 , y 1 ) and E * 2 (x 2 , y 2 ), the former is a stable node while the latter is an unstable saddle, this process can be seen more clearly from Figure  4. Secondly, the internal equilibrium point E * 2 (x 2 , y 2 ) will coincide with the boundary equilibrium point E 1 (5, 0) with the value of β increasing and passing through β TC . And this collision of the two equilibrium points changes E 1 (5, 0) from a stable node to an unstable saddle. Similar to the above approach, now we fix an additional parameter β = 0.17488281, then let parameter m vary within a certain range as Figure 3(b). At the beginning, that is to say when m < m H p = 1.385035335, both two internal equilibrium points E * 1 and E * 2 are unstable, the former is an unstable focus and the latter is an unstable saddle. Furthermore, it is worth mentioning that there is a limit cycle in a small neighborhood containing E * 1 , which is surrendered by one of the unstable trajectory of saddle E * 2 . At the same time, we can calculate that the first Lyapunov number is l 1 = −770.954244π < 0, which means that the limit cycle is stable. Thus, the unstable focus will become a center when the value of m gets greater and reaches m H p . that is to say, E * 1 will turn into a stable focus, as the value of m continues to increase and exceeds m H p . In a word, the detailed dynamic evolution process of Hopf bifurcation can be seen in Figure 5. Furthermore, it is easy to see from Figure 5(d) that if the value of m is less than m H p , model (2.3) has limit cycles around E * 1 , which are represented by the colorful circles in the diagram, if the value of m is larger than m H p , model (2.3) has a stable equilibrium point. Moreover, it is also worth emphasizing that the amplitude of limit cycle is increasing as the value of m is decreasing, which means that if the aggregation area of algae is smaller, the area where algae and fish periodically oscillate and coexist is more wider. And with the passage of time, fish and algae will eventually reach a state of coexistence within a small range of m > m H p .  3) with parameters fixed as mentioned in the main text. (a) Internal equilibrium points E * 1 and E * 2 are not exist when 0.1738 = β < β S N . (b) When β = β S N , there exists a special internal equilibrium point, which is a saddle-node. (c) Exist two internal equilibrium points E * 1 and E * 2 , the former is a stable node and the latter is an unstable saddle, when 0.1758 = β > β S N . (d) Partical enlarged view of the whole process of saddle-node bifurcation. The black point represents a saddle-node, red for E * 1 and blue for E * 2 .
In order to explore how Allee effect affect the dynamic behavior of model (2.3), we choose Allee threshold n as the control parameter to simulate the Hopf bifurcation with k = 5, r = 0.15, d = 0.21, m = 1.385035335, β = 0.17488281, the detailed results are shown in Figure 6. It is can see clearly that the equilibrium point E * 1 is a stable focus and E * 2 is an unstable saddle when 0.1998 = n < n H p = 0.2. That is to say, when the population densities are within a certain range, the algae and fish will coexist at equilibrium point E * 1 . When the value of n is larger than n H p = 0.2, the equilibrium point E * 1 will lose stability, and a stable limit cycle will appear, which implies that a supercritical Hopf bifurcation takes place. Moreover, when the value of n increases, the amplitude of limit cycle will be larger and larger. Obviously, this situation represents that the periodic oscillation coexistence mode between algae and fish will gradually take shape when n exceeds the Hopf bifurcation threshold value. And in a certain small range, the larger the value of n, the more conducive it is for the coexistence of periodic oscillation of algae and fish. At the same time, it is also worth mentioning that the value of Allee threshold n seriously affects the dynamic behavior of model (2.3).  Now, it can be found from Figure 3(b) that model (2.3) can go through two transcritical bifurcations as the value of m increase. When m H p < m < m TC1 = 3.547677046, the equilibrium points E 1 and E * 1 are stable node, the equilibrium point E * 2 is an unstable saddle. As the value of m passes through m TC1 , a transcritical bifurcation takes place, which can cause the collision of saddle E * 2 with stable node E 1 . This collision transforms the stability of boundary equilibrium point E 1 , and makes it an unstable saddle. Then if the value of m increases greater than m TC2 = 3.622895828, model (2.3) occurs a transcritical bifurcation at E 1 again, but this time is the consequence of the collision of E 1 with E * 1 , which can prompt saddle E 1 regain its stability and back to a node. These switching of stability with respect to two Transcritical bifurcations can be better explained through the phase portraits in Figure 7.  