In this paper, the dynamical properties of Smith type diffusion model with Dirichlet boundary conditions are studied. The properties of hyperbolic fixed points and non-hyperbolic fixed points of the model are analyzed. By using the central manifold theorem, the bifurcation phenomenon of the model is studied. The results show that flip, transcritical, pitchfork and Fold-flip bifurcations exist at non-hyperbolic fixed points.
For many years, the relationship between growth rate and density for population has been the object of discussion and experiment, and researchers have proposed many mathematical models to describe this relationship. In 1963, Smith [
d x d t = x H ( x ) , (1)
where H ( x ) is average growth rate of population and
H ( x ) = r ( k − x k + r c x )
k is the mass of x at saturation, r is the growth rate of x without food restriction, c is the replacement rate of x at saturation. In 2020, Liu et al. based on Smith model (1) and considering linear yield, proposed the following model (see [
d x d t = x ( r ( k − x ) k + b x − h ) , (2)
where h (>0) is a linear harvest rate and b = r/c. They studied phenomena such as harvest behavior and equilibrium bifurcation caused by Allee effects.
Since diffusion can significantly change the spatial distribution of species, in recent years, many researchers have paid attention to the model with diffusion effects (see [
μ i t + 1 = p μ i t 1 + q μ i t + d Δ 2 μ i − 1 t
under the Dirichlet boundary condition (see [
μ 0 t = 0 = μ m + 1 t (3)
and studied the existence of steady-state solutions and bifurcation of the model.
In this paper, inspired by papers [
μ i t + 1 = μ i t ( r ( k − μ i t ) k + b μ i t − h ) + d Δ 2 μ i − 1 t (4)
where d > 0 is the diffusion coefficient, t ∈ Z + = { 0 , 1 , 2 , ⋯ } , Δ 2 is the second order difference operator, and Δ 2 μ i − 1 t = μ i + 1 t − 2 μ i t + μ i − 1 t , i ∈ { 1 , 2 , ⋯ , m } . Supposing that the Dirichlet boundary condition (3) is as μ 0 t = 0 = μ 3 t and binary variables as
x t = μ 2 t , y t = μ 1 t , t ∈ Z + = { 0 , 1 , 2 , ⋯ } ,
then, we rewrite model (4) (with replacing n with t) as the following two-dimensional discrete time element population model
{ x n + 1 = x n + δ x n ( r ( k − x n ) k + b x n − h ) + d ( − 2 x n + y n ) , y n + 1 = y n + δ y n ( r ( k − y n ) k + b y n − h ) + d ( x n − 2 y n ) , (5)
where n ∈ Z + = { 0 , 1 , 2 , ⋯ } and δ > 0 . Obviously, the system (5) has three fixed points: zero fixed point E 0 = ( 0 , 0 ) and two positive fixed points
E 1 = ( 1 , 2 − δ d ( r ( k − 1 ) k + b − h ) ) , E 2 = ( 2 − δ d ( r ( k − 1 ) k + b − h ) , 1 ) .
In the following, the hyperbolic and non-hyperbolic properties of three fixed points are analyzed, and the flip, transcritical and pitchfork bifurcations generated at E0 are also studied.
We write the model (5) as a plane map on ℝ 2 as follows
( x y ) ↦ ( ( 1 − δ h − 2 d ) x + δ x r ( k − x ) k + b x + d y ( 1 − δ h − 2 d ) y + δ y r ( k − y ) k + b y + d x ) . (6)
For any fixed point ( x i , y i ) , the Jacobian matrix of (6) is
J ( E 0 ) = ( A 0 d d A 0 )
where A 0 = 1 − δ ( h − r ) − 2 d , and
J ( E i ) = ( e + A i d d e + B i )
where
e = 1 − δ h − 2 d , A i = δ r ( k − x i ) k + b x i − δ r k ( b + 1 ) x i ( k + b x i ) 2 , B i = δ r ( k − y i ) k + b y i − δ r k ( b + 1 ) y i ( k + b y i ) 2
for i = 1 , 2 .
