Open Access

Bifurcation analysis of a diffusive model of pioneer and climax species interaction

Advances in Difference Equations20112011:52

https://doi.org/10.1186/1687-1847-2011-52

Received: 21 April 2011

Accepted: 7 November 2011

Published: 7 November 2011

Abstract

A diffusive model of pioneer and climax species interaction is considered. We perform a detailed Hopf bifurcation analysis to the model, and derive conditions for determining the bifurcation direction and the stability of the bifurcating periodic solutions.

Keywords

pioneer and climax Hopf bifurcation diffusive model

1 Introduction

We consider the following model:
u t = d 1 Δ u + u f ( c 11 u + v ) , v t = d 2 Δ v + v g ( u + c 22 v ) ,
(1.1)
where x Ω, t > 0, and u, v represent a measure of a pioneer and a climax species, respectively. f(z), the growth rate of the pioneer population, is generally assumed to be smoothly deceasing, and has a unique positive root at a value z 1 so that the crowding is particularly harmful for pioneer species. But for the climax population, it is different from pioneer population. Climax fitness increases at low total density but decreasing at higher densities. So that, it has an optimum value of density for growing. Hence, g(z), the growth rate of the climax population, is assumed to be non-monotone, has a hump, and possesses two distinct positive roots at some values z 2 and z 3, with z 2 < z 3 and g'(z 2) > 0 > g'(z 3). For the reason above, we set
f ( c 11 u + v ) = z 1 - c 11 u - v , g ( u + c 22 v ) = - ( z 2 - u - c 22 v ) ( z 3 - u - c 22 v )
(1.2)

in this article.

Equation (1.1) is often used to describe forestry models. Examples can be found in [1, 2] and references therein. The dynamics of pioneer-climax models have been studied widely. Systems described by ordinary differential equations are under the hypothesis of homogeneous environment. The stability of positive equilibrium and bifurcation, especial Hopf bifurcation are the subject of many investigations. More recently, the environmental factors are introduced to the pioneer-climax systems. Models including diffusivity (i.e. systems described by reaction-diffusion equations) have been considered. The existence of positive steady state solutions are the subject of investigations. In addition, traveling wave solutions are the most interesting problem. The readers can get some results from [3]. In bifurcation problems, Buchanan [4] has studied Turing instability in a pioneer/climax population interaction model. He determined the values of the diffusional coefficients for which the model undergoes a Turing bifurcation, and he show that a Turing bifurcation occurs when an equilibrium solution becomes unstable to perturbations which are nonhomogeneous in space but remains stable to spatially homogeneous perturbations. Hopf bifurcation for diffusive pioneer-climax species interaction has not been studied. Our study will be performed in Hopf bifurcation.

The rest of this article are structured in the following way: in Section 2, the conditions of the existence of positive equilibrium are given. The critical values of the parameter for Hopf bifurcation occurring are also searched. And the stability and direction of the bifurcating periodic solutions at λ 1 are studied. In Section 3, some conclusions are stated.

2 Hopf bifurcation analysis

In this section, we consider the following model:
u t = d 1 Δ u + u ( z 1 - c 11 u - v ) , v t = d 2 Δ v - v ( z 2 - u - c 22 v ) ( z 3 - u - c 22 v ) .
(2.1)
Clearly, it has one trivial equilibrium (0, 0), and three semitrivial equilibria (z 1/c 11,0), (0, z 2/c 22), and (0, z 3/c 22). There also has two nontrivial equilibria E 1, E 2:
E 1 = z 2 - c 22 z 1 1 - c 11 c 22 , z 1 - c 11 z 2 1 - c 11 c 22 , E 2 = z 3 - c 22 z 1 1 - c 11 c 22 , z 1 - c 11 z 3 1 - c 11 c 22 .
As in [4], in the following, we will limit our analysis to the case z 3 > z 2 and z 1 > c 11 z 2, z 2 > c 22 z 1. Immediately, the condition c 11 c 22 < 1 follows as a consequence, and then E 1 is a constant positive equilibrium. If there has additional condition that z 1 > c 11 z 3, then E 2 is an another constant positive equilibrium. E 1, E 2 are also positive equilibria for Equation (2.1) without diffusion, and when E 2 exists, it is unstable. In fact, the linear system at E 2 = (u*, v*) has the form
u t v t = L u v = c 11 u * f ( c 11 u * + v * ) u * f ( c 11 u * + v * ) v * g ( z 3 ) c 22 v * g ( z 3 ) .
For f' (c 11 u* + v*) = -1 and g'(z 3) = z 2 - z 3, then the trace and determinant of L are
tr L = - c 11 u * + c 22 v * ( z 2 - z 3 ) < 0 , det L = ( 1 - c 11 c 22 ) u * v * ( z 2 - z 3 ) < 0 ,

