Abstract: In this paper, we extend a Leslie–Gower-type predator–prey system with ratio-dependent Holling III functional response considering the cost of antipredator defence due to fear. We study the impact of the fear effect on the model, and we find that many interesting dynamical properties of the model can occur when the fear effect is present. Firstly, the relationship between the fear coefficient K and the positive equilibrium point is introduced. Meanwhile, the existence of the Turing instability, the Hopf bifurcation, and the Turing–Hopf bifurcation are analyzed by some key bifurcation parameters. Next, a normal form for the Turing–Hopf bifurcation is calculated. Finally, numerical simulations are carried out to corroborate our theoretical results.
Keywords: predator–prey system, the fear effect, Turing instability, Turing–Hopf bifurcation, normal form.
Dynamic analysis of a Leslie–Gower-type predator–prey system with the fear effect and ratio-dependent Holling III functional response
Vilniaus Universitetas
Esta obra está bajo una Licencia Creative Commons Atribución 4.0 Internacional.
Recepción: 03 Noviembre 2021
Revisado: 23 Abril 2022
Publicación: 29 Junio 2022
As one of the most common mutual relationships between two populations in nature, predator–prey relationship plays a significant role in ecology and mathematical biology [2]. Since the dynamic behaviours of predator–prey models were formulated by Lotka and Volterra, many experts and scholars have studied various types of predator–prey models over the past decades, thus establishing the theoretical basis for interspecific interactions [1, 4, 12, 14]. In 1948, Leslie and Gower introduced a functional response called Leslie– Gower, which was based on the Logistic model proposed by Verhulst. This functional response can be adapted to describe the correlation between a reduction in the number of predators and the per capita availability of the predator’s preferred food [2, 17]. The Leslie–Gower predator–prey system typically takes the following form:
(1)where 
is the prey-to-predator conversion factor, and the term 
is known as the Leslie–Gower term, which means the scarcity of prey leads to a loss of predator density.
The functional response is an important component of the predator–prey model describing the way in which predator–prey interactions occur. In proposing this function, an essential role is played by the behavior of prey and predator [17,18]. Some common types of functional responses are listed here. For example, Holling I–III functional response [20], Beddington–DeAngelis functional response [10], Leslie–Gower functional response [11], Crowley–Martin functional response [9], and ratio-dependent functional response [6].
The ratio-dependent functional response, a predator-dependent functional response, is an important form in characterizing the functional responses of predator and has a better modeling for a predator–prey system [25]. Based on the existed studies and the above considerations, many experts and scholars have replaced 
in (1) with a ratio-dependent functional response and considered its spatial distribution patterns and dispersal mechanisms. Making a suitable nondimensional scaling, system (1) reduces to
(2)where 
and 
denote, respectively, the densities of the prey and the predator species at the space location 
and time 
is the Laplacian operator. The positive constants 
and 
represent separately the diffusion coefficients of the prey and the predator. Parameters
and 
are all positive constants. The nonlinear term 
in (2) is called ratio-dependent Holling type III functional response [5, 21], and it indicates the growth rate of predator per capita, which is a function of the ratio of the number of prey to predator. The term 
is known as the Leslie–Gower term [8], and it means the scarcity of prey leads to a loss of predator density [16]. There have been several reports on the dynamic behavior of this ratio-dependent predator–prey system (2). The Turing and Turing–Hopf bifurcations of system (2) were investigated and the spatiotemporal patterns of system (2) in two-dimensional space were discovered by Chen and Wu [3]. In [16], under homogeneous Neumann boundary conditions, the existence conditions of Hopf bifurcation, Turing instability of spatial uniformity and Turing–Hopf bifurcation in one- dimensional space were shown. In Chang and Zhang [2], they used the Leray–Schauder degree theory and the implicit function theorem to report the nature of the spatially homogeneous Hopf bifurcation and the nonexistence and existence of nonconstant steady states.
