Hopf Bifurcations and Oscillatory Patterns of a Homogeneous Reaction-Diffusion Singular Predator-Prey Model

Abstract

A kind of homogeneous reaction-diffusion singular predator-prey model with no-flux boundary condition is considered. By using the abstract simplified Hopf bifurcation theorem due to Yi et al. 2009, we performed detailed Hopf bifurcation analysis of this particular pattern formation system. These results suggest the existence of oscillatory patterns if the system parameters fall into certain parameter ranges. And all these oscillatory patterns are proved to be unstable.

1. Introduction

In this paper, we consider the following reaction-diffusion singular predator-prey model: 𝑢_𝑡=𝑑₁𝑢_𝑥𝑥()()𝑣+𝛼1−𝑢𝑢−𝑣,𝑥∈0,ℓ𝜋,𝑡>0,_𝑡=𝑑₂𝑣_𝑥𝑥𝑣+𝛽𝑣1−𝑢()𝑢(),𝑥∈0,ℓ𝜋,𝑡>0,𝑥,0=𝑢₀(𝑥),𝑣(𝑥,0)=𝑣₀(𝑥)(),𝑢,𝑥∈0,ℓ𝜋_′𝑥(𝑥,𝑡)=𝑣^′_𝑥(𝑥,𝑡)=0,𝑥=0,ℓ𝜋,𝑡>0, (1) where 𝑢(𝑥,𝑡) and 𝑣(𝑥,𝑡) are the population densities of the prey and predator at time 𝑡 and position 𝑥,  𝛼 and 𝛽 are dimensionless positive parameters; ℓ∈(0,∞). The underlying spatially homogeneous problem was derived in [1] to model prey-predator interactions in fragile (insular) environments. The spatially structured system (1) supplemented with initial data and no-flux boundary conditions was introduced in [2]. For more information on the system (1), we refer the reader to [1, 2] for great details.

Mathematically, the model was considered by Ducrot and Langlais [3], where the authors first provided a suitable notion of global travelling wave weak solution. They mainly studied the existence of travelling waves solutions for predator invasion in such environments. Under suitable conditions on the diffusion coefficients and on species growth rates they were able to prove that the travelling wave solutions were actually positive on a half line and identically zero elsewhere.

The present paper is targeting to consider the Hopf bifurcations of the reaction-diffusion system (1), by using the simplified Hopf bifurcation theorem due to [4], which is widely used to prove the existence of oscillatory patterns of different kind of pattern formation systems, including Gierer-Minhardt model [5], Degn-Harrison model [6], bimolecular model [7], hair growth controlling model [8], and the Sel’kov model [9].

Our results show that, under suitable choice of system parameters, system (1) will undergo spatially homogeneous and nonhomogeneous oscillatory phenomena. And once the oscillatory patterns exist, they are always unstable. These unstable patterns cannot be observed by numerical simulations; thus, numerical simulations corresponding to our analytical analysis are unavailable in the paper, even though, the analytical results we obtained allow for the clearer understanding of the rich dynamics of the system.

The rest of this paper is structured in the following way. In Section 2, we perform Hopf bifurcation analysis of the system. In Section 3, we draw some concluding remarks to end up our discussion. Throughout the paper, we denote by ℕ the set of all the positive integers and ℕ₀=ℕ∪{0}.

2. Stability and Hopf Bifurcation Analysis

To begin with, for convenience of our discussion, we copy (1) in the following: 𝑢_𝑡=𝑑₁𝑢_𝑥𝑥()()𝑣+𝛼1−𝑢𝑢−𝑣,𝑥∈0,ℓ𝜋,𝑡>0,_𝑡=𝑑₂𝑣_𝑥𝑥𝑣+𝛽𝑣1−𝑢()𝑢(),𝑥∈0,ℓ𝜋,𝑡>0,𝑥,0=𝑢₀(𝑥),𝑣(𝑥,0)=𝑣₀(𝑥)(),𝑢,𝑥∈0,ℓ𝜋_′𝑥(𝑥,𝑡)=𝑣^′_𝑥(𝑥,𝑡)=0,𝑥=0,ℓ𝜋,𝑡>0. (2)

System (2) has a unique equilibrium solution (𝑢_𝛼,𝑣_𝛼), with 𝑢_𝛼=𝑣_𝛼=∶𝛼−1𝛼, (3) which is in the first quadrant if and only if 𝛼>1.

