Hopf bifurcation in a delayed reaction diffusion predator-prey model with weak Allee effect on prey and fear effect on predator

1.   Introduction

One of the most important factors in ecology is the predator-prey interaction, which is usually complex and diverse. Scholars have long been committed to using mathematical methods to explain and predict it ^{[1,2,3,4,5,6,7]}. The population dynamics of predator and prey could easily be affected in nature ^[8,9,10]. In this work, we investigate a self-diffusive Leslie-Gower model with a time delay, fear effect on the predator, weak Allee effect, and nonlocal competition on the prey.

Many researchers only pay attention to the direct killing of prey by predators. However, with the development of society and the advancement of biomathematics, researchers ^{[11,12,13,14,15,16]} have a deeper understanding of the interactions between predator and prey, which are complex and multiple in nature. The existence of predators not only directly affects the density and growth rate of prey populations by eating prey, but also indirectly affects prey populations by influencing their dynamical behavior. Some experiments have shown that prey will make various anti-predation responses to protect itself when it faces a predator, such as physiological changes, vigilance, foraging behavior. The phenomenon that the prey is always alert to possible attacks, is called fear effect. It is a physiological change related to the behavior and stress of a prey population in the presence of a predator.

Until now, most literatures have studied the fear effect on prey but few literatures have studied it on predator. Some scientists ^[17,18,19] have conducted experiments to study the fear effect on predators. They have found that the fear effect caused by a large predator could lead to a similar predation effect on medium-sized predators, leading the medium-sized predators to produce the anti-predation responses. This anti-predation behavior produced by the medium-sized predator will directly affect its consumption of prey. We can maintain the balance of the ecosystem by controlling the number of large predators to avoid excessive consumption of prey at the bottom of the food chain by the medium-sized predator.

When the density of the species population is low, the change law of the species population will be greatly affected by the internal mating of the population. W. Allee proposed the famous Allee effect for the first time ^[20]. Additionally, many scholars were concerned about the Allee effect in the predator-prey model ^{[21,22,23,24,25,26,27,28,29]}. Since the Allee effect usually occurs when the population density is either small or sparse. Thus, the Allee effects are strongly related to the extinction vulnerability of populations. The Allee effect is generally divided into either the strong or weak Allee effect. Whether the Allee effect is weak or strong depends on the opposing strengths of positive and negative density dependence. A strong Allee effect involves an Allee threshold. The Allee threshold is either a critical population size or a density below which the per capita population growth rate becomes negative. As mentioned in ^[23], the term $(u-m)$ is added to the logistic growth function $r u \left(1-\frac{u}{K}\right)$ to investigate the influence of Allee threshold on prey. In ^[24], Courchamp revealed that studies of the strong Allee effect help support the relationship between the species populations at low densities and the population growth rate. Authors in ^[29] studied the following model:

Here $u$ and $v$ represent the prey and predator density, respectively. $K$, $m$, $c$, $r$, and $e$ denote the environmental capacity, Allee threshold, maximal per capita consumption rate, and the intrinsic growth rates of the prey and predator, respectively. $n$ is a measure of food quality that the prey provides for the conversion into predator birth. $k$ measures the fear effect on the predator.

Unlike the strong Allee effect, there is no threshold for the weak Allee effect. Many literatures ^[30,31,32] have investigated the ecological model with the weak Allee effect. Authors in ^[21] have mentioned the following model:

where $\beta$ is the weak Allee constant. Let $(u, \; v, \; t) = (K\bar{u}, \; Kn\bar{v}, \; \frac{\bar{t}}{r})$ and ignore the bar. Then, Eq (1.2) becomes

where $b = \frac{\beta}{K}$, $q = \frac{Knc}{r}$, $p = Knk$, $s = \frac{e}{r}$.

In nature, due to the particularity of species breeding conditions and the necessity of gestational length, the species population density and birth rate of the species population at this stage are affected by the past period. Additionally, the energy conversion between predator and prey is not instantaneous. The influence of past history on the per capita growth rate of predators cannot be ignored ^{[33,34,35,36]}; therefore, we consider a time delay parameter $\tau$ for the predator-prey model. In addition, we assume that the distribution of population is uniform in model (1.3), which is generally not the situation in nature. In nature, due to the widespread self-diffusion phenomenon, few species populations have homogeneous spatial distribution ^[37,38,39]. Since the existence of diffusion phenomenon, the population model often shows some more abundant dynamic phenomena.