In proposing model (2), the fear effect of prey for predator has been neglected. All prey respond to the risk of predation and then exhibit a variety of antipredator responses. The common ones are habitat change, foraging, vigilance, and other different physiological changes [15,19,22–24]. Some previous experimental studies have confirmed this result. In addition, the fear of predation has immense impact on prey species. The mortality of prey is directly increased by predation. Moreover, the fear of prey may affect the physiological condition of juvenile prey, which is harmful to its survival in adulthood, for example, reducing the reproduction of prey [19, 22]. Furthermore, strong behavioral responses can be elicited by the fear of predation enough to affect the population and life history of entire prey populations [13]. According to their ideas [22–24], we extend model (2) by multiplying the production term by a factor 
taking in account the cost of antipredator defence due to fear [22], then model (2) becomes
(3)where parameter 
refers to the level of fear.
The organizational structure of this paper is as follows. In Section 2, both the existence of positive equilibrium state and the linear stability of system (3) are analysed. Further-more, the bifurcation analysis is investigated in Section 3, where the existence of the Turing bifurcation, the Hopf bifurcation, and the Turing–Hopf bifurcation are shown by choosing the 
and 
as bifurcation parameters, respectively. Moreover, the normal form of the Turing–Hopf bifurcation with 
and 
as parameters for system (3) near the unique positive constant equilibrium is obtained in Section 4. Finally, in Section 5, numerical simulations are carried out to verify the obtained theoretical conclusions.
Firstly, the existence of coexisting equilibrium is analyzed by considering the following equation:
(4)Obviously, (1, 0) is a boundary equilibrium point of system (3). In this article, we mainly study the relevant properties of the coexistence equilibrium of system (3). By using the second equation of (4), we are able to obtain 
. Substituting this into the first equation of (4), we obtain the expression 
.
Theorem 1. The following statements are correct:
-
Assume that
holds, then 
must have a positive real solution. -
The equilibrium of prey
is a strictly decreasing function with respect to 
when
.
Proof. (i) Obviously, 
is a quadratic function of one variable 
with the downward opening, and its symmetry axis is 
. We can also find 
. So there must be a positive solution 
to meet 
when
(ii) In the case with the fear factor, i.e., 
, for system (3), when 
, we can obtain the equilibrium of prey
Simple computation shows
that which indicates that 
is a strictly decreasing function with respect to 
. That is, increasing the level of 
can decrease the value of the prey equilibrium 
.
Theorem 1 means that system (3) has a positive equilibrium point. We suppose that 
is the positive equilibrium point of system (3) for the remainder of the article. Furthermore, it also means that if the model has a unique positive equilibrium, the fear coefficient 
has an effect on the positive equilibrium point: as the fear coefficient 
increases, the positive equilibrium point of prey 
gradually decreases (see Fig. 1). Next, we conduct a linear stability analysis of system (3). For the positive equilibrium 
, the linearized system of (3) is
(5)where
and
It is obvious to know that the characteristic equation of the linearized system (5) is
(6)where
Figure 1
Curve of relationship between K and u∗. Keeping the other parameters fixed at the values: m = 0.35, β = 1.28.
(7)and the eigenvalues of system (3) are given by
(8)Then we make the following hypotheses:
If Assumptions (A1) and (A2) are both valid, then when 
, there is 
and 
, that is to say, the real parts of the eigenvalues of system (3) are all less than zero. Therefore, the following theorem is obtained.
Theorem 2. Suppose that (A1) and (A2) hold. Then the ordinary differential equation system corresponding to system (3) is locally asymptotically stable at the positive equilibrium point 
.
In this section, the existence conditions of the Turing instability are analyzed. Under the assumptions that (A1) and (A2) are established, it is known from Theorem 2 that there is 
. Then 
(see (7)) is selected as the Turing bifurcation line parameter. Let 
and discuss in the following three situations:
Case 1. 
Case 2. 
and 
, where 
Case 3. 
and 
, where 
After the analysis and discussion of the above three situations, we can get the following theorem.