In the following, we always assume that 𝛼>1 holds. We fix the parameter 𝛽 and choose 𝛼 as the bifurcation parameter.

The linearized operator of system (2) evaluated at (𝑢_𝛼,𝑣_𝛼) is given by 𝐿(𝛼)𝑑∶=¹𝜕²𝜕𝑥²+(2−𝛼)−1𝛽𝑑₂𝜕²𝜕𝑥²−𝛽. (4)

It follows from [4, 10] that the eigenvalues of 𝐿(𝛼) are given by these of the operator 𝐿_𝑛(𝛼), which is defined by 𝐿_𝑛(𝛼)−𝑑∶=¹𝑛²ℓ²+()𝑑2−𝛼−1𝛽−²𝑛²ℓ²−𝛽, (5) whose characteristic equation is 𝜆²−𝜆𝑇_𝑛(𝛼)+𝐷_𝑛(𝛼)=0,𝑛=0,1,2,…, (6) where 𝑇_𝑛(𝛼)=−𝑑2−𝛼−𝛽¹+𝑑₂𝑛²ℓ²(𝛼)−𝑑∶=𝐴¹+𝑑₂𝑛²ℓ²,𝐷_𝑛(𝛼)=𝑑₁𝑑₂𝑛⁴ℓ⁴+𝑛²ℓ²𝑑₁𝛽−𝑑₂(2−𝛼)+𝛽(𝛼−1). (7)

We have the following theorem on the stability of the unique positive equilibrium solution (𝑢_𝛼,𝑣_𝛼).

Theorem 1. Suppose that one of the following conditions holds(1)𝛽≥1 and 𝛼>1,(2)0<𝛽<1 and 𝛼>2−𝛽and that 𝑑₁,𝑑₂>0 such that 𝑑₁𝑑₂≥1𝛽 (8) is satisfied. Then, (𝑢_𝛼,𝑣_𝛼) is always locally asymptotically stable in the reaction-diffusion equation (2).

Proof. On one hand, for any 𝛼>1, we have 𝐷_𝑛(𝛼)>𝑑₁𝑑₂𝑛⁴ℓ⁴+𝑛²ℓ²𝑑₁𝛽−𝑑₂, (9) which implies that 𝐷_𝑛(𝛼)>0 always holds for any 𝛼∈(1,∞) if 𝑑₁𝑑₂≥1𝛽 (10) holds.
On the other hand, if 𝛽≥1 and 𝛼>1, or 0<𝛽<1 but 𝛼>2−𝛽 holds, then we can obtain that, for any 𝑛∈ℕ₀ and 𝛼>0, we always have 𝑇_𝑛(𝛼)<0. Thus, (𝑢_𝛼,𝑣_𝛼) is always locally asymptotically stable in the reaction-diffusion equation (2).

Now we consider the Hopf bifurcations of the system. According to [4], a point 𝛼^𝐻∈(1,∞) is a Hopf bifurcation point if and only if there exists 𝑛∈ℕ₀, such that 𝑇_𝑛𝛼^𝐻=0,𝐷_𝑛𝛼^𝐻𝑇>0;_𝑗𝛼^𝐻≠0,𝐷_𝑗𝛼^𝐻≠0for𝑗≠𝑛; (11) and 𝜏^′(𝛼^𝐻)≠0, where 𝜏(𝛼) is the real parts of the unique pair of complex eigenvalues 𝜏(𝛼)±𝑖𝜔(𝛼) near the imaginary axis.

Thus, any potential Hopf bifurcation point 𝛼^𝐻 must be in the interval (1,2−𝛽], where we assume that 0<𝛽<1.