which imply that L has a positive eigenvalue, and then E 2 is unstable. Hence, the researchers are concerned more about the dynamics at E 1. In the corresponding diffusion system, the dynamics at E 1 is richer than that at E 2. Hence, we take our attention to the equilibrium E 1. In [4], Turing instability has been studied thoroughly. The effect on the stability due to the diffusion is analyzed. In this article, we pay attention to Hopf bifurcation bifurcated by E 1. We investigate on the effect on the stability due to the diffusion. In other words, diffusion driving Hopf bifurcation is studied.

Denote λ = z 2 - c 22 z 1. With the conditions above, we have that λ < z 3 - c 22 z 1 and 0 < λ < (1 - c 11 c 22) z 1/c 11. Hence, the domain of the parameter λ is 0 < λ < min{z 3 - c 22 z 1, (1 - c 11 c 22) z 1/c 11}. In this article, we choose λ as a main bifurcation parameter and consider the complicated dynamic behavior near the fixed point E 1 with the effect of diffusion.

For convenience, we first transform the equilibrium E 1 = (u*, v*) to the origin via the translation û = u - λ ( 1 - c 11 c 22 ) , v ^ = v - ( z 1 - c 11 λ ( 1 - c 11 c 22 ) ) and drop the hats for simplicity of notation, then system (2.1) is transformed into
u t = d 1 Δ u + a 11 u + a 12 v + f ( u , v ) , v t = d 2 Δ v + a 21 u + a 22 v + g ( u , v ) ,
(2.2)
where
a 11 = - c 11 u * , a 12 = - u * , a 21 = z ¯ v * , a 22 = c 22 z ¯ v * ,
and
z ¯ = z 3 - u * - c 22 v * , f ( u , v ) = - c 11 u 2 - u v , g ( u , v ) = ( z ¯ - 2 c 22 v * ) u v + ( c 22 z ¯ - c 22 2 v * ) v 2 - v * u 2 - u 2 v - 2 c 22 u v 2 - c 22 2 v 3 .
In the following, we consider system (2.2) on spatial domain Ω = (0, ℓπ), ℓ + with Dirichlet boundary condition
u ( 0 , t ) = u ( π , t ) = 0 , v ( 0 , t ) = v ( π , t ) = 0 , t > 0 .
Define the real-valued Sobolev space
X : = { ( u , v ) u , v H 2 ( 0 , π ) , ( u , v ) x = 0 , π = 0 } ,

and the complexification of X by X = X + iX = {x 1 + ix 2|x 1, x 2 X}.

The linearized operator of system (2.2) evaluated at (0, 0) is
L : = a 11 + d 1 2 x 2 a 12 a 21 a 22 + d 2 2 x 2
and accordingly we define (denote μ n , n are the eigenvalues of the eigenvalue problem -Δϕ = μϕ, ϕ(0) = ϕ(ℓπ) = 0)
L n : = a 11 - d 1 μ n a 12 a 21 a 22 - d 2 μ n .
Then, the characteristic equation of L n (λ) is
β 2 - β T n + D n = 0 , n = 1 , 2 , ,
(2.3)
where
T n = a 11 + a 22 - ( d 1 + d 2 ) μ n , D n = a 11 a 22 - a 12 a 21 - ( d 1 a 22 + d 2 a 11 ) μ n + d 1 d 2 μ n 2 .
More immediately, let T n , D n be expressed by expression with parameter λ:
T n ( λ ) = - ( d 1 + d 2 ) μ n + c 11 c 22 1 - c 11 c 22 λ 2 - 2 c 22 z 1 + c 11 c 22 z 3 - c 22 z 1 + c 11 1 - c 11 c 22 λ + c 22 z 1 ( z 3 - c 22 z 1 ) , D n ( λ ) = d 1 d 2 μ n 2 - d 1 c 11 c 22 1 - c 11 c 22 λ 2 - c 22 d 1 2 z 1 + c 11 z 3 - z 1 1 - c 11 c 22 λ + c 22 d 1 z 1 ( z 3 - c 22 z 1 ) - d 2 c 11 λ 1 - c 11 c 22 μ n + c 11 1 - c 11 c 22 λ 3 - 2 z 1 + c 11 z 3 - z 1 1 - c 11 c 22 λ 2 + z 1 ( z 3 - c 22 z 1 ) λ .