Theorem 3. Suppose (A1) and (A2) hold. Then the following statements are correct:
-
In Case 1 or Case 2, the positive equilibrium point
is locally asymptotically stable. -
In Case 3, if there does not exist a
such that 
then the positive equilibrium point 
of system (3) is locally asymptotically stable; conversely, if there exists at least a 
such that 
then the positive equilibrium point 
of system (3) is Turing unstable
Proof. Suppose that (A1) and (A2) hold. Under the condition that the parameters of Case 1 or Case 2 are satisfied, there is 
, then if 
, system (3) has the eigenvalue of the negative real part. Then statement (i) is proved. When the parameter relationship belongs to Case 3 and there is not 
such that 
then a similar method can be used to prove the conclusion. When the parameter relationship belongs to Case 3 and there is 
such that 
,then the real part of the eigenvalue 
of system (3) will be positive, which means that the positive equilibrium point 
of system (3) becomes no longer stable. Then the statement (ii) is proved.
Next, we further analyze the necessary conditions and the boundary for the occurrence of Turing instability. Assume that
(B1) 
where 
(B2) 
where 
We can find that the curve 
increases monotonically with 
until the first inflection point 
, after which the curve 
decreases linearly until the second inflection point 
, and the value of 
is less than zero when 
is greater than the second inflectiopoint 
. (see Fig. 2(a)). To simplify the study, we only study the part of the curve
before the first inflection point 
. In this part, 
is monotonically increasing in 
and intersects 
at the point 
. We take 
Hence, we have the following lemma.
Lemma 1. Suppose that (A1) and (A2) hold, then assumptions (B1) and (B2) hold if and only if 
.
Proof. Let 
, then we can rewrite 
Obviously, the symmetry axis is 
, and at this time, 
can be taken to the minimum 
.
Since 
, then 
. If we want 
with the condition 
must satisfy 
Then we can obtain 
When 
takes a value on the axis of symmetry, 
can be taken to the minimum, and we can get 
To ensure that 
is required. Then we can gain 
.In the part of the curve 
before the first inflection point 
increases monotonically with 
and intersects with 
at the point 
, where 
Denote
where 
, then 
when 
. We can find that the curve 
increases monotonically with 
until the first inflection point 
, after which the curve 
decreases linearly until the second inflection point 
, and the value of 
is less than zero when 
is greater than the second inflection point 
(see Fig. 2(b)). For simplicity, only the part of the curve 
before the first inflection point 
is studied in this section. Since both 
and 
are positive parameters, this qualification makes sense.
Lemma 2. Suppose that (A1) and (A2) hold, function 
, has the following properties:
-
increases monotonically with 
and intersects 
, and 
decreases monotonically as 
increases, where
. -
For
, the equation 
only has one root 
, and 
-
Define
, then 
. Moreover, 
if and only if 
.
Theorem 4. Suppose that (A1) and (A2) hold.
-
For any given
, when 
, system (3) occurs Turing bifurcation at the positive equilibrium point 
. -
, is the critical curve of Turing instability
, system (
.
, Turing instability occurs in system (
.In Fig. 3(a), we characterize a graph of functions 
, and
which will help us understand the results of Lemmas 1 and 2. In Fig. 3(b), we present a graph of the Turing bifurcation line 
.
Figure 2
(a) The curve of d2 = ε2(d1)d1. (b) The curve of d2 = dn(d1) at n = 2. Keeping the other parameters fixed at the values: m = 0.35, β = 1.28, K = 0.01, l = 2.
Figure 3
(a) The figure of functions d2 = ε1d1, d2 = ε2(d1)d1, and d2 = dn(d1), 0 < d1 n = 1, 2, 3 in d1, d2-plane. (b) The Turing bifurcation line d2 = d2∗(d1) (0 < d1 ≤ dA). Keeping the other parameters fixed at the values: m = 0.35, β = 1.28, K = 0.01, l = 2.
In this section, the existence conditions of the Hopf bifurcation with 
as the parameter are first analyzed. Denote
Obviously, 
under hypothesis (A2). Denote
After analysis, we can get the following theorem.