For any Hopf bifurcation point 𝛼^𝐻∈(1,2−𝛽], let 𝜏(𝛼)±𝑖𝜔(𝛼) be the eigenvalues of 𝐿_𝑛(𝛼) near 𝛼=𝛼^𝐻 then we have 𝜏(𝛼)=𝐴(𝛼)2−𝑑₁+𝑑₂𝑛²2ℓ²,𝜔(𝛼)=𝐷_𝑛(𝛼)−𝜏²(𝛼). (12)

Then, for any 𝛼∈(1,2−𝛽], we have 𝜏^′(𝛼)1=−2. (13) This implies that the transversality condition 𝜏^′(𝛼)≠0 is always satisfied for any 𝛼∈(1,2−𝛽].

Suppose that 0<𝛽<1 holds and define ℓ_𝑛=𝑛1−𝛽/𝑑₁+𝑑₂,𝑛∈ℕ₀. (14)

Then, for any ℓ_𝑛<ℓ≤ℓ_𝑛+1 and 1≤𝑗≤𝑛, we define 𝛼^𝐻_𝑗 as the roots of 𝐴(𝛼)=𝑑₁+𝑑₂𝑗²ℓ² (15) satisfying 1<𝛼^𝐻_𝑗<𝛼^𝐻₀∶=2−𝛽, and these points satisfy 1<𝛼^𝐻_𝑛<𝛼^𝐻_𝑛−1<⋯<𝛼^𝐻₁<𝛼^𝐻₀=2−𝛽 (16) and lim_𝑗→∞𝛼^𝐻_𝑗=1. Clearly 𝑇_𝑗(𝛼^𝐻_𝑗)=0 and 𝑇_𝑖(𝛼^𝐻_𝑗)≠0 for 𝑖≠𝑗.

Now we derive a condition from the parameters so that 𝐷_𝑛(𝛼)>0 for all 𝛼∈(1,2−𝛽] and 𝑛∈ℕ₀. In fact, from (8) in Theorem 1, it follows that 𝐷_𝑛(𝛼)>0 always holds.

Based on the discussions above, we have the following Hopf bifurcation results for the reaction-diffusion model (2).

Theorem 2. Suppose that the constants 𝑑₁,𝑑₂,𝛽 satisfying the condition (8) and ℓ_𝑛 are defined as in (14). Then, for any ℓ in (ℓ_𝑛,ℓ_𝑛+1], there exist 𝑛 points 𝛼^𝐻_𝑗(ℓ), 1≤𝑗≤𝑛, satisfying 1<𝛼^𝐻_𝑛<𝛼^𝐻_𝑛−1<⋯<𝛼^𝐻₁<𝛼^𝐻₀=2−𝛽, (17) such that the system (1) undergoes a Hopf bifurcation at 𝛼=𝛼^𝐻_𝑗 or 𝛼=𝛼^𝐻₀. Moreover,(1)the bifurcating periodic solutions from 𝛼=𝛼^𝐻₀ are spatially homogeneous, which coincides with the periodic solution of the corresponding ODE system; (2)the bifurcating periodic solutions from 𝛼=𝛼^𝐻_𝑗 are spatially nonhomogeneous.

Next we consider the bifurcation direction and stability of the bifurcating periodic solutions.

Theorem 3. For the system (2), the direction of the Hopf bifurcation at 𝛼=𝛼^𝐻₀ is forward, and the bifurcating (spatial homogeneous) periodic solutions are unstable.