In order to discuss the properties of E0, we define
β = h − r , L 1 = { d | d = 2 − δ β } , L 2 = { d | d = 1 3 ( 2 − δ β ) } ,
L 3 = { d | d = − δ β , β < 0 } , L 4 = { d | d = − 1 3 δ β , β < 0 } ,
D 1 = { ( d , β ) | 0 < d < − 1 3 δ β , β < 0 } ,
D 2 { ( d , β ) | 0 < 1 3 ( 2 − δ β ) < d < − δ β , β < 0 } ,
D 3 = { ( d , β ) | d > 2 − δ β > 0 } ,
K 1 = { ( d , β ) | 0 < − δ β < d < 1 3 ( 2 − δ β ) , β ≤ 0 } ,
K 2 = { ( d , β ) | 0 < d < min { − δ β , 1 3 ( 2 − δ β ) } } ,
K 3 = { ( d , β ) | max { 0 , − δ β , 1 3 ( 2 − δ β ) } < d < 2 − δ β } .
Theorem 2.1 The fixed point E0 has the following properties:
1) when ( d , β ) ∈ D 1 , 2 , 3 , E0 isan unstable node;
2) when ( d , β ) ∈ K 1 , E0 isa stable node;
3) when ( d , β ) ∈ K 2 , 3 , E0 is a saddle;
4) when ( d , β ) ∈ L i ( i = 1 , 2 , 3 , 4 ) , E0 is non-hyperbolic.
Proof. The characteristic equation corresponding to the Jacobian matrix of fixed point E0 is
F ( λ ) = [ λ − ( 1 − δ β − 2 d ) ] 2 − d 2 .
Thus, we get
λ 1 = 1 − δ β − d , λ 2 = 1 − δ β − 3 d ,
and
λ 1 > λ 2 .
Therefore, we easily prove that
1) when ( d , β ) ∈ D 1 , 2 , 3 , we have | λ 1 | > 1 , | λ 2 | > 1 . Thus E0 is an unstable node;
2) when ( d , β ) ∈ K 1 , we have | λ 1 | < 1 , | λ 2 | < 1 . Thus E0 is a stable node;
3) when ( d , β ) ∈ K 2 , 3 , we have | λ 1 | > 1 , | λ 2 | < 1 or | λ 1 | < 1 , | λ 2 | > 1 . Thus E0 is astable node saddle point;
4) when ( d , β ) ∈ L i ( i = 1 , 2 , 3 , 4 ) , we have | λ 1 | = 1 or | λ 2 | = 1 . Thus E0 is a non-hyperbolic fixed point.
In order to discuss the stabilitiesof fixed point E i ( i = 1 , 2 ) , we give the following lemma of which the proof is obvious and will be omitted.
Lemma 2.1 Suppose F ( λ ) = λ 2 + Q λ + S and λ 1 , λ 2 are two roots of F ( λ ) = 0 . Then
1) when − 2 < Q < 2 , Q 2 ≥ 4 S and F ( − 1 ) > 0 , then 0 − 1 < λ 1 , 2 < 1 ;
2) when Q < − 2 and Q 2 ≥ 4 S , then λ 1 , 2 > 1 ;
3) when Q > 2 and Q 2 ≥ 4 S , then λ 1 , 2 < − 1 ;
4) when F ( 1 ) > 0 and F ( − 1 ) < 0 , then | λ 1 | > 1 and | λ 2 | < 1 ;
5) when F ( 1 ) = 0 and Q ≠ 0 , − 2 , then λ 1 = 1 and λ 2 ≠ 1 ;
6) when F ( 1 ) = 0 and Q = 0 or −2, then λ 1 , 2 = 1 ;
7) when F ( − 1 ) = 0 and Q ≠ 2 , then λ 1 = − 1 and λ 2 ≠ − 1 ;
8) when F ( − 1 ) = 0 and Q = 2 , then λ 1 , 2 = − 1 ;
9) when 0 < S < 1 and Q 2 < 4 S , then λ 1 , 2 is a pair of complex roots and | λ 1 , 2 | < 1 ;
10) when S > 1 and Q 2 < 4 S , then λ 1 , 2 is a pair of complex roots and | λ 1 , 2 | > 1 ;
11) when S = 1 and − 2 < Q < 2 , then λ 1 , 2 is a pair of complex roots and | λ 1 , 2 | = 1 .
Let
μ i = − 1 δ r ( A i + B i ) , ω i = ( e + A i ) ( e + B i ) ,
for i = 1 , 2 .
Theorem 2.2 All of topology types for the fixed points E i ( i = 1 , 2 ) are listed in
Proof. Thecharacteristic equation corresponding to the Jacobian matrix of fixed point E 1 , 2 is
F ( λ ) = λ 2 + Q λ + S = 0 ,
where
Q = − tr ( J ( E i ) ) = δ r μ i − 2 e , S = det ( J ( E i ) ) = ω i − d 2 ,
for i = 1 , 2 .
