Complex dynamics of a nonlinear discrete predator-prey system with Allee e ﬀ ect

: The transition between strong and weak Allee e ﬀ ects in prey provides a simple regime shift in ecology. In this article, we study a discrete predator-prey system with Holling type II functional response and Allee e ﬀ ect. First, the number of ﬁ xed points of the system, local stability, and global stability is discussed. The population changes of predator and prey under strong or weak Allee e ﬀ ects are proved using the nullclines and direction ﬁ eld, respectively. Second, using the bifurcation theory, the bifurcation conditions for the system to undergo transcritical bifurcation and Neimark-Sacker bifurcation at the equilibrium point are obtained. Finally, the dynamic behavior of the system is analyzed by numerical simulation of bifurcation diagram, phase diagram, and maximum Lyapunov exponent diagram. The results show that the system will produce complex dynamic phenomena such as periodic state, quasi-periodic state, and chaos.


Introduction
Population refers to a collection of the same kind of individuals living in a certain area at the same time.In nature, each population belongs to a certain level, and often, there are both predators at the upper level and preys at the lower level.As a result, in order to grasp the link between populations at all levels, assess and predict population persistence and extinction, a population model to explain the population system's evolution law must be established.
The general population model can be divided into two categories: differential equation model and difference equation model.In general, when the population number is relatively large or generations overlap, it can be described by differential equation.The difference equation is suitable for describing the population with long life cycle, few population, and non-overlapping generations.For example, herbs often flower in their first year and then die, roots and all, after setting seed, and a small fraction (<1%) of the 22,000 species of teleost fish are semelparous and die soon after spawning (see [1] for more examples).In 1976, May [2] showed that although the first-order difference equation model is simple, it can also show a series of surprising dynamic behaviors, such as from stable point to a bifurcation hierarchy of stable cycles, and finally produce chaos.However, in a continuoustime model, a minimum of three species are needed for exhibiting chaos [3].It can be seen that the discrete model will show more interesting dynamic behavior.Therefore, the discrete dynamic system model has attracted the attention of many scholars [4][5][6][7][8][9].AlSharawi et al. [10] considered the influence of vigilance of prey on dynamics of a discrete-time predator-prey system.They studied the stability, persistence, flip bifurcation, and Neimark-Sacker bifurcation of the discrete system.The results show that with the increase of prey vigilance, the density of predator population continuously decreases and high vigilance will have a detrimental role for the prey population.Streipert et al. [11] deduced a discrete predator-prey model through the first principles in economics.They extended standard phase plane analysis by introducing the next iterate root-curve associated with the nontrivial prey nullcline.The stability of the system is proved by combining the curve with the nullclines and direction field.Finally, it is proved that the system will have a transcritical bifurcation at the boundary equilibrium point and a Neimark-Sacker bifurcation at the internal equilibrium point.
The Allee effect, a reduction of the per capita growth rate of a population of biological species at densities smaller than a critical value, was first introduced by Allee in 1931 [12].Allee effects are mainly classified into two ways: strong and weak Allee effects.There is a critical value for the strong Allee effect.When the population density is less than this critical value, the population shows a negative growth trend, and then, the population will become extinct.When the population density is higher than this critical value, the population growth is positive, and then, it can develop.The weak Allee effect does not have a critical threshold.Its characteristic is that the individual growth rate at low density is lower than that at high density and always keeps a positive value, so the two populations can coexist permanently.Empirical evidence of Allee effects has been observed in many natural species, for example, plants [13,14], insects [15], marine invertebrates [16], and birds and mammals [17], etc.Therefore, many researchers have studied the bifurcation and stability analysis for discrete-time predatorprey system with the Allee effect [7,[18][19][20][21].However, in these articles, they did not consider the impact of strong and weak Allee effects on the local stability of the equilibrium point of the predation system, nor did they prove the global stability of the equilibrium point.So, in this article, we consider a discrete-time predator-prey model with Holling type II functional response and Allee effect in preys, which is given by where x and y represent the