Proof. By Theorem 2.1 of [4], in order to determine the stability and bifurcation direction of the bifurcating periodic solution, we need to calculate Re(𝑐₁(𝛼^𝐻₀)). When 𝛼=𝛼^𝐻₀, we put 𝑎𝑞=⁰𝑏₀=1𝛽−𝜔₀𝑖,𝑞^∗=𝑎^∗₀𝑏^∗₀=𝜔₀+𝛽𝑖2𝜔₀−𝑖ℓ𝜋2𝜔₀,ℓ𝜋 (18) where 𝜔₀=𝛽(1−𝛽).
Now translate (2) into the following system by the translation 𝑢=𝑢−𝑢_𝛼 and 𝑣=𝑣−𝑣_𝛼, and still let 𝑢 and 𝑣 denote 𝑢 and 𝑣 for the convenience of notation. We have 𝑢_𝑡−𝑑₁𝑢_𝑥𝑥=𝛼1−𝑢+𝑢_𝛼𝑢+𝑢_𝛼−𝑣+𝑣_𝛼,()𝑣𝑥∈0,ℓ𝜋,𝑡>0,_𝑡−𝑑₂𝑣_𝑥𝑥=𝛽𝑣+𝑣_𝛼1−𝑣+𝑣_𝛼𝑢+𝑢_𝛼,()𝑢()𝑥∈0,ℓ𝜋,𝑡>0,𝑥,0=𝑢₀(𝑥),𝑣(𝑥,0)=𝑣₀(𝑥)(),𝑢,𝑥∈0,ℓ𝜋_′𝑥(𝑥,𝑡)=𝑣^′_𝑥(𝑥,𝑡)=0,𝑥=0,ℓ𝜋,𝑡>0. (19)
Following [4], we define 𝑓(𝛼,𝑢,𝑣)=𝛼1−𝑢+𝑢_𝛼𝑢+𝑢_𝛼−𝑣+𝑣_𝛼,𝑔(𝛼,𝑢,𝑣)=𝛽𝑣+𝑣_𝛼1−𝑣+𝑣_𝛼𝑢+𝑢_𝛼. (20)
By (2.19) of [4], we have 𝑐₀=𝑒₀,𝑑=2𝛽−2₀=2𝛽𝛽−21−𝛽1−𝛽+𝜔₀𝑖²,𝑓₀=𝛽𝛽−2,𝑔₀ℎ=0,₀=2𝛽2−𝛽²1−𝛽3−2𝛽+2𝜔₀𝑖. (21)
Following [4], we define 𝑄_𝑞𝑞=𝑐₀𝑑₀,𝑄_𝑞𝑞=𝑒₀𝑓₀,𝐶_𝑞𝑞𝑞=𝑔₀𝑓₀. (22)
Note that ⟨⋅,⋅⟩ is the complex-value inner product defined as 𝑈₁,𝑈₂=₀^ℓ𝜋𝑢₁𝑢₂+𝑣₁𝑣₂𝑑𝑥 (23) with 𝑈_𝑖=(𝑢_𝑖,𝑣_𝑖)^𝑇,  𝑖=1,2.
Then we obtain easily that 𝑞^∗,𝑄_𝑞𝑞=𝛽−2𝜔₀1−2𝛽𝜔₀−2𝛽²𝑖,𝑞^∗,𝑄_𝑞𝑞=𝛽−2𝜔₀1+2𝛽𝜔₀+2𝛽²𝑖,𝑞^∗,𝑄_𝑞𝑞=𝑞^∗,𝑄_𝑞𝑞𝑞=𝛽−2,_∗,𝐶_𝑞𝑞𝑞=𝛽2−𝛽²𝜔₀1−𝛽−2𝜔₀+3−2𝛽. (24)
Hence it is straightforward to calculate that 𝐻₂₀=𝑐₀𝑑₀−⟨𝑞^∗,𝑄_𝑞𝑞⟩𝑎₀𝑏₀−𝑞^∗,𝑄_𝑞𝑞𝑎₀𝑏₀=00,𝐻₁₁=𝑒₀𝑓₀−⟨𝑞^∗,𝑄_𝑞𝑞⟩𝑎₀𝑏₀−𝑞^∗,𝑄_𝑞𝑞𝑎₀𝑏₀=00. (25)
By (25), we have 𝑤₂₀=2𝑖𝜔₀𝐼−𝐿(𝛼^𝐻₀)⁻¹𝐻₂₀=00,𝑤₁₁=−𝐿𝛼^𝐻₀⁻¹𝐻₁₁=00, (26) where 𝐼 is the identity matrix. The equalities in (26) lead to 𝑞^∗,𝑄_𝑤11𝑞=𝑞^∗,𝑄_𝑤20𝑞=0. (27)
Therefore, we can calculate that Re𝑐₁𝛼^𝐻₀𝑖=Re2𝜔₀𝑞^∗,𝑄_𝑞𝑞⋅𝑞^∗,𝑄_𝑞𝑞+𝑞^∗,𝑄_𝑤11𝑞+12𝑞^∗,𝐶_𝑞𝑞𝑞+12𝑞^∗,𝑄_𝑤20𝑞=Re2−𝛽²2𝜔²₀2𝛽²+1−2𝛽𝜔₀𝑖+𝛽2−𝛽2𝜔₀1−𝛽−2𝜔₀+3−2𝛽𝑖=𝛽2−𝛽>0. (28)
From (13), it follows that 𝜏^′(𝛼^𝐻₀)=−1/2<0, and then by Theorem 2.2 of [4], the direction of the Hopf bifurcation is forward and the bifurcating periodic solutions are unstable since Re(𝑐₁(𝛼^𝐻₀))>0.