In order to study how the parameters m and β synergistically affect the dynamic behavior of model (2.3), the numerically simulation of B-T bifurcation with k = 5, n = 0.2, r = 0.15, d = 0.21 will be carried out. By calculation, we obtain m BT = 1.374006280 and β BT = 0.1748528137, and we have ∂ (σ 00 , σ 01 ) ∂ (ξ 1 , ξ 2 ) ξ 1 =ξ 2 =0 = 0.00009876 -43590.55576 95.31449003 -7.25883992 = 4.154811591 · 10 6 0, 20 = 0.19835682 + 0.30459173ξ 1 + 1.52158465ξ 2 + 0.41550653ξ 2 1 + 1.88040682ξ 1 ξ 2 , θ 11 = −5.51455564+24.53907955ξ 1 +21.15093188ξ 2 +12.40764023ξ 2 1 −100.4590524ξ 1 ξ 2 −121.6857571ξ 2 2 . Therefore, we know that the transformation of parameters is nonsingular, and 20 > 0 and θ 00 0 for small ξ 1 and ξ 2 . Moreover, the local expressions of bifurcation curves S N, H p and HL around the origin are revealed up to second-order approximately as: (1) The saddle-node bifuecation curve satisfies σ 01 (ξ 1 , ξ 2 ) 0, and S N = (ξ 1 , ξ 2 ) | 0.00009876ξ 1 − 43590.55576ξ 2 + 0.06615614ξ 2 1 + 855595.4398ξ 1 ξ 2 + 1003142.522ξ 2 2 , (2) The Hopf bifurcation curve satisfies σ 00 (ξ 1 , ξ 2 ) < 0 and  Meanwhile, the corresponding bifurcation curves is depicted in Figure 8, and the small image in Figure 8 is a partial enlargement of the saddle-node bifurcation curve, which should be exactly coincided with horizontal line ξ 2 = 0. Here we need to explain that the error in the image is inevitable through simulation, but this error does not affect the readability of the numerical simulation results. Obviously, these three curves will divide the visible area into four regions as I, II, III and IV. It is easy to see form Figure 9(a) that when (ξ 1 , ξ 2 ) = (0, 0), model (2.3) has an unique internal equilibrium point, which is a cusp of codimension 2. If we fix ξ 1 = 0.01 for the seek of convenience, then we have the following results.
(1) There is no internal equilibrium point when ξ 2 = −0.0001 in region I, it can see from Figure 9(b) that the fish will finally extinct as long as the algae population exceeds a certain threshold. (2) Model (4.1) has gone through a saddle-node bifurcation when ξ 2 = 0.00001, (ξ 1 , ξ 2 ) can go through the curve S N into region II, there exists a stable focus E * 1 and an unstable saddle E * 2 , which is shown in Figure 9(c).
(3) With the value of ξ 2 varying from 0.00001 to 0.00003, (ξ 1 , ξ 2 ) moves from region I to II and model (4.1) occurs a supercritical Hopf bifurcation (see Figure 9(d)), which deprives the stability of E * 1 and produces a stable limit cycle surrounding E * 1 . (4) As the value of ξ 2 increases to 0.00005307, (ξ 1 , ξ 2 ) is exactly locating on the curve HL, the stable limit cycle grows and goes through the saddle E * 2 and converts to an unstable homoclinic orbit, which can be seen in Figure 9(e). (5) The homoclinic orbit disappears when ξ 2 = 0.00007 and (ξ 1 , ξ 2 ) is in region IV, and there exists an unstable focus E * 1 and a saddle E * 2 (see Figure 9(f)). From the above numerical example, it can be seen that the aggregation effect and conversion rate have a great influence on model (2.3), and the two populations show a series of rich properties such as extinction, coexistence and periodic oscillation near the B-T bifurcation point.
Based on the above numerical simulation analysis, we can know that the values of three key parameters m, n and β have an important influence on the dynamic behaviors of model (2.3), which can not only essentially change the dynamic characteristics, but also affect the survival and extinction of algae and fish. It is clearly visible from Figures 3(a) and 4 that the value of key parameter β can lead to the taking place of a saddle-node bifurcationthe in model (2.3), which can promote algae and fish to form a new stable coexistence model. It must be stated from Figures 3(b), 5 and 7 that the value of the key parameter m is very important for the occurrence of subcritical Hopf bifurcation and transcritical bifurcation of model (2.3), which is not only related to whether algae and fish can survive for a long time, but also can urge algae and fish to form a stable periodic oscillation coexistence mode. It is more worthy of our clear understanding from Figures 8 and 9 that the synergistic action mechanism of key parameters m and β plays an important role in changing the dynamic characteristics and internal essential laws of model (2.3), which can not only impel algae and fish to form two new types of periodic oscillation coexistence modes (Limit cycle coexistence mode and homoclinic orbit coexistence mode), but also lead to the extinction of algae and fish, or the extinction of fish and the outbreak of algae bloom. All in all, it is necessary to consider Allee effect and aggregation effect in building aquatic ecological model to further ascertain the dynamic relationship between algae and fish.  Two internal equilibrium points appear through a saddle-node bifurcation, E * 1 is a stable focus and E * 2 is an unstable saddle, when (ξ 1 , ξ 2 ) = (0.01, 0.00001) lies in region II. (d) A stable limit cycle arises surrounding E * 1 with the happening of Hopf bifurcation as (ξ 1 xi 2 ) into region III. (e) When (ξ 1 , ξ 2 ) = (0.01, 0.00005307) lies on the curve HL, the limit cycle will get larger and approaches to the saddle E * 2 then becomes a homoclinic orbit. (f) The homoclinic break as (ξ 1 , ξ 2 ) into region IV, E * 1 and E * 2 are unstable, the former is a focus while the latter is a saddle.