densities of prey and predator population, respectively, and is the generalized Holling type II functional response.In System (1), r denotes the intrinsic growth rate of the prey population, K denotes the environmental carrying capacity of the prey, d is the predator's mortality rate, < < α 0 1 represents the conversion efficiency of intake of prey into new predators, β denotes the maximal predator capita consumption rate, a represents the half-saturation constant, and c denotes the handling time.The parameters r, K , β, c, and d are assumed to be positive.The parameter A represents the threshold of multiplicative Allee effect.When < < A K 0 , it indicates the the strong Allee effect; when − < < K A 0, it indicates the weak Allee effect.Note that = A 0 is the transition between the weak and strong Allee effects.In this article, we study a discrete predator-prey system with Holling type-II functional response and Allee effect.There are several highlights in our analysis: (a) the influence of Allee effect parameter A on the local stability of equilibrium point is analyzed; (b) the conditions of global stability of discrete System (1) at the positive equilibrium point are obtained; (c) the population changes of predator and prey under strong or weak Allee effect are proved using the nullclines and direction field, respectively; and (d) the transcritical bifurcation and Neimark-Sacker bifurcation of System (1) are completely and rigorously analyzed.
This article is organized as follows.In Section 2, we analyze the dynamics of System (1), including the local stability of the equilibrium points, the global stability of the positive equilibrium point, and the bifurcation analysis at the equilibrium point.In Section 3, we verify our analytical results through numerical simulations.In Section 4, this article is ended with a brief conclusion.
which has the Jacobian matrix where x y , ( ) denotes the fixed points of System (1).To obtain the stable results of (2), we consider the algebraic equation where It is well-known that it has two roots of the form ( ) ) be the unique positive equilibrium point of System (1), then the following proposi- tions hold: Nonlinear discrete predator-prey system  3 (3) It is a saddle if Jacobian matrix can be evaluated at E 0, 0 0 ( ) as .
The eigenvalues of the Jacobian are = − λ e d 1 and = − λ e rAK 2 at trivial equilibrium point E 0, 0 0 ( ).The results regarding dynamical behaviors are listed in Table 1.
From Table 1, we can obtain the following theorem.
Theorem 1. E 0, 0 0 ( ) is always locally asymptotically stable under strong Allee effect, while it is always unstable under weak Allee effect.
The Jacobian matrix computed at E K , 0 The eigenvalues of the Jacobian are ] .The properties of semi- trivial equilibrium point E K , 0 K ( ) are summarized in Table 2.
Table 1: Properties of origin equilibrium point E 0, 0 0 ( ) From Table 2, we can obtain the following theorem.
) is locally asymptotically stable.
Jacobian matrix can be evaluated at E A, 0 A ( ) as The eigenvalues of the Jacobian are = + − λ rAK rA 1 The results regarding dynamical behaviors are listed in Table 3. From Table 3, we can obtain the following theorem.
Then characteristic equation of J E* ( ) is given by Table 3: Properties of semi-trivial equilibrium point E A, 0 A ( )

| | Source
Nonlinear discrete predator-prey system  5 where ) is non-decreasing in and non-increasing in y for all x y , ( ).Moreover, g x y , ( ) is non- decreasing in both arguments x and y for all x y , ( ).
( ) be a positive solution of the system Through calculation, we can obtain Hence, the unique positive equilibrium point . □ From Theorems 4 and 5, we can obtain the following conclusions.In fact, x n is increasing if the forward operator, △ = − , is positive.For System (1), the forward operators are and From (10), we see that ( ), then the sequence of iterates, x n is decreasing, and if < y l x n n ( ), then x n is increasing.Similarly, from (11), it follows that y n is increasing as long as > − respectively.We refer to these curves as nullclines.These two curves divide the first quadrant into three or four regions = R i 1, 2, 3, 4 i ( ).We define the regions  , System (1) has a unique positive equilibrium point.By observing the arrow direction in the image, we can find that when the number of prey population is small, the prey will be extinct and the predators will also be extinct.This is due to − < .Thus, the trivial equilibrium E 0 is stable under the strong Allee effect, which is consistent with the conclusion of Theorem 1.
In Figure 2, when − < < K A 0, for ( )}.Similarly, we can find that when the prey population is small, under the influence of the weak Allee effect, the prey population will not be extinct, and the population will increase.However, the number of predator population will gradually decrease and become extinct.This is different from the strong Allee effect.And by observing Figures 1 and 2, it can be found that when > x K, the growth rate of the prey population shows a negative growth, which is consistent with the biological significance.
and System (1) undergoes a transcritical bifurcation at the trivial equilibrium point E 0 .