According to [5], we have

Lemma 2.1. Hopf bifurcation occurs at a certain critical value λ 0 if there exists unique n such that
T n ( λ 0 ) = 0 , D n ( λ 0 ) > 0 a n d T j ( λ 0 ) 0 , D j ( λ 0 ) 0 f o r j n ;
(2.4)

and for the unique pair of complex eigenvalues near the imaginary axis α(λ) ± iω (λ), the transversality condition α'(λ 0) ≠ 0 holds.

Let us consider the sign of D n (λ) first. Denote λ ¯ = min { z 3 - c 22 z 1 , ( 1 - c 11 c 22 ) z 1 c 11 } . Clearly, λ ¯ = z 3 - c 22 z 1 if c 11 z 3 > z 1 and λ ¯ = ( 1 - c 11 c 22 ) z 1 c 11 if c 11 z 3 > z 1. We will prove that there exists N 1 such that D n (λ) > 0 for all λ ( 0 , λ ¯ ) and n > N 1 under some simple conditions.

Lemma 2.2. If z 1c 11 z 3 / 2 or z 1 ≥ 2c 11 z 3, then D n (λ) > 0 for all λ ( 0 , λ ¯ ) and n > N 1, where N 1 such that μ n > c 22 z 1 (z 3 - c 22 z 1)/d 2 for n > N 1.

Proof. First, we claim that D n (0) > 0, D n ( λ ¯ ) > 0 for all n > N 1. Directly calculating, we have
D n ( 0 ) = d 1 d 2 μ n 2 - c 22 d 1 z 1 ( z 3 - c 22 z 1 ) μ n > 0 , D n ( λ ¯ ) = d 1 d 2 μ n 2 + d 2 μ n c 11 ( z 3 - c 22 z 1 ) 1 - c 11 c 22 > 0 if λ ¯ = z 3 - c 22 z 1 , d 1 d 2 μ n 2 + d 2 μ n z 1 > 0 if λ ¯ = ( 1 - c 11 c 22 ) z 1 c 11 .
Next, we prove that for all λ ( 0 , λ ¯ ) , D n (λ) > 0 if D n (0) > 0, D n ( λ ¯ ) > 0 satisfied. From the expression of D n (λ), we have D n (λ) → +∞ when λ → +∞ and D n (λ) → - ∞ when λ → - ∞, and D n (λ) has two inflection points for any fixed n . We only need to prove that 0 and λ ¯ are in the same side of the second inflection point. Differentiating D n (λ) with respect to λ for fixed n, we have
D n ( λ ) = a λ 2 + b λ + c ,
where
a = 3 c 11 1 - c 11 c 22 , b = - 2 z 1 - 2 c 11 ( z 3 - c 22 z 1 ) 1 - c 11 c 22 - 2 d 1 μ n c 11 c 22 1 - c 11 c 22 , c = z 1 ( z 3 - c 22 z 1 ) - c 22 d 1 μ n 2 z 1 + c 11 z 3 - z 1 1 - c 11 c 22 + d 2 μ n c 11 1 - c 11 c 22 .
The axis of symmetry of D n ( λ ) is
λ min = 1 3 ( z 3 - c 22 z 1 ) + 1 - c 11 c 22 c 11 z 1 + c 22 d 1 μ n > 0 .

If z 1c 11 z 3/2, then λ min λ ¯ = ( 1 - c 11 c 22 ) z 1 c 11 . Else if z 1 ≥ 2c 11 z 3, then λ min λ ¯ = z 3 - c 22 z 1 . That is, 0 < λ ¯ λ min , 0 and λ ¯ are in the same side of the second inflection point and the proof is complete.

