Stability and Bifurcation Analysis on a Predator–Prey System with the Weak Allee Effect

In this paper, the dynamics of a predator-prey system with the weak Allee effect is considered. The sufficient conditions for the existence of Hopf bifurcation and stability switches induced by delay are investigated. By using the theory of normal form and center manifold, an explicit expression, which can be applied to determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions, are obtained. Numerical simulations are performed to illustrate the theoretical analysis results.


Introduction
The Allee effect refers to the phenomenon that individuals of many species benefit from the presence of conspecifics, which implies when the per capita growth rate becomes bigger with the increase of the density and gets smaller as long as density passes through a critical value [1]. When the growth rate decreases but still keeps positive at a low population density, the Allee effect is called weak. It is easy to see that this situation is totally different from the logistic growth. During the last decade this phenomenon attracted a lot of attention with the rise of conservation biology.
The dynamics with the Allee effect, such as the stability of positive equilibrium [2][3][4][5][6][7], the existence of periodic solution [8][9][10][11][12], or bifurcation analysis [13][14][15], on nonlinear systems have been extensively investigated. Especially, Boukal and L. Berec [16] critically reviewed and classified models of single-species population dynamics with the demographic Allee effect. Meanwhile in consideration of the carrying capacity of environment with respect to the prey in the per capita growth rate of the population, they also proposed the following differential equation with the weak Allee effecṫ where N(t) denotes the prey population at moment t, 0 < ε < 1 is the per capita growth rate of the population and K > 0 is the carrying capacity of the environment. In biology, the group defense enriches the ability of the prey to defend or escape from the predators, and thus it helps decrease the predation. Tabares et al. [17] proposed the coupled system of equationṡ (3) which describes the dynamics of interactions between a predator and a prey species with weak Allee effect. They studied the influence of the weak Allee effect with the density function G : R + → R + defined by G(t) = a 2 te −at , a > 0 and got analogous stability results as (2). For all biological processes in nature, the time delay is an inherent phenomenon. Hence, it is practical to assume that the growth rate of the prey population is determined by its recent history. So taking into account the maturation time of the prey, we define the density function G(t) by and it makes system (2) the following delayed predator-prey model with weak Allee effect.
In this paper, we study the effect of the delay τ on the positive equilibrium of system (4), and investigate the normal form for the Hopf bifurcation. In Section 2, stability of the equilibrium and existence of Hopf bifurcations are given. In Section 3, the direction and stability of the bifurcating periodic solutions of the system will be analyzed. Finally, in Section 4, numerical simulations are performed to illustrate our theoretical results.
Let ω 2 = z, then (16) can be rewritten as Since the constant term in (17) is positive, f (z) = 0 has at least one positive root if and only if there exists z * > 0 such that In fact, . Through direct calculation , we get We rewrite (19) as Denote l 4 : a = 4bεγ(1−bγ) 2 1−bγ+2εb 3 γ 2 −2bε(1−bγ) 2 and l 5 : f (z * ) = 0, then curve l 4 divides D 4 into two regions D 5 and D 6 , curve l 5 divides D 3 into two regions D 7 and D 8 (see Figure 1b). This means that Equation (17) has two positive roots in region D 7 . Thus, we can draw the following conclusions.  A positive root of f (z) = 0 corresponds to a pair of purely imaginary roots of Equation (3). Thus we get the following results. Lemma 3. Assume (10), (11), (18) and (20) hold.

From (24) and (25), it is easy to verify that
Therefore, together with the define of τ ± j and ω ± , means that the lemma is true. Now we have found the stability switches. In another words, the stability of the equilibrium switching from stability to instability when τ passes through τ + j (j = 0, 1, 2, · · · , m) and back to stability when τ passes through τ − j (j = 0, 1, 2, · · · , m − 1). When τ > τ + m , the equilibrium is instability for ever.

Direction and Stability of Hopf Bifurcations
In Section 2, some conditions which guarantee system (5) and its modification (6) undergo Hopf bifurcation at some critical values of τ are obtained. We now apply the normal form and center manifold theory introduced by Hassard et al. [20] to study the direction of these Hopf bifurcations and stability of the bifurcated periodic solutions.
It is well-known from [20] that the sign of µ 2 decides the direction of Hopf bifurcation, the sign of β 2 determines the stability of bifurcating periodic solutions and T 2 defines the period of bifurcating periodic solutions. Thus we have the following conclusion.

Conclusions
In this paper, we have studied the dynamics of a predator-prey system with the weak Allee effect. Firstly, we investigated the effect of the time delay τ on the stability of the positive equilibrium of system (5). Then, by using the normal form and center manifold theorems, we explored the existence of the stability switches, the Hopf bifurcation, the bifurcating direction and the stability of the bifurcating periodic solutions when the characteristic equation of system (6) has two pairs of purely imaginary roots. Meanwhile, we found the characteristic equation of system (6) has one pair of purely imaginary roots which are double roots when f (z) = 0 has one positive root. We believe system (6) perhaps has more interesting dynamics in this case which is more complicated than our present work and will be considered in the future. Furthermore, we can incorporate other time delays, such as predator maturation time, etc., into the mathematical model to investigate their dynamics by other methods, from which we may be able to obtain more interesting results.