Therefore, from lemma 2.1, we easily obtain the conclusion of
Theorem 3.1 When ( d , β ) ∈ L 1 = { d | d = 2 − δ β } , system(5) undergoes a supercritical flip bifurcation at point E 0 ( 0 , 0 ) , i.e., as δ β goes from less than 2 to more than 2, system (5) bifurcates out a stable period-2 orbit at the fixed point E 0 ( 0 , 0 ) .
Proof. When ( d , β ) ∈ L 1 = { d | d = 2 − δ β } , characteristic values λ 1 = − 1 , λ 2 = − 5 + 2 δ β and | λ 2 | ≠ 1 . Denote ε = d + δ β − 2 and select ε as the bifurcation parameter. Therefore the mapping (6) becomes
| conditions | topology types |
|---|---|
| 2 e − 2 δ r < μ i < 2 e + 2 δ r , max { 0 , δ r μ i − 2 e + d 2 − 1 } < ω i ≤ 1 4 ( δ r μ i − 2 e ) 2 + d 2 | stable node |
| or μ i < 4 ( 1 − d 2 ) + 2 e 2 δ r , max { d 2 , 1 4 ( δ r μ i − 2 e ) 2 + d 2 } < ω i ≤ 1 + d 2 | |
| μ i < 2 e − 2 δ r , 1 + d 2 < ω i ≤ 1 4 ( δ r μ i − 2 e ) 2 + d 2 | unstable node |
| or μ i > 2 e + 2 δ r , 1 + d 2 < ω i ≤ 1 4 ( δ r μ i − 2 e ) 2 + d 2 | |
| μ i > 2 e + 1 − d 2 δ r , 0 < ω i < δ r μ i − 2 e + d 2 − 1 | saddle |
| ω i > max { 1 + d 2 , 1 4 ( δ r μ i − 2 e 2 ) + d 2 } | unstable focus |
| ω i = δ r μ i − 2 e + d 2 − 1 | non-hyperbolic |
| 2 e − 2 δ r < μ i < 2 e + 2 δ r ω i = d 2 + 1 |
( x y ) ↦ ( − ( ε + d + δ r + 1 ) x + δ x r ( k − x ) k + b x + d y − ( ε + d + δ r + 1 ) y + δ y r ( k − y ) k + b y + d x ) . (7)
The Taylor expansion of mapping (7) is
( x y ) ↦ ( − ε − d − 1 d d − ε − d − 1 ) ( x y ) + ( − δ r ( b + 1 ) k x 2 − δ r ( b + 1 ) k y 2 ) + O ( | ( x , y ) | 3 ) . (8)
We get the Jacobian matrix
J ( E 0 ) = ( − ε − d − 1 d d − ε − d − 1 ) ,
characteristic values
λ 1 = − ε − 1 , λ 2 = − ε − 2 d − 1
and the corresponding eigenvectors
( 1 , 1 ) τ , ( 1 , − 1 ) τ . (9)
From the eigenvectors the following transformation is obtained
( x y ) = ( 1 1 1 − 1 ) ( u v )
and it can transform (8) into the following mapping (ε is treatedas an independent variable)
( u v ε ) ↦ ( − ε − 1 0 0 0 − ε − 2 d − 1 0 0 0 1 ) ( u v ε ) + ( − δ r ( b + 1 ) k ( u 2 + v 2 ) − 2 δ r ( b + 1 ) k u v 0 ) + O ( | ( u , v ) | 3 ) . (10)
From center manifold theorem, the stability of mapping (10) in small neighborhood of ( u , v ) = ( 0 , 0 ) can be determined by single parameter mapping, which satisfies
W l o c ( 0 , 0 ) = { ( u , v , ε ) ∈ R 3 | v = h ( u , ε ) , h ( 0 , 0 ) = 0 , D h ( 0 , 0 ) = 0 } . (11)
Suppose the central manifold is as follows
v = h ( u , ε ) = A u 2 + B u ε + C ε 2 + O ( | ( u , ε ) | 3 ) . (12)
Then
h ( ( − ε − 1 ) u + − δ r ( b + 1 ) k ( u 2 + h 2 ( u , ε ) ) , ε ) = ( − ε − 2 d − 1 ) h ( u , ε ) + − 2 δ r ( b + 1 ) k u h ( u , ε ) + O ( | ( u , ε ) | 3 ) (13)
From (10), (12) and (14), we get
A = B = C = 0 .