Another very important point is that the limited resources in nature makes the unlimited growth of species impossible, which will inevitably lead to competition among the prey population. Since the spatial distributions for the predator and prey population are inhomogeneous and disperse, this competition is usually nonlocal. Many scholars have studied the influence of it on the dynamic behavior of species ^{[40,41,42,43,44,45,46]}. In ^[47,48], the authors modified the $\frac{u}{K}$ as $\frac{1}{K} \int_{\Omega}G(x, y)u(y, t)dy$ with some kernel function $G(x, y)$ to describe this competition. Due to the above factors, we added time delay and self-diffusion terms into Eq (1.3), and considered the nonlocal competition:

Here $\frac{\partial u(x, t)}{\partial t}$ and $\frac{\partial v(x, t)}{\partial t}$ represent the density gradients of the prey populations and predator populations, respectively. $d_1, \; d_2 > 0$ denote the diffusion coefficients of prey and predator, respectively. The notation $\Delta$ denotes the Laplace operator, and the notation $\Omega$ denotes a bounded domain with a smooth boundary $\partial\Omega$. $\tau$ describes either a gestation period or reaction time. The integral term $\int_{\Omega}G(x, y)u(y, t)dy$ in the first equation of (1.4) accounts for the nonlocal competition among the prey individuals. The kernel function is of the following form:

which can be regarded as a measurement of the competition pressure at location $x$ from the individuals at another location $y$. In this case, the competition strength among all prey individuals is the same across the habitat.

This paper mainly studies the significant effects of the weak Allee effect on prey and the fear effect on predators in the predator-prey system. We shall separately study the influence of the weak Allee effect and the fear effect on the spatial bifurcating periodic solutions.

This paper is organized as follows. In Section 2, we investigate the existence and stability of a coexisting equilibrium. In Section 3, we analyze the existence of a Hopf bifurcation. In Section 4, we consider the property of the Hopf bifurcation. In Section 5, we conduct a series of numerical simulations to illustrate the theoretical results. In Section 6, we elaborate a short conclusion.

2.   Equilibria and stability

2.1.   The existence of the equilibria

A discussion of the equilibria has been given in ^[21], but for the completeness of the paper, we still give the following lemma.

Lemma 2.1. The equilibria of system (1.4) admit the following statements:

(ⅰ) The system (1.4) always has a distinct boundary equilibria given by $\mathbb{E}_0(1, 0)$ for all positive parameters.

(ⅱ) When $bq > 1$ and

(ⅱa) if $p-q-1 > 0$ and $\; 0 < b < b^*$, then system (1.4) has two positive equilibria $\mathbb{E}_{1}(u_-, v_-)$ and $\mathbb{E}_{2}(u_+, v_+)$,

(ⅱb) if $p-q-1 > 0$ and $b = b^*$, then system (1.4) has a unique positive equilibrium $\mathbb{E}_3(u_3, v_3)$,

(ⅱc) if $p-q-1 > 0$, $b > b^*$ or $p-q-1 \leq 0$, then system (1.4) has no positive equilibrium.

(ⅲ) When $bq = 1$ and

(ⅲa) if $p-q-1 > 0$, then system (1.4) has a unique positive equilibrium $\mathbb{E}_{2}(u_+, v_+)$,

(ⅲc) if $p-q-1 \leq 0$, then system (1.4) has no positive equilibrium.

(ⅳ) When $0 < bq < 1$, then system (1.4) has a unique positive equilibrium $\mathbb{E}_{2}(u_+, v_+)$.

In the above, $b^* = \frac{(1+p+q)^2-4pq}{4pq}$, $u_{\pm} = v_{\pm} = \frac{(p-1-q)\pm \sqrt{(1-p+q)^2-4p(bq-1)}}{2p}$, and $u_3 = v_3 = \frac{(p-1-q)}{2p}$. Considering the biological significance of the system equilibrium, the rest of our discussion is focused on positive equilibrium.

Furthermore, we focus on the dynamics of system (1.3) in a neighbordhood of each equilibrium. Figure 1 shows the phase portraits of system (1.3) $b = 0.3$, $q = 0.9$, $p = 0.8$, and $s = 0.29$. The "green dot" represents the boundary point $\mathbb{E}_0$, the "blue dot" represents the equilibrium $\mathbb{E}_1$, and the "red dot" represents the unique positive equilibrium $\mathbb{E}_2$. By calculating, we obtain the real parts of eigenvalues of equilibrium $\mathbb{E}_2$ of $-0.195771$ and $-0.195771$. Therefore, the equilibrium $\mathbb{E}_2$ is a stable node. We selected the equilibrium point $\mathbb{E}_2$ for numerical simulations under the confirmed biological significance.

2.2.   The stability analysis of the equilibria

Assume that $\Omega = (0, l\pi)$ and $G(x, y) = \frac{1}{l\pi}$. Let $\mathbb{N}$ denote the positive integer set, and $\mathbb{N}_0$ denote the nonnegative integer set. Without loss of generality, let us say that the positive equilibrium point is $E_*(u_*, v_*)$. Then, linearize system (1.4) at $E_*(u_*, v_*)$.

where

and $\hat{u} = \frac{1}{l\pi}\int_{0}^{l\pi}u(y, t)dy$.

Naturally, the characteristic equation is as follows:

where

Then, we make the following hypothesis:

Furthermore, we come to the following situations.

Theorem 2.2. Assume $\tau = 0$. Then, the following statements are true for system (2.1).

(ⅰ) If ${\bf (H_1)}$