Proof. For =
A 0, the equilibria E 0 and E A coalesce.The Jacobian evaluated at E 0 given in ( 5) has eigenvalues  , and the trivial equilibrium point E A is non-hyperbolic.Similarly, according to the literature [23], it manifests that System (1) exhibits a transcritical bifurcation at E A .□ Theorem 10.The interior equilibrium point E* loses its stability via Neimark-Sacker bifurcation if Proof.Neimark-Sacker bifurcation occurs in the system when a pair of complex eigenvalues with unit modulus [24], i.e.,

It is obtained
This gives the condition for the Neimark-Sacker bifurcation.□ Now, we discuss the Neimark-Sacker bifurcation of the equilibrium point E x y * *, * ( ).Here, we choose α as a bifurcation parameter.Neimark-Sacker bifurcation in a discrete system is the birth of a closed invariant curve from an equilibrium point.The bifurcation can be supercritical when closed invariant curve is stable and subcritical, when it is unstable.
Then, the dynamic analysis of System (1) is analyzed when the parameters change in the small field of Ω NS .Select parameter ∈ r K A a c β α d , , , , , , , Ω NS ( ) , and consider the following system: where is a small perturbation parameter.
Nonlinear discrete predator-prey system  9 Let = − = − u x x v y y *, *, then we obtain where The characteristic equation of System (17 is as follows: Therefore, eigenvalues λ 1 and λ 2 of the equilibrium point (0,0) of System (17) do not lay in the intersection of the unit circle with the coordinate axes when = α* 0 and the condition (18) holds.
, we use the following transformation: and System (17) becomes where System ( 17) undergoes the Neimark-Sacker bifurcation if the following quantity is not zero: where Remark 1.When the system has the Neimark-Sacker bifurcation, it will dispose of the expenditure invariant curve from the fixed point, which indicates that prey and predator can coexist, and the dynamic behavior can be periodic or quasi-periodic.

Numerical simulations
This section will show the bifurcation diagram, phase diagram, and maximum Lyapunov exponent diagram with the Allee effect model to verify the correctness of theoretical analysis.
Nonlinear discrete predator-prey system  11

Weak Allee effect
Assuming that the parameter is = ).Lemma 2 shows that when < α 0.2857, the coexistence equilibrium point does not exist; when > α 0.2857, there is a unique coexistence equilibrium point.It can be seen from Theorem 8 and Figures 3(a

Strong Allee effect
When = A 0.1 and other parameters remain unchanged, it can also be obtained that when < α 0.2857, the coexistence equilibrium point does not exist; when > α 0.2857, there is a unique coexistence equilibrium point, and when = α 0.2857, System (1) has a transcritical bifurcation at the boundary equilibrium point E K .According to Theorem 10, when = α 0.3729, System (1) has the Neimark-Sacker bifurcation at the coexistence equilibrium point.Figure 6  It can be found from Figures 3 and 6 that under the weak Allee effect, the growth rate of the population at low density is always positive and the population will not become extinct.Under the strong Allee effect, the population will become extinct at low density.This is consistent with the conclusion obtained from Figures 1 and 2.  , the boundary equilibrium point E K will have a transcritical bifurcation, and when the coexistence equilibrium E* exists and loses stability, System (1) will have a Neimark-Sacker bifurcation.The numerical simulation reveals that when the energy conversion rate of predator α increases gradually, System (1) will produce periodic, quasi-periodic windows, and chaos.
and decreasing if < − x n ad αβ cd .By Lemma 1, solutions remain in the first quadrant for all nonnegative initial conditions.We divide the first quadrant into regions based on the component-wise monotonicity obtained by solving =