Next, we seek the critical points λ ( 0 , λ ¯ ) such that T n = 0. Define
T ( λ , p ) : = - ( d 1 + d 2 ) p + c 11 c 22 1 - c 11 c 22 λ 2 - 2 c 22 z 1 + c 11 c 22 z 3 - c 22 z 1 + c 11 1 - c 11 c 22 λ + c 22 z 1 ( z 3 - c 22 z 1 ) .
Then, T n (λ) = 0 is equivalent to T ( λ , p ) = 0 . Solving p from T ( λ , p ) = 0 , we have
p ( λ ) = 1 d 1 + d 2 c 11 c 22 1 - c 11 c 22 λ 2 - 2 c 22 z 1 + c 11 c 22 z 3 - c 22 z 1 + c 11 1 - c 11 c 22 λ + c 22 z 1 ( z 3 - c 22 z 1 ) .
Immediately,
p ( 0 ) = 1 d 1 + d 2 c 22 z 1 ( z 3 - c 22 z 1 ) > 0 , p ( λ ¯ ) = - 1 d 1 + d 2 c 11 ( z 3 - c 22 z 1 ) 1 - c 11 c 22 < 0 if λ ¯ = z 3 - c 22 z 1 , - z 1 d 1 + d 2 < 0 if λ ¯ = ( 1 - c 11 c 22 ) z 1 c 11 .

Lemma 2.3. Denote N 2 be the number such that μ N 2 p ( 0 ) < μ N 2 + 1 . Then, there exists N 2 points λ i , i = 1,2, ..., N 2, satisfying λ ¯ > λ 1 > λ 2 > > λ N 2 0 , such that T i (λ j ) < 0 for i < j, and T i (λ j ) > 0 for i < j, i = 1,2, ..., 1 ≤ jN 2.

Lemma 2.4. Suppose λ i , 1 ≤ iN 2 be defined as in Lemma 2.3. If α(λ i ) ± (λ i )be the unique pair of complex eigenvalues near the imaginary axis, then α'(λ i ) < 0.

Theorem 2.5. Suppose the condition of Lemma 2.2 is satisfied and λ i , 1 ≤ iN 2 be defined as in Lemma 2.3. Then, Hopf bifurcation occurs at λ i if
μ i < d 2 - d 1 d 1 ( d 1 + d 2 ) c 11 λ i 1 - c 11 c 22 , 1 i min { N 1 , N 2 } ,
(2.5)

where N 1, N 2 are defined as before.

Proof. We need to show that D n (λ i ) > 0, n , then Lemma 2.1 could be used. First, T i (λ i ) = 0 gives
( d 1 + d 2 ) μ i + c 11 λ i 1 - c 11 c 22 = c 11 c 22 1 - c 11 c 22 λ i 2 - c 22 2 z 1 + c 11 z 3 - z 1 1 - c 11 c 22 λ i + c 22 z 1 ( z 3 - c 22 z 1 ) .
Now, D n (λ i ) could be expressed as
D n ( λ i ) = d 1 d 2 μ n 2 - d 1 2 μ i + d 1 d 2 μ i + ( d 1 - d 2 ) c 11 λ i 1 - c 11 c 22 μ n + c 11 1 - c 11 c 22 λ i 3 - 2 z 1 + c 11 z 3 - z 1 1 - c 11 c 22 λ i 2 + z 1 ( z 3 - c 22 z 1 ) λ i .
Define
D ( λ i , p ) = d 1 d 2 p 2 - d 1 2 μ i + d 1 d 2 μ i + ( d 1 - d 2 ) c 11 λ i 1 - c 11 c 22 p + c 11 1 - c 11 c 22 λ i 3 - 2 z 1 + c 11 z 3 - z 1 1 - c 11 c 22 λ i 2 + z 1 ( z 3 - c 22 z 1 ) λ i .
Clearly, D ( λ i , 0 ) > 0 and the axis of symmetry of D ( λ i , p ) is
p min = d 1 2 μ i + d 1 d 2 μ i + ( d 1 - d 2 ) c 11 λ i ( 1 - c 11 c 22 ) 2 d 1 d 2 .

The condition in the theorem ensure p min < 0, which lead to D ( λ i , p ) > 0 for p > 0. Hence, D n (λ i ) > 0 and λ i are Hopf bifurcation points.

Remark 2.6. Theorem 2.5 gives a sufficient condition for Hopf bifurcation occurring. From the proof of Theorem 2.5, we see that the inequality (2.5) is stringent. We consider that D ( λ i , p ) is continuous with respect to p, but D n (λ i ) is a set of discrete values. Hence, we need not to ensure that the inequality (2.5) is always satisfied in some simple case. For instance, N 2 = 1. Example 2.8 exactly demonstrates this feature.

