Stability and Hopf Bifurcation Analysis for a Gause-Type Predator-Prey System with Multiple Delays

This paper is concerned with a Gause-type predator-prey system with two delays. Firstly, we study the stability and the existence of Hopf bifurcation at the coexistence equilibrium by analyzing the distribution of the roots of the associated characteristic equation. A group of sufficient conditions for the existence of Hopf bifurcation is obtained. Secondly, an explicit formula for determining the stability and the direction of periodic solutions that bifurcate from Hopf bifurcation is derived by using the normal form theory and center manifold argument. Finally, some numerical simulations are carried out to illustrate the main theoretical results.


Introduction
Multispecies predator-prey models have been studied by many scholars [1][2][3][4][5][6][7].Guo and Jiang [7] studied the following three-species food-chain system: where (), (), and () are the population densities of the prey, the predator and the top predator at time .The prey grows with intrinsic growth rate  and carrying capacity  in the absence of predation.The predator captures the prey with capture rate  and Holling type II functional response /(1 + ).The top predator captures its prey (the predator) with capture rate  and Holling type I functional response .The predator and the top predator contribute to their growth with the conversion rates  and , respectively.The parameters ℎ and  are the death rates of the predator and the top predator, respectively.All the parameters , , , ℎ, , , , ,  and in system (1) are assumed to be positive.The constant  ≥ 0 represents the time delay due to the gestation of the prey.Guo and Jiang [7] investigated the bifurcation phenomenon and the properties of periodic solutions of system (1).
Predator-prey systems with single delay as system (1) have been investigated extensively [8][9][10][11][12].However, there are some papers on the bifurcations of a population dynamics with multiple delays [13][14][15][16].Gakkhar and Singh [15] studied the effects of two delays on a delayed predator-prey system with modified Leslie-Gower and Holling type II functional response and established the existence of periodic solutions via Hopf bifurcation with respect to both delays.Motivated by the work of Guo and Jiang [7] and Gakkhar and Singh [15], we consider the following predator-prey system with two delays: where  1 denotes the time delay due to the gestation of the predator and  2 denotes the time delay due to the gestation of the top predator.This paper is organized as follows.In the next section, we will consider the stability of the positive equilibrium of system (2) and the existence of local Hopf bifurcation at the positive equilibrium.In Section 3, we can determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions from the Hopf bifurcation.Some numerical simulations are also given to illustrate the theoretical prediction in Section 4.
Equation (8) becomes It follows that where Letting Obviously,  20 ≥ 0. Thus, we assume that (15) has at least one positive solution.Without loss of generality, we assume that it has three positive roots, which are denoted as V 11 , V 12 , and V 13 .Then (13) has three positive roots  1 = √ V 1 ,  = 1, 2, 3.
From (12), we can get Then, we denote Next, we verify the transversality condition.Differentiating the two sides of (11) with respect to  1 and noticing that  is a function of  1 , we can get Therefore From ( 13), we have Therefore with Obviously, if ) holds, the transversality condition is satisfied.In conclusion, we have the following results.
Equation ( 8) becomes Let  =  2 ( 2 > 0) be a root of ( 23), then we have which follows that with becomes Similar as in Case 2, we assume that ( 31 ) : (27) has at least one positive solution.Without loss of generality, we assume that it has three positive roots, which are denoted by V 21 , V 22 and V 23 .Then (25) has three positive roots From (24), we get Then, we denote Similar as in Case 2, we know that if condition ( 32 ) : then, Re[/ 2 ] −1 = 20 ̸ = 0. Namely, if condition ( 32 ) holds, the transversality condition is satisfied.Therefore, we have the following results.Therefore, we have the following theorem.It is considered that with (8),  2 in its stable interval and  1 is considered as a parameter.

Direction and Stability of the Hopf Bifurcation
In this section, we will employ the normal form method and center manifold theorem introduced by Hassard et al. [17] to determine the direction of Hopf bifurcation and stability of the bifurcated periodic solutions of system (2) with respect to  1 for  2 ∈ (0,  20 ).Without loss of generality, we assume that   2 <  * 1 , where   2 ∈ (0,  20 ).Let  1 =  * 1 + ,  ∈ .Then  = 0 is the Hopf bifurcation value of system (2).Rescaling the time delay  → (/ 1 ), then system (2) can be rewritten as where By Riesz representation theorem, there exists a 3 × 3 matrix function (, ) : [−1, 0] →  3 whose elements are of bounded variation, such that In fact, we can choose For  ∈ ([−1, 0],  3 ), we define Then system (44) can be transformed into the following operator equation: The adjoint operator  * of  is defined by associated with a bilinear form where () = (, 0).From the above discussion, we know that ± * 1  * 1 are the eigenvalues of (0) and they are also eigenvalues of  * (0).We assume that Then, by a simple computation, we can obtain Then we have ⟨ * , ⟩ = 1.

Numerical Simulation and Discussion
In this section, we present some numerical simulations to illustrate the analytical results obtained in the previous sections.
Guo and Jiang [7] have obtained that the three species in system (2) with only one time delay can coexist, however, we get that the species could also coexist with some available time delays of the predator and the top predator.This is valuable from the view of ecology.As the future work, we shall consider the following more general and more complicated system with multiple delays:     where  1 is feedback delay of the prey and  2 ,  3 are the time delays due to the gestation of the predator and the top predator, respectively.
2. That is, the transversality condition is satisfied.Hence, we have the following theorem.Theorem 4.
Time t