HOPF BIFURCATION AND STEADY-STATE BIFURCATION FOR A LESLIE-GOWER PREY-PREDATOR MODEL WITH STRONG ALLEE EFFECT IN PREY

. It is well known that the Leslie-Gower prey-predator model (with-out Allee eﬀect) has a unique globally asymptotically stable positive equilib- rium point, thus there is no Hopf bifurcation branching from positive equilibrium point. In this paper we study the Leslie-Gower prey-predator model with strong Allee eﬀect in prey, and perform a detailed Hopf bifurcation analysis to both the ODE and PDE models, and derive conditions for determining the steady-state bifurcation of PDE model. Moreover, by the center manifold theory and the normal form method, the direction and stability of Hopf bifurcation solutions are established. Finally, some numerical simulations are presented. Apparently, Allee eﬀect changes the topology structure of the original Leslie- Gower model.

1. Introduction. The dynamical relationship between prey and predator has been a research hotspot in mathematical biology. So far, a lot of worthwhile prey-predator models have been proposed from theoretical and practical perspectives. In particular, Leslie [11,12] introduced a prey-predator model where the carrying capacity of predator's environment is proportional to the number of prey which is known as the second Leslie-Gower prey-predator model [23]. The parameters r 1 , r 2 , a 1 , a 2 , b 1 are positive constants; r 1 and r 2 are respectively the growth rate of the prey H and the predator P ; b 1 measures the strength of competition among individuals of species H; a 1 is the maximal per capita consumption rate, i.e., the maximum number of prey that can be eaten by a predator in each time unit; a 2 is a measure of food quality that the prey provides for conversion into predator population. The system (1) has a unique coexisting equilibrium ( r1a2 a1r2+a2b1 , r1r2 a1r2+a2b1 ), which is globally asymptotically stable [10]. The term a 2 P/H is called the Leslie-Gower term. It measures the loss in the predator population due to rarity (per capita P/H) of its favorite food. The system (1) assumes that the predator subsistence depends on prey population exclusively. However, in literature [2], the predator P will switch over to other preys when its favorite food H is seriously scarce, and its growth will be limited when its favorite food is not available in abundance. This situation can be taken care of by adding a positive constant c to the denominator. Hence, the second equation of (1) becomes dP/dt = [r 2 − a 2 P/(H + c)]P , which is also known as modified Leslie-Gower model. For this scenario, its unique interior equilibrium still is globally asymptotically stable under certain conditions. The PDE version of the Leslie-Gower model also has been studied extensively, see [7,17,24,29].
In 1931, Warder Clyde Allee proposed that intraspecific cooperation might lead to inverse density dependence. This idea was extended in his famous book [1] on animal ecology in 1949. Allee observed that many animal and plant species suffer a decrease of the per capita birthrate as their populations reach small sizes or low densities, which is now generally known as the Allee effect. This phenomenon has become crucial for population dynamics since in fact it has a surprising number of ramifications towards different branches of ecology [4,25]. Recently, there have been some works concerning the Allee effect in the classical dynamic population models [5,8,13,18,19,26]. Many interesting dynamical properties caused by the Allee effect are found which differ from the original system. Actually, Allee effect induces the appearance of a new equilibrium point which changes the stability of other equilibrium points. It shows that Allee effect changes the dynamic properties of original system significantly. In mathematical terms, Allee effect is expressed by modifying the natural growth function (usually the logistic growth function). The most common mathematical form describing this phenomenon for a single species is given by the equation [3] where x is the population size, r is the intrinsic growth rate and k is the environmental carrying capacity. The parameter m is a threshold population level. The equation (2) represents the strong Allee effect and weak Allee effect when m > 0 and m ≤ 0, respectively. In this paper, we are interested in Leslie-Gower prey-predator model incorporating the strong Allee effect in prey, which is generally described by the Kolmogorov type differential system: where u and v represent the densities of prey and predator, respectively. The positive constants β and µ have the same meaning with a 1 and r 2 in system (1), respectively. The parameter b ∈ (0, 1) represents Allee threshold value.
On the other hand, the spatial component of ecological interactions has been recognized as an important factor in studying how ecological communities are shaped [15,20,21,22,28,29]. Considering spatially inhomogeneous distribution, a reactiondiffusion model corresponding to (3) can be written as follows where Ω is a bounded domain in R n (n ≥ 1) with smooth boundary ∂Ω; the parameters d 1 and d 2 are respectively the diffusion coefficients of u and v; ν is the unit outward normal vector on the boundary ∂Ω. The homogeneous Neumann boundary condition means that the system (4) is self-contained and has zero population flux across the boundary ∂Ω. Ni and Wang give some estimates and the dynamical properties of the solution to (3) in [18] and investigate the stabilities of nonnegative constant steady states and the existence and non-existence of non-constant positive steady states of (4) in [19]. For a more comprehensive study of the dynamic properties of systems (3) and (4), motivated by the papers [6,14,33,27], we will carry out the Hopf bifurcation analysis of (3) and (4) and provide the steady-state bifurcation analysis of (4) in this paper. The introduction of Allee effect will lead the systems (3) and (4) to have two equilibrium points at the interior of the first quadrant under certain conditions. Bifurcation analysis for the system with Allee effect have been very rare all the time due to the complexity of this situation. We have known that the Leslie-Gower prey-predator model without Allee effect (i.e. the system (1)) has a unique globally asymptotically stable positive equilibrium point for all parameters [10], thus there is no Hopf bifurcation branching from positive equilibrium point. However, in this paper, we find that the system (3) has Hopf bifurcation from one positive equilibrium point and the system (4) has Hopf bifurcation from two positive equilibrium points. Apparently, Allee effect changes the topology structure of the original Leslie-Gower system. Now, we state some existing results (see [18,19]) on systems (3) and (4) to prepare for the later research. For (3) and (4), there are two semi-trivial steady states (b, 0), (1, 0) and some positive constant steady candidates: then the following relations hold naturally, A(λ (1) ) < βλ (1) , A(λ (2) ) > βλ (2) , A(λ (3) ) = βλ (3) .
2. Hopf bifurcation of ODE problem (3). In this section, we shall study the existence, stability and direction for Hopf bifurcation of (3).
Summarizing the above analysis, we obtain the main result of this section: Theorem 2.1. Assume that the parameters β, µ > 0 and 0 < b < 1.
1. If 0 < µ < b 0 , then the system (3) undergoes a Hopf bifurcation from (λ (1) , λ (1) ) when λ (1) = λ    The detailed natures of Hopf bifurcation need further analysis of the normal form of (3). In the following, we adopt the same notations and computations as in [31,32] to study the direction and stability of Hopf bifurcation.