In the following, we take attention to the stability and direction of bifurcating periodic solutions bifurcated at λ 1.

We give the detail of the calculation process of the direction of Hopf bifurcation at λ 1 in the following. It is obvious that ±, with ω = D 1 ( λ 1 ) , are the only pair of simple purely imaginary eigenvalues of L(λ 1). We need to calculate the Poincaré norm form of (2.2) for λ = λ 1:
ż = i ω z + z j = 1 M c j ( z z ¯ ) j ,
where z is a complex variable, M ≥ 1and c j are complex-valued coefficients. The direction of Hopf bifurcation at λ 1 is decided by the sign of Re(c 1), which has the following form:
c 1 = i 2 ω g 20 g 11 - 2 g 11 2 - 1 3 g 02 2 + 1 2 g 21 .
In the following, we will calculate g 20, g 11, g 02, and g 21. We recall that
f ( u , v ) = - c 11 u 2 - u v , g ( u , v ) = ( z ¯ - 2 c 22 v * ) u v + ( c 22 z ¯ - c 22 2 v * ) v 2 - v * u 2 - u 2 v - 2 c 22 u v 2 - c 22 2 v 3 .
Notice that the eigenvalues μ n = n 2/ℓ2, n = 1,2, ..., the corresponding eigenfunction are sin(nx/ℓ) in our problem. Hence, we set q = (a, b) T sin(x/ℓ) be such that L(λ 1)q = iωq and let q* = M(a*, b*) T sin(x/ℓ) be such that L(λ 1) T q* = -iωq*, and moreover, 〈q*, q〉 = 1 and q * , q ¯ = 0 . Here
u , v = 0 π ū T v d x , u , v X
be the inner dot and
a = b * = 1 , b = i ω + d 1 μ 1 - a 11 a 12 , a * = - i ω + d 2 μ 1 - a 22 a 12 , M = 2 π ω i a 12 .
Express the partial derivatives of f(u, v) and g(u, v) at (u, v) = (0, 0) with respect to λ when λ 1, we have
f u u = - c 11 , f u v = - 1 , g u v = z 3 - 3 c 22 z 1 + λ 1 ( 3 c 11 c 22 - 1 ) 1 - c 11 c 22 , g u u = - z 1 + c 11 λ 1 1 - c 11 c 22 , g v v = c 22 ( z 3 - 2 c 22 z 1 ) + c 22 λ 1 ( 2 c 11 c 22 - 1 ) 1 - c 11 c 22 , g v v v = - c 22 2 , g u u v = - 1 , g u v v = - 2 c 22 ,
and the others are equal to zero. As stated in [5, 6], we need to calculate Q q q , Q q q ¯ , and C q q q ¯ , which are defined as
Q q q = sin 2 ( x ) c d , Q q q ¯ = sin 2 ( x ) e f , C q q q ¯ = sin 3 ( x ) g h ,
where
c = f u u a 2 + 2 f u v a b + f v v b 2 , d = g u u a 2 + 2 g u v a b + g v v b 2 , e = f u u a 2 + f u v ( a b ¯ + ā b ) + f v v b 2 , f = g u u a 2 + g u v ( a b ¯ + ā b ) + g v v b 2 , g = f u u u a 2 a + f u u v ( 2 a 2 b + a 2 b ¯ ) + f u v v ( 2 b 2 a + b 2 ā ) + f v v v b 2 b , h = g u u u a 2 a + g u u v ( 2 a 2 b + a 2 b ¯ ) + g u v v ( 2 b 2 a + b 2 ā ) + g v v v b 2 b .
From direct calculation, we have
q * , Q q q = 4 M ¯ 3 ( ā * c + d ) , q * , Q q q ¯ = 4 M ¯ 3 ( ā * e + f ) , q ¯ * , Q q q = 4 M 3 ( a * c + d ) , q ¯ * , Q q q ¯ = 4 M 3 ( a * e + f ) .
(2.6)