Theorem 5. Suppose (A2) holds. System (3) undergoes a Hopf bifurcation at 
when 
for 
. Moreover, the bifurcating periodic solution is spatially homogeneous when 
and spatially nonhomogeneous when 
for 
and 
Proof. Let 
be the roots of Eq. (6).
-
When
we can get 
for 
, then Eq. (6) has a pair of pure imaginary roots 
-
When
is near 
, from Eq. (8) we can get
Then we can obtain 
That is to say, for each 
, the transversal condition holds. This completes the proof.
Next, the existence conditions of the Turing–Hopf bifurcation with 
and 
as parameters are analyzed. System (3) will undergo a Turing–Hopf bifurcation when the following conditions are satisfied:
-
When
, Eq. (6) has a pair of pure imaginary roots 
This phenomenon can be produced when 
-
When
, Eq. (6) has a single zero root. This phenomenon can be produced when 
.
In this section, we assume (A2) always holds. Denote
such that
From the Theorem 5 we can know that system (3) will have a Hopf bifurcation at the positive equilibrium point 
when 
. Therefore, 
when 
are the Turing bifurcation lines, and 
is the Hopf bifurcation line. When
, we hope to find the first intersection of these two types of bifurcation lines in the first quadrant as the Turing–Hopf bifurcation point. After the above analysis, we can get the following theorem.
Theorem 6. Suppose hypothesis (A2) holds. For system (3), the following statements are correct:
-
If
, system (3) does not undergo Turing–Hopf bifurcation. -
If
, system (3) undergoes Turing–Hopf bifurcation at the point 
, and the positive equilibrium 
of system (3) is locally asymptotically stable when 
, where
Proof. In 
-plane, the Turing bifurcation curves are
The Hopf bifurcation curve is 
.
-
If
then there is no intersection point between Turing bifurcation curves 
and the Hopf bifurcation curve 
in the first quadrant. This indicates that system (3) does not undergo Turing–Hopf bifurcation. -
If
then the Turing bifurcation curve 
and the Hopf bifurcation curve 
intersect at point 
. This point is called the Turing–Hopf bifurcation point. In addition, when
, it is easy to prove that 
for 
. Then the positive equilibrium point 
is locally asymptotically stable. Next, let us verify the transversality conditions. Suppose 
with 
when 
, and 
with 
when 
, then the transversality conditions are as follows:
This completes the proof.
In this section, the existence conditions of the Hopf bifurcation with 
as the parameter are first analyzed. Denote
where
and
Obviously, 
under hypothesis (A2). Denote 
. Then we make the following hypotheses:
(C1) 
(C2) 
After analysis, we can get the following theorem.
Theorem 7. Suppose (A2), (C1) and (C2) hold. System (3) undergoes a Hopf bifurcation at 
when 
for 
. Moreover, the bifurcating periodic solution is spatially homogeneous when 
and spatially nonhomogeneous when 
for 
´ and 
.
Next, the existence conditions of the Turing–Hopf bifurcation with 
and 
as parameters are analyzed. In this section, we assume (A2) always holds. Denote
such that
From Theorem 7 we can know that system (3) will have a Hopf bifurcation at 
when 
. Therefore, 
when 
are the Turing bifurcation lines, and 
is the Hopf bifurcation line. When 
, we hope to find the first intersection of these two types of bifurcation lines in the first quadrant as the Turing–Hopf bifurcation point. After the above analysis, we can get the following theorem.