Figure 1 :
Figure 1: Phase diagram of System (1) when < < A K 0 .The red dashed line corresponds to the predator nullclines, and the blue curves correspond to the prey nullclines.A horizontal arrow pointing to the right (left) represents > < + x x -0 0 n n 1 ( ), and a vertical arrow pointing up (down) represents > < + y y -0 0 n n 1 ( ).Subfigure (a) is a schematic image if < K ad αβ cd -, (b) is a schematic image if < <A 0 ad αβ cd -, while (c) shows the case when < < A K ad αβ cd -

1 and = λ 1 2 ,( 1 ) 2 ,( 1 )
the trivial equilibrium point E 0 is non-hyperbolic.It can be concluded that the central manifold of the map →− x x rx K x exp[ ( )].According to the literature[22], it manifests that System (1) exhibits a transcritical bifurcation at E 0 .undergoes a transcritical bifurcation at the semi-trivial equilibrium point E K .E K and E* coalesce.The Jacobian evaluated at E K given in (5and the trivial equilibrium point E K is non-hyperbolic.Similarly, according to the literature[23], it manifests that System (1) exhibits a transcritical bifurcation at E K .undergoes a transcritical bifurcation at the semitrivial equilibrium point E A .

Figure 2 : 1 ( 1 (
Figure 2: Phase diagram of System (1) when < < K A -0.The red dashed line corresponds to the predator nullclines, and the blue curves correspond to the prey nullclines.A horizontal arrow pointing to the right (left) represents > < + x x -0 0 n n 1 ( ), and a vertical arrow pointing

, 1 ,
and = d 0.1, α is the bifurcation parameter, and the initial value of the system is = ) and (b) when = α 0.2857, System (1) has a transcritical bifurcation at the boundary equilibrium point E K .And according to Theorem 10, when = α 0.3981, System (1) will have a supercritical Neimark-Sacker bifurcation at the coexistence equilibrium point.The bifurcation diagram and the maximum Lyapunov exponent diagram are shown in Figure 3. Combined with Figures 3 and 4, when < α 0.3981, the fixed point is stable.When = α 0.39, the phase and evolution diagrams of predator and prey with time are given in Figures 4(b) and 5(a).It can be seen from Theorem 6 that the coexistence equilibrium points of System (1) are globally asymptotically stable.However, when > α 0.3981, the coexistence equilibrium loses its stability, and a stable invariant loop appears.At this time, the periodic solution of System (1) appears (Figures 4(c) and (d) and 5(b)).When α increases, it can be seen from Figures 3(c), 4(e) and (f), and 5(c) that System (1) will have quasi-periodic solutions and chaos.
(a) and (b) is the bifurcation graph of α on [0, 1], and Figure 6(c) is the maximum Lyapunov exponent graph corresponding to Figure 6(a).
Figure6(d-f) is a locally enlarged view of Figure6(a-c).It can be seen from Figures 6 and that when < α 0.3729, the fixed point is stable; when > α 0.3729, the fixed point loses its stability and a stable invariant loop appears.At this time, System (1) generates a periodic solution (Figures7(d) and 8(b)).When α increases, System (1) produces quasi-periodic solutions and chaotic phenomena.However, when we continue to increase α, the population will become extinct (Figure8(c)).Combining Theorem 6 and Figure8(a), when < < α 0.2875 0.3729, the coexistence equilibrium point of System (1) exists and is globally asymptotically stable.