Then, we have (the detail meaning of the following parameters are stated in [6, 5])
H 20 = Q q q - q * , Q q q q - q ¯ * , Q q q q ¯ = 1 2 ( 1 - cos ( 2 x ) ) c d - q * , Q q q a b - q ¯ * , Q q q ā b ¯ sin ( x ) = k = 1 - 8 ( 2 k - 1 ) ( 2 k + 1 ) ( 2 k - 3 ) π c d sin ( ( 2 k - 1 ) x ) - q * , Q q q 1 b - q ¯ * , Q q q 1 b ¯ sin ( x )
(2.7)
and
H 11 = Q q q ¯ - q * , Q q q ¯ q - q ¯ * , Q q q ¯ q ¯ = 1 2 ( 1 - cos ( 2 x ) ) e f - q * , Q q q ¯ a b - q ¯ * , Q q q ¯ ā b ¯ sin ( x ) = k = 1 - 8 ( 2 k - 1 ) ( 2 k + 1 ) ( 2 k - 3 ) π e f sin ( ( 2 k - 1 ) x ) - q * , Q q q ¯ 1 b - q ¯ * , Q q q ¯ 1 b ¯ sin ( x ) .
(2.8)
Therefore, we can obtain w 20, w 11 as
w 20 = [ 2 i ω I - L ( λ 1 ) ] - 1 H 20 and w 11 = - [ L ( λ 1 ) ] - 1 H 11 .
Clearly, the calculation of (2iωI - L(λ 1))-1 and [L(λ 1)]-1 are restricted to the subspaces spanned by the eigenmodes sin(kx/ℓ), k = 1,2, .... One can compute that
( 2 i ω I - L k ( λ 1 ) ) - 1 = ( α 1 k + i α 2 k ) - 1 2 i ω - a 22 + d 2 μ k a 12 a 21 2 i ω - a 11 + d 1 μ k , L k - 1 ( λ 1 ) = 1 α 3 k a 22 - d 2 μ k - a 12 - a 21 a 11 - d 1 μ k ,
where
α 1 k = - 4 ω 2 + a 11 a 22 - a 12 a 21 - ( d 1 a 22 + d 2 a 11 ) μ k + d 1 d 2 μ k 2 , α 2 k = - 2 ω ( a 11 + a 22 ) + 2 ω ( d 1 + d 2 ) μ k , α 3 k = a 11 a 22 - a 12 a 21 - ( d 2 a 11 + d 1 a 22 ) μ k + d 1 d 2 μ k 2 .
Then,
w 20 = k = 1 8 s i n ( ( 2 k 1 ) x / ) ( 2 k 1 ) ( 2 k + 1 ) ( 2 k 3 ) π ( 2 i ω I L 2 k 1 ( λ 1 ) ) 1 ( c d ) ( 2 i ω I L 1 ( λ 1 ) ) 1 [ q * , Q q q ( a b ) q ¯ * , Q q q ( a ¯ b ¯ ) ] s i n ( x / ) = k = 1 8 s i n ( ( 2 k 1 ) x / ) ( 2 k 3 ) 1 ( 4 k 2 1 ) ( α 1 2 k 1 + i α 2 2 k 1 ) π ( ( 2 i ω a 22 + d 2 μ 2 k 1 ) c + a 12 d a 21 c + ( 2 i ω a 11 + d 1 μ 2 k 1 ) d ) 1 α 11 + i α 21 ( ( 2 i ω a 22 + d 2 μ 1 ) ξ 1 + a 12 ξ 2 a 21 ξ 1 + ( 2 i ω a 11 + d 1 μ 1 ) ξ 2 ) s i n ( x / ) , w 11 = k = 1 8 s i n ( ( 2 k 1 ) x / ) α 3 2 k 1 ( 4 k 2 1 ) ( 2 k 3 ) π ( ( a 22 d 2 μ 2 k 1 ) e + a 12 f a 21 e ( a 11 d 1 μ 2 k 1 ) f ) 1 α 31 ( ( a 22 d 2 μ 1 ) ξ 3 + a 12 ξ 4 a 21 ξ 3 ( a 11 d 1 μ 1 ) ξ 4 ) s i n ( x / ) ,
where
ξ 1 = q * , Q q q a - q ¯ * , Q q q ā = 4 c 3 ( ā * M ¯ - a * M ) + 4 d 3 ( M ¯ - M ) , ξ 2 = q * , Q q q b - q ¯ * , Q q q b ¯ = 4 c 3 ( b ā * M ¯ - b ¯ a * M ) + 4 d 3 ( b M ¯ - b ¯ M ) , ξ 3 = q * , Q q q ¯ a - q ¯ * , Q q q ¯ ā = 4 e 3 ( ā * M ¯ - a * M ) + 4 f 3 ( M ¯ - M ) , ξ 4 = q * , Q q q ¯ b - q ¯ * , Q q q ¯ b ¯ = 4 e 3 ( b ā * M ¯ - b ¯ a * M ) + 4 f 3 ( b M ¯ - b ¯ M ) .
Then,
Q w 20 q ¯ = k = 1 Q w 20 q ¯ 1 k Q w 20 q ¯ 2 k sin x sin ( 2 k - 1 ) x + Q w 20 q ¯ 10 Q w 20 q ¯ 20 sin 2 x = k = 1 f u u w 20 1 k + f u v b ¯ w 20 1 k + f u v w 20 2 k g u u w 20 1 k + g u v b ¯ w 20 1 k + g u v w 20 2 k + g v v b ¯ w 20 2 k sin x sin ( 2 k - 1 ) x + f u u w 20 10 + f u v b ¯ w 20 10 + f u v w 20 20 g u u w 20 10 + g u v b ¯ w 20 10 + g u v w 20 20 + g v v b ¯ w 20 20 sin 2 x , Q w 11 q = k = 1 Q w 11 q 1 k Q w 11 q 2 k sin x sin ( 2 k - 1 ) x + Q w 11 q 10 Q w 11 q 20 sin 2 x , = k = 1 f u u w 11 1 k + f u v b w 11 1 k + f u v w 11 2 k g u u w 11 1 k + g u v b w 11 1 k + g u v w 11 2 k + g v v b w 11 2 k sin x sin ( 2 k - 1 ) x + f u u w 11 10 + f u v b w 11 10 + f u v w 11 20 g u u w 11 10 + g u v b w 11 10 + g u v w 11 20 + g v v b w 11 20 sin 2 x ,
where
ξ 1 = q * , Q q q a q ¯ * , Q q q a ¯ = 4 c 3 ( a ¯ * M ¯ a * M ) + 4 d 3 ( M ¯ M ) , ξ 2 = q * , Q q q b q ¯ * , Q q q b ¯ = 4 c 3 ( b a ¯ * M ¯ b ¯ a * M ) + 4 d 3 ( b M ¯ b ¯ M ) , ξ 3 = q * , Q q q ¯ a q ¯ * , Q q q ¯ a ¯ = 4 e 3 ( a ¯ * M ¯ a * M ) + 4 f 3 ( M ¯ M ) , ξ 4 = q * , Q q q ¯ b q ¯ * , Q q q ¯ b ¯ = 4 e 3 ( b a ¯ * M ¯ b ¯ a * M ) + 4 f 3 ( b M ¯ b ¯ M ) .
and
w 20 10 = ( 2 i ω - a 22 + d 2 μ 1 ) ξ 1 + a 12 ξ 2 α 1 1 + i α 2 1 , w 20 20 = a 21 ξ 1 + ( 2 i ω - a 11 + d 1 μ 1 ) ξ 2 α 1 1 + i α 2 1 , w 11 10 = - ( a 22 - d 2 μ 1 ) ξ 3 + a 12 ξ 4 α 3 1 , w 11 20 = a 21 ξ 3 - ( a 11 - d 1 μ 1 ) ξ 4 α 3 1 .
Notice that
0 π sin 4 ( x ) d x = 3 π 8 , 0 π sin 2 ( x ) sin ( ( 2 k - 1 ) x ) d x = - 4 ( 2 k - 1 ) ( 2 k + 1 ) ( 2 k - 3 ) ,
we have
q * , C q q q ¯ = 3 M ¯ h π 8 , q * , Q w 20 q ¯ = k = 1 - 4 M ¯ ( 2 k - 1 ) ( 2 k + 1 ) ( 2 k - 3 ) ( ā * Q w 20 q ¯ 1 k + Q w 20 q ¯ 2 k ) + 4 M 3 ( ā * Q w 20 q ¯ 10 + Q w 20 q ¯ 20 ) , q * , Q w 11 q = k = 1 - 4 M ¯ ( 2 k - 1 ) ( 2 k + 1 ) ( 2 k - 3 ) ( ā * Q w 11 q 1 k + Q w 11 q 2 k ) + 4 M 3 ( ā * Q w 11 q ¯ 10 + Q w 11 q ¯ 20 ) .
Hence, we have
g 20 = q * , Q q q = 4 M ¯ 3 ( ā * c + d ) , g 11 = q * , Q q q ¯ = 4 M ¯ 3 ( ā * e + f ) , g 02 = q * , Q q ¯ q = 4 M ¯ 3 ( ā * c ¯ + d ¯ ) ,
and
g 21 = 2 q * , Q w 11 q + q * , Q w 20 q ¯ + q * , C q q q ¯ = k = 1 - 4 M ¯ ( ( 2 Q w 11 q 1 k + Q w 20 q ¯ 1 k ) ā n * + ( 2 Q w 11 q 2 k + Q w 20 q ¯ 2 k ) ) ( 2 k - 1 ) ( 2 k + 1 ) ( 2 k - 3 ) + 4 M ¯ ( ( 2 Q w 11 q 10 + Q w 20 q ¯ 10 ) ā * + ( 2 Q w 11 q 20 + Q w 20 q ¯ 20 ) ) 3 + 3 M ¯ h π 8 .
Then, it follows that
c 1 = i 2 ω ( g 20 g 11 - 2 g 11 2 - 1 3 g 02 2 ) + 1 2 g 21 = 8 2 i 9 ω [ M ¯ 2 ( ā * c + d ) ( ā * e + f ) - 2 M 2 ā * e + f 2 - 1 3 M 2 ā * c + d 2 ] + k = 1 - 2 M ¯ ( ( 2 Q w 11 q 1 k + Q w 20 q ¯ 1 k ) ā n * + ( 2 Q w 11 q 2 k + Q w 20 q ¯ 2 k ) ) ( 2 k - 1 ) ( 2 k + 1 ) ( 2 k - 3 ) + 2 M ¯ ( ( 2 Q w 11 q 10 + Q w 20 q ¯ 10 ) ā * + ( 2 Q w 11 q 20 + Q w 20 q ¯ 20 ) ) 3 + 3 M ¯ h π 16 .