Remark 4. (1) The direction of Hopf bifurcations at 𝛼=𝛼^𝐻_𝑗 with 𝑗≥1 can also be calculated as what we did in Theorem 3, using the abstract results due to [4]. However, the calculations are very complicated. Thus, here we did not calculate the bifurcation direction of Hopf bifurcations at 𝛼=𝛼^𝐻_𝑗 with 𝑗≥1.
(2) The spatial nonhomogeneous periodic solutions at 𝛼=𝛼^𝐻_𝑗 with 𝑗≥1 found in Theorem 2 are clearly unstable since the steady state (𝑢_𝛼,𝑣_𝛼) is unstable.
(3) Since all the periodic solutions are unstable, the simulations of the oscillatory patterns are unavailable here.

3. Concluding Remarks

In this paper, we considered a kind of diffusive Gaucel-Langleis model. By using the simplified Hopf bifurcation theory due to [4], we were capable of investigating the existence of Hopf bifurcations of the system, which indicates the existence of oscillatory patterns of the system. Our main bifurcation and stability analysis results in the paper can be summarized as follows.(1) If 𝛽 is sufficiently large, say, if 𝛽≥1 holds, then Hopf bifurcations will never be possible. In fact, we can show that if one of the following conditions holds(a)𝛽≥1 and 𝛼>1,(b)0<𝛽<1 and 𝛼>2−𝛽,   and 𝑑₁,𝑑₂>0 such that 𝑑₁/𝑑₂≥1/𝛽 is satisfied, then, (𝑢_𝛼,𝑣_𝛼) is always locally asymptotically stable in the reaction-diffusion equation (2) (Theorem 1).(2) However, if 𝛽 is not that large, say, 0<𝛽<1, then, Hopf bifurcation is possible in suitable parameter ranges. In fact, by choosing 𝛼 as the bifurcation parameter, we can obtain that if the constants 𝑑₁,𝑑₂,𝛽 satisfying the condition (14) and ℓ_𝑛 are defined as in (14), then, for any ℓ in (ℓ_𝑛,ℓ_𝑛+1], there exist 𝑛 points 𝛼^𝐻_𝑗(ℓ),  1≤𝑗≤𝑛, satisfying 1<𝛼^𝐻_𝑛<𝛼^𝐻_𝑛−1<⋯<𝛼^𝐻₁<𝛼^𝐻₀=2−𝛽, (29) such that the system (1) undergoes a Hopf bifurcation at 𝛼=𝛼^𝐻_𝑗 or 𝛼=𝛼^𝐻₀, (Theorem 2). Moreover,(a)the bifurcation direction of Hop bifurcations at 𝛼=𝛼^𝐻₀ is forward and the bifurcating periodic solutions are unstable (Theorem 3),(b)the bifurcation direction of Hopf bifurcations at 𝛼=𝛼^𝐻_𝑗 with 𝑗≥1 can also be calculated as what we did in Theorem 3, using the abstract results due to [4]. However, the calculations are very complicated. This is beyond the scope of this paper, (c)the spatial nonhomogeneous periodic solutions at 𝛼=𝛼^𝐻_𝑗 with 𝑗≥1 found in Theorem 2 are clearly unstable since the steady state (𝑢_𝛼,𝑣_𝛼) is unstable (Remark 4).

Acknowledgment

The authors are grateful to the anonymous referee for the comments and suggestions which definitely led to an improved presentation of the paper.