Then
v = h ( u , ε ) = O ( | ( u , ε ) | 3 ) . (14)
The mapping (10) restricted to the central manifold (14) is
T : u ↦ f ( u , ε ) = ( − ε − 1 ) u + − δ r ( b + 1 ) k u 2 + O ( | ( u , ε ) | 3 ) . (15)
Thus
f ( 0 , 0 ) = 0 , ∂ f ∂ u ( 0 , 0 ) = − 1 , ( ∂ f ∂ ε ∂ 2 f ∂ u 2 + 2 ∂ 2 f ∂ u ∂ ε ) ( 0 , 0 ) = − 2 ≠ 0 ,
[ 1 2 ( ∂ 2 f ∂ u 2 ) 2 + 1 3 ( ∂ 3 f ∂ u 3 ) ] ( 0 , 0 ) = 2 δ 2 r 2 ( b + 1 ) 2 k 2 ≠ 0.
Therefore, from [
Theorem 3.2 When ( d , β ) ∈ L 2 = { d | d = 1 3 ( 2 − δ β ) } and d ≠ 1 , δ β ≠ − 1 , system(5) undergoes a subcritical flip bifurcation at point E 0 ( 0 , 0 ) .
Proof. When ( d , β ) ∈ L 2 = { d | d = 1 3 ( 2 − δ β ) } and d ≠ 1 , δ β ≠ − 1 , we have
λ 1 = 1 3 ( 1 − 2 δ β ) , λ 2 = − 1
and | λ 1 | ≠ 1 . Similar to Theorem 3.1, the proof of this theorem can be obtained and will be omitted.
Theorem 3.3 When ( d , β ) ∈ L 3 = { d | d = − δ β , β < 0 } and d ≠ 1 , system(5) undergoes atranscritical bifurcation at point E 0 ( 0 , 0 ) .
Proof. When ( d , β ) ∈ L 3 = { d | d = − δ β , β < 0 } and d ≠ 1 , we have
λ 1 = 1 , | λ 2 | = | 1 + 2 δ β | ≠ 1 .
Set d = − δ β + ε and chose ε as the bifurcation parameter, thus mapping (6) is written in the following form
( u v ε ) ↦ ( 1 − ε 0 0 0 1 − ε − 2 d 0 0 0 ε ) ( u v ε ) + ( − δ r ( b + 1 ) k ( u 2 + v 2 ) − 2 δ r ( b + 1 ) k u v 0 ) . (16)
Similar to the proof of theorem 3.1, one-dimensional equations under the restriction of a central manifold is obtained
T : u ↦ f ( u , ε ) = ( 1 − ε ) u + − δ r ( b + 1 ) k u 2 + O ( | ( u , ε ) | 3 ) . (17)
We have
f ( 0 , 0 ) = 0 , ∂ f ∂ u ( 0 , 0 ) = 1 , ∂ f ∂ ε ( 0 , 0 ) = 0 ,
∂ 2 f ∂ u 2 ( 0 , 0 ) = − 2 δ r ( b + 1 ) k ≠ 0 , ∂ 2 f ∂ u ∂ ε ( 0 , 0 ) = − 1 ≠ 0.
Therefore, from [
Theorem 3.4 When ( d , β ) ∈ L 4 = { d | d = − 1 3 δ β , β < 0 } , system(5) undergoes a pitchfork bifurcation at point E0.
Proof. When ( d , β ) ∈ L 4 = { d | d = − 1 3 δ β , β < 0 } , we have
λ 1 = 1 − 2 3 δ β , λ 2 = 1 .
Set d = 1 3 ( − δ β + ε ) and chose ε as the bifurcation parameter, thus mapping (6) is written in the following form
( u v ε ) ↦ ( 2 d − ε + 1 0 0 0 − ε + 1 0 0 0 ε ) ( u v ε ) + ( − δ r ( b + 1 ) k ( u 2 + v 2 ) − 2 δ r ( b + 1 ) k u v 0 ) . (18)
Similar to the proof of theorem 3.1, one-dimensional equations under the restriction of a central manifold is obtained
T : u ↦ f ( u , ε ) = ( − ε + 1 ) v + − δ 2 r 2 ( b + 1 ) 2 k 2 d v 3 + O ( | ( u , ε ) | 3 ) . (19)
Then we have
f ( 0 , 0 ) = 0 , ∂ f ∂ u ( 0 , 0 ) = 1 , ∂ f ∂ ε ( 0 , 0 ) = 0 ,
∂ 2 f ∂ v 2 ( 0 , 0 ) = 0 , ∂ 2 f ∂ u ∂ ε ( 0 , 0 ) = − 1 ≠ 0 , ∂ 3 f ∂ v 3 ( 0 , 0 ) = − 6 δ 2 r 2 ( b + 1 ) 2 k 2 d ≠ 0.
Therefore, from [
Theorem 3.5 When d = 1 and δ β = − 1 , system (5) undergoes fold-flip bifurcation at point E0.
Proof. When d = 1 and δ β = − 1 , we have λ 1 = 1 , λ 2 = − 1 . Let
q = d − 1 , r = δ β + 1
and chose q and r as the bifurcation parameter. Then mapping (6) may be
( x y ) ↦ ( ( − 2 q − r − δ r ) x + δ x r ( k − x ) k + b x + ( q + 1 ) y ( − 2 q − r − δ r ) y + δ y r ( k − y ) k + b y + ( q + 1 ) x ) . (20)
The Taylor expansion of mapping (20) is
( x y ) ↦ ( − 2 q − r q + 1 q + 1 − 2 q − r ) ( x y ) + ( − δ r ( b + 1 ) k x 2 − δ r ( b + 1 ) k y 2 ) + O ( | ( x , y ) | 3 ) . (21)
We get the Jacobian matrix
J ( E 0 ) = ( − 2 q − r q + 1 q + 1 − 2 q − r ) .
The characteristic values of J ( E 0 ) are
λ 1 = − q − r + 1 , λ 2 = − 3 q − r − 1
and the corresponding eigenvectors are
q 1 = ( 1 , 1 ) T , q 2 = ( − 1 , 1 ) T , p 1 = ( 1 2 , 1 2 ) T , p 2 = ( − 1 2 , 1 2 ) T
which satisfy
{ J ( E 0 ) q 1 = λ 1 q 1 , J ( E 0 ) q 2 = λ 2 q 2 , J ( E 0 ) T p 1 = λ 1 p 1 , J ( E 0 ) T p 2 = λ 2 p 2 , 〈 p 1 , q 1 〉 = 〈 p 2 , q 2 〉 = 1 , 〈 p 1 , q 2 〉 = 〈 p 2 , q 1 〉 = 0 ,
where 〈 ⋅ , ⋅ 〉 is scalar product. Therefore, any vector X = ( x , y ) T can be uniquely expressed as
X = ξ 1 q 1 + ξ 2 q 2
where ξ 1 and ξ 2 can be calculated by the following equation
{ ξ 1 = 〈 p 1 , x 〉 , ξ 2 = 〈 p 2 , x 〉 .
Then the mapping (21) can be rewritten in the following form with the new coordinates ξ 1 and ξ 2
( ξ 1 ξ 2 ) ↦ ( β 1 + 1 0 0 β 2 − 1 ) ( ξ 1 ξ 2 ) + ( 1 2 ! g 20 ξ 1 2 + 1 2 ! g 02 ξ 2 2 h 11 ξ 1 ξ 2 ) + O ( | ( ξ 1 , ξ 2 ) | 3 ) , (22)
where
β 1 = − q − r , β 2 = − 3 q − r ,
g 20 = − 2 δ r ( b + 1 ) k , g 02 = − 2 δ r ( b + 1 ) k , h 11 = − δ r ( b + 1 ) k ≠ 0 ,
g 11 = h 20 = h 02 = g i j = h i j = 0 , for i + j = 3.
Then from the result of [
This work has been supported by Guangdong Basic and Applied Basic Research Foundation (Grant No. 2022A1515010964, 2022A1515010193), the Key Project of Science and Technology Innovation of Guangdong College Students (Grant No. pdjh2023b0325).
The authors declare no conflicts of interest regarding the publication of this paper.
Yan, Y.-M., Yang, M.-H., Lin, Y., Chen, L.-D. and Zhou, X.-L. (2024) The Dynamical Properties of a Class of Discrete Smith Diffusion Model. International Journal of Modern Nonlinear Theory and Application, 13, 1-10. https://doi.org/10.4236/ijmnta.2024.131001