Theorem 8. Suppose hypothesis (A2), (C1), and (C2) hold. For system (3), the following statements are correct:
-
If
system (3) does not undergo Turing–Hopf bifurcation. -
If
system (3) undergoes Turing–Hopf bifurcation at the point 
The proofs of Theorems 7 and 8 are similar to those of Theorems 5 and 6, respectively, and will not be repeated here
In this section, we calculate the normal forms of the Turing–Hopf bifurcation for the reaction–diffusion system (3) at the coexistence equilibrium 
. Firstly, we introduce the parameters 
and 
by letting 
and 
, which satisfy that the reaction–diffusion system (3) will undergo Turing–Hopf bifurcation at the positive equilibrium point 
when 
and 
. Then system (3) can be transformed into
(9)For system (9), 
is still the positive equilibrium point. Let us make the transformations 
and
to move 
to the origin. After omitting the horizontal bar, system (9) becomes
(10)Then according to [7], for system (10), we can get
where 
. Then we can obtain
with
and 
The characteristic matrix corresponding to system (10) is
According to Theorem 
are eigenvalues of 
, and 
is a simple eigenvalue for 
with other eigenvalues having negative real parts. Then, by straightforward calculations, we have
Therefore, 
and 
. satisfying 
, where 
is the identity matrix. By [7], we can compute the following parameters:
where
According to [7], the normal form restricted to the third order on the central manifold of the reaction–diffusion system (3) at the Turing–Hopf singularity is
(11)Through the parameter transformation
, the normal form equation (11) can be rewritten into real coordinates form (the third-order term is truncated, and the azimuth angle is removed item 
)
In this section, we perform the numerical simulations. Taking
, we have
(12)Through calculation, we get the unique positive equilibrium point 
then hypothesis (A2) holds. In addition,
is the Hopf bifurcation curve in 
, 
-plane.
are the Turing bifurcation curves. Then the normal form restricted on center manifold for the reaction–diffusion system (12) at Turing–Hopf singularity is
Then we have
(13)Considering 
, system (13) has equilibria
By [7], we know: 
is the coexistence equilibrium; 
are spatially inhomogeneous steady states; 
is spatially homogeneous periodic solution; 
are spatially inhomogeneous periodic solutions.
Then the following critical bifurcation curves can be obtained:
From Fig. 4 it can be seen that the first intersection of the Turing curves 
and the Hopf curve 
with 
is chosen as the Turing–Hopf bifurcation point. Therefore, system (12) undergoes Turing–Hopf bifurcation at the point 
=(0.4052,0.0411). Then the parameter plane partition diagram and phase diagram can be obtained as shown in Fig. 5.
Figure 4
Turing–Hopf bifurcation point (r∗, dn∗ ) in r, d2-plane.
Figure 5
The bifurcation set (a) and the phase portraits (b) for Turing–Hopf bifurcation of system (12).
Proposition 1. For given 
, the plane of parameters 
, 
is divided into six regions by bifurcation curves 
. For each region, different dynamic phenomenon are generated by system (3). When 
, system (3) has a locally asymptotically stable positive equilibrium point 
(see Fig. 6). Conversely, when 
, the positive equilibrium point 
becomes unstable. When 
, system (3) has a pair of stable spatially inhomogeneous steady states (see Fig. 7). The spatial patterns and bistability are shown by system (3). When 
, there are a pair of stable spatially inhomogeneous periodic solutions and a pair of unstable spatially inhomogeneous steady states (see Fig. 8). When 
, there are a pair of stable spatially inhomogeneous periodic solutions, a pair of unstable spatially inhomogeneous steady states, and an unstable spatially homogeneous periodic solution (see Figs. 9 and 10). When 
, there are a stable spatially homogeneous periodic solution and a pair of unstable spatially inhomogeneous steady states. There are heteroclinic solutions connecting the unstable spatially inhomogeneous steady state to stable spatially homogeneous periodic solution (see Fig. 11). When 
, system (3) has a stable spatially homogeneous periodic solution (see Fig. 12).
Figure 6
When (σ1, σ2) = (1, 0) lies in region D1, the positive constant equilibrium (u∗, v∗) = (0.0513, 0.0513) is asymptotically stable. The initial value is u(x, 0) = 0.05, v(x, 0) = 0.05.
Figure 7
When (σ1, σ2) = (1, 0.2) lies in region D2, the positive constant equilibrium (u∗, v∗) = (0.0513, 0.0513) is unstable, and there are two stable spatially inhomogeneous steady states like cos(2x). (a) The initial value is u(x, 0) = 0.051338728 − 0.01 cos(2x), v(x, 0) = 0.051338728 + 0.01 cos(2x); (b) the initial value is u(x, 0) = 0.051338728 + 0.01 cos(2x), v(x, 0) = 0.051338728 − 0.01 cos(2x).