Conclusions
Although the dynamic behaviors of the predator-prey model with Allee effect in the single or both species have been studied by many mathematicians and biologists, there was relatively few literature on the combination of it with algal aggregation. Within the research framework of mutual restriction between algae and fish, we proposed a modified algae and fish model to probe bifurcation dynamic behaviors. Firstly, we studied all possible equilibrium points of model (2.3) and their stability, which could not only directly give the ideal coexistence mode of algae and fish, but also provide a theoretical basis for the subsequent discussion of bifurcation dynamics. Secondly, when there is only one internal equilibrium point in model (2.3), we gave a threshold condition about aggregation effect parameter m and Allee effect parameter n to ensure that model (2.3) has a limit cycle, this result implied that the relationship between aggregation effect parameter m and Allee effect parameter n was very important for whether algae and fish could exist periodic oscillation survival mode. Finally, by selecting the absorption-conversion rate β, aggregation effect parameter m and Allee effect threshold n as control parameters respectively, we theoretically derived some key threshold conditions to compel that model (2.3) could undergo transcritical, saddle-node, Hopf and B-T bifurcation, these bifurcation dynamic behaviors could force the internal essential changes in the dynamic relationship and coexistence mode between algae and fish. Moreover, it had to be said that these theoretical results not only could summarize and develop the previous theoretical research results, but also further promote the rapid development of bifurcation dynamics in aquatic ecosystem.
In order to verify the feasibility of the theoretical analysis results and visually explore dynamic relationship between algae and fish, a large number of bifurcation numerical simulation results were implemented. Through analysis and comparison, it could be seen that in model (2.3) aggregation effect parameter m and Allee effect parameter n played an important role in the occurrence and evolution of bifurcation dynamics, which also indirectly showed that the coexistence mode of algae and fish depended heavily on aggregation effect and Allee effect. Furthermore, through numerical analysis of B-T bifurcation behavior, it was worth us to clarify that value relationship of the absorption-conversion rate β and aggregation effect parameter m could promote that model (2.3) experienced saddle-node, Hopf and Homoclinic bifurcation, these bifurcation behaviors could not only force algae and fish to form three new coexistence modes, but also was the internal driving force to dynamically adjust their coexistence mode. In a word, it should be emphasized that when we used the mathematical ecological model to study the dynamic relationship between algae and fish, aggregation effect and Allee effect were one of the ecological and environmental factors, which needed to be considered urgently.
The theoretical and numerical results of this paper can get the following four results: (1) If the dynamic behavior of transcritical bifurcation can occur in model (2.3), which means that algae and fish can change from a single population survival mode to a dual population sustainable survival mode. (2) If the dynamic behavior of saddle-node bifurcation can occur in model (2.3), which hints that algae and fish can change from a dual population unsteady coexistence mode to a dual population steady coexistence mode. (3) If the dynamic behavior of Hopf bifurcation can occur in model (2.3), which shows that algae and fish can change from a dual population steady-state coexistence mode to a dual population periodic oscillation mode. (4) If the dynamic behavior of B-T bifurcation can occur in model (2.3), which suggests that the coexistence mode of algae and fish is easily affected by ecoenvironmental factors, and the coexistence mode can change back and forth between an equilibrium point steady-state mode, a stable periodic oscillation mode and an unstable survival mode. Therefore, the four kinds of bifurcation dynamic behaviors occurred in model (2.3) can represent the dynamic change characteristics of the coexistence mode of algae and fish under the dynamic change of ecoenvironmental factors. In other words, the interaction mechanism between algae and fish in nature can be described and explained by the bifurcation dynamic behavior, dynamic evolution process and internal essential characteristics of model (2.3).
In the follow-up research works, we will firstly deepen theoretical research of bifurcation dynamics in some modified ecological models, especially B-T bifurcation. Since neither algae nor fish will remain in a fixed space for many factors (mate choice, food supplies, population density, etc.), it is meaningful to consider their changes of spatial diffusibility [35,36]. Finally explore how to control algae bloom by fish under human external control. In a word, all the results are expected to be helpful in the process of study bifurcation dynamic behavior in some aquatic ecosystems.