Define a matrix
Clearly, By the transformation the system (10) becomes In order to obtain the stability of Hopf bifurcation periodic solutions, we need to calculate the sign of a(λ 0 ) given by where all partial derivatives are evaluated at the bifurcation point, i.e., (x, y, λ (1) ) = (0, 0, λ 0 ). It is easy to calculate that By tedious but simple calculations, we can obtain From (9), the above calculation of a(λ 0 ) and the Poincaré-Andronov-Hopf Theorem, the desired result can be deduced.
3. Hopf bifurcation of PDE problem (4). In this section, we shall investigate the existence, stability and direction of Hopf bifurcation solutions to (4). Here we only consider the one dimensional case. Without loss of generality, Ω is restricted on the one-dimensional space domain (0, lπ) with l ∈ R + .
2. The condition d −1 1 d 2 b 0 ≤ µ < b 0 implies that the set Σ 2 is not empty. The remainder of the argument is analogous to that in case 1, so we omit it. Theorem 3.1 gives conditions under which the system (4) occurs Hopf bifurcation from (λ (1) , λ (1) ). Actually, the similar conclusions for (λ (2) , λ (2) ) also can be obtained by the same method, but it is very complicated since too many cases of b. In addition, note that Theorem 3.1 only gives the number of Hopf bifurcation points but does not give the order of them. Therefore, we will not repeat above process for (λ (2) , λ (2) ). In the following, we will use another method to obtain not only more detailed existence conditions of Hopf bifurcation from (λ (1) , λ (1) ) and (λ (2) , λ (2) ) but also the exact order of bifurcation points. Define 3, the following states hold true.
Hence, the conclusion (i) is evident from what we have proved.

Remark 2. For cases 1 and 4 in Theorem 3.3, the condition
and [·] is integral function, i.e., [x] is the largest integer less or equal to x. Now, we adopt the method and the same notations in [9,33] to calculate the bifurcation direction and stability of bifurcating periodic solutions from (λ (1) , λ (1) ).
Theorem 3.4. For the system (4), the following statements hold true.
To determine Λ, we analyze the basic properties of p ± (µ) defined in (0, µ * ] in the following lemma.