Theorem 2.7. Suppose the conditions in Theorem 2.7 are satisfied. Then, the positive constant equilibrium E 1 is asymptotically stable when λ ( λ 1 , λ ¯ ) . Hopf bifurcation occurs at λ 1, and the bifurcating periodic solutions are in the left(right) neighborhood of λ 1 and stable(unstable) if Re(c 1) < 0(> 0).

Example 2.8. Suppose ℓ = 1(i. e. Ω = (0, π)). d 1 = 1/10, d 2 = 3/10, z 1 = z 2 = 1, z 3 = 3/2 and c 11 = 1/3. Let c 22 be the bifurcation parameter. We found that there has only one Hopf bifurcation point λ = 0.0833. E 1 is stable for 0.0833 < λ < 1.1667. For λ < 0.0833, Hopf bifurcation occurs and the bifurcating periodic solutions are stable. In other words, c 22 = 0.9167 is the critical value for Hopf bifurcation. We give the simulation for c 22 0.9167 ± 0.02 in the follows. If c 22 = 0.9167 - 0.02, E 1 is stable (Figure 1). If c 22 = 0.9167 + 0.02, there exists periodic solution, which is stable (Figure 2).
Figure 1

E 1 is asymptotically stable for c 22 = 0.9167 - 0.02. The initial value is (u 0, v 0) = (0.1, 0.2) * sin x.

Figure 2

Periodic solution appear for c 22 = 0.9167 + 0.02. The initial value is (u 0, v 0) = (0.1, 0.2) * sin x.

3 Conclusion

In this article, we take λ as a main bifurcation parameter, study stability of the constant positive equilibrium E 1, which exists for λ ( 0 , λ ¯ ) . The critical values for Hopf bifurcation occurring are found out, and the stability and direction of bifurcating periodic solutions bifurcated at λ 1 are studied. By the method of the reference [5] and our early work [6], we give the detail of the calculation of the norm form for system (2.2). In addition, we claim that the bifurcating periodic solutions are all spatially nonhomogeneous, since the problem is subject to Dirichlet fixed boundary conditions.

Declarations

Acknowledgements

We express the special gratitude to the reviewers for the helpful comments given for this article. This research was supported by the National Natural Science Foundation of China (No. 11031002).

Authors’ Affiliations

(1)
Department of Mathematics, Harbin Institute of Technology

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© Liu and Wei; licensee Springer. 2011

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