Figure 8
When (σ1, σ2) = (0.1, 0.1) lies in region D3, the positive constant equilibrium (u∗, v∗) = (0.0513, 0.0513) is unstable, and there are stable spatially inhomogeneous periodic solutions. The initial value is u(x, 0) = 0.051338728 + 0.05 sin(5x), v(x, 0) = 0.051338728. (a) and (b) are transient behaviours for u and v, respectively; (c) and (d) are long-term behaviours for u and v, respectively.
Figure 9
When (σ1, σ2) = ( 0.1, 0.2) lies in region D4, the positive constant equilibrium (u∗, v∗) = (0.0513, 0.0513) is unstable, and there are stable spatially inhomogeneous periodic solutions. The initial value is u(x, 0) = 0.051338728 + 0.01 sin(5x), v(x, 0) = 0.051338728. (a) and (b) are transient behaviours for u and v, respectively.
Figure 10
When (σ1, σ2) = ( 0.1, 0.2) lies in region D4, the positive constant equilibrium (u∗, v∗) = (0.0513, 0.0513) is unstable, and there are stable spatially inhomogeneous periodic solutions. The initial value is u(x, 0) = 0.051338728 + 0.01 sin(5x), v(x, 0) = 0.051338728. (a) and (b) are long-term behaviours for u and v, respectively.
Figure 11
When (σ1, σ2) = ( 0.1, 0) lies in region D5, the positive constant equilibrium (u∗, v∗) = (0.0513, 0.0513) is unstable, and there are heteroclinic solutions connecting the unstable spatially inhomogeneous steady state to stable spatially homogeneous periodic solution. The initial value is u(x, 0) = 0.051338728 0.01 cos(2x), v(x, 0) = 0.051338728 + 0.01 cos(2x). (a) and (b) are transient behaviours for u and v, respectively; (c) and (d) are long-term behaviours for u and v, respectively.
Figure 12
When (σ1, σ2) = ( 0.1, 0.03) lies in region D6, the positive constant equilibrium (u∗, v∗) = (0.0513, 0.0513) is unstable, and there is a stable spatially homogeneous periodic solution. The initial value is u(x, 0) = 0.051338728 + 0.01, v(x, 0) = 0.051338728 − 0.01.
Figure 13
The numerical simulations of system (3) with K = 0.1153 and the initial condition at (0.05, 0.05).
In this section, we perform the numerical simulations of Hopf bifurcation with 
as the parameter. Taking 
, we have 
. Through calculation, we get the unique positive equilibrium point
, then hypothesis (A2), (C1), and (C2) hold. Figure 13 shows the numerical simulations of system (3) with Hopf bifurcation.
In this paper, we have formulated a Leslie–Gower-type predator–prey system with the fear effect and ratio-dependent Holling III functional response to consider the cost of fear on prey demography. The main focus of this study is to investigate the influence of antipredator behaviour due to fear of predators analytically and numerically. We obtain many interesting results. Mathematically, we analyze the existence and stability of the positive equilibria of the model and give conditions for the existence of the Turing instability, the Hopf bifurcation, and the Turing–Hopf bifurcation by selecting 
, 
and 
as the bifurcating parameters, respectively. In addition, we calculate the normal form of system (3) and divide the plane of parameters 
into six regions 
. In each region, we demonstrate that the predator–prey model exhibits complex spatiotemporal dynamics including spatially homogeneous periodic solutions, spatially inhomogeneous periodic solutions, and spatially inhomogeneous steady-state solutions. Ecologically, we show that if the model has a unique positive equilibrium, the fear effect can reduce the density of predator and prey: as the level of fear 
increases, both the predator and prey density gradually decrease. Moreover, depending on the predator and prey’s different states in each region, we can give solutions accordingly.
The authors wish to express their gratitude to the editors and the reviewers for the helpful comments.
Figure 1
Curve of relationship between K and u∗. Keeping the other parameters fixed at the values: m = 0.35, β = 1.28.
Figure 2
(a) The curve of d2 = ε2(d1)d1. (b) The curve of d2 = dn(d1) at n = 2. Keeping the other parameters fixed at the values: m = 0.35, β = 1.28, K = 0.01, l = 2.
Figure 3
(a) The figure of functions d2 = ε1d1, d2 = ε2(d1)d1, and d2 = dn(d1), 0 < d1 n = 1, 2, 3 in d1, d2-plane. (b) The Turing bifurcation line d2 = d2∗(d1) (0 < d1 ≤ dA). Keeping the other parameters fixed at the values: m = 0.35, β = 1.28, K = 0.01, l = 2.
Figure 4
Turing–Hopf bifurcation point (r∗, dn∗ ) in r, d2-plane.
Figure 5
The bifurcation set (a) and the phase portraits (b) for Turing–Hopf bifurcation of system (12).
Figure 6
When (σ1, σ2) = (1, 0) lies in region D1, the positive constant equilibrium (u∗, v∗) = (0.0513, 0.0513) is asymptotically stable. The initial value is u(x, 0) = 0.05, v(x, 0) = 0.05.
Figure 7
When (σ1, σ2) = (1, 0.2) lies in region D2, the positive constant equilibrium (u∗, v∗) = (0.0513, 0.0513) is unstable, and there are two stable spatially inhomogeneous steady states like cos(2x). (a) The initial value is u(x, 0) = 0.051338728 − 0.01 cos(2x), v(x, 0) = 0.051338728 + 0.01 cos(2x); (b) the initial value is u(x, 0) = 0.051338728 + 0.01 cos(2x), v(x, 0) = 0.051338728 − 0.01 cos(2x).
Figure 8
When (σ1, σ2) = (0.1, 0.1) lies in region D3, the positive constant equilibrium (u∗, v∗) = (0.0513, 0.0513) is unstable, and there are stable spatially inhomogeneous periodic solutions. The initial value is u(x, 0) = 0.051338728 + 0.05 sin(5x), v(x, 0) = 0.051338728. (a) and (b) are transient behaviours for u and v, respectively; (c) and (d) are long-term behaviours for u and v, respectively.
Figure 9
When (σ1, σ2) = ( 0.1, 0.2) lies in region D4, the positive constant equilibrium (u∗, v∗) = (0.0513, 0.0513) is unstable, and there are stable spatially inhomogeneous periodic solutions. The initial value is u(x, 0) = 0.051338728 + 0.01 sin(5x), v(x, 0) = 0.051338728. (a) and (b) are transient behaviours for u and v, respectively.
Figure 10
When (σ1, σ2) = ( 0.1, 0.2) lies in region D4, the positive constant equilibrium (u∗, v∗) = (0.0513, 0.0513) is unstable, and there are stable spatially inhomogeneous periodic solutions. The initial value is u(x, 0) = 0.051338728 + 0.01 sin(5x), v(x, 0) = 0.051338728. (a) and (b) are long-term behaviours for u and v, respectively.
Figure 11
When (σ1, σ2) = ( 0.1, 0) lies in region D5, the positive constant equilibrium (u∗, v∗) = (0.0513, 0.0513) is unstable, and there are heteroclinic solutions connecting the unstable spatially inhomogeneous steady state to stable spatially homogeneous periodic solution. The initial value is u(x, 0) = 0.051338728 0.01 cos(2x), v(x, 0) = 0.051338728 + 0.01 cos(2x). (a) and (b) are transient behaviours for u and v, respectively; (c) and (d) are long-term behaviours for u and v, respectively.
Figure 12
When (σ1, σ2) = ( 0.1, 0.03) lies in region D6, the positive constant equilibrium (u∗, v∗) = (0.0513, 0.0513) is unstable, and there is a stable spatially homogeneous periodic solution. The initial value is u(x, 0) = 0.051338728 + 0.01, v(x, 0) = 0.051338728 − 0.01.
Figure 13
The numerical simulations of system (3) with K = 0.1153 and the initial condition at (0.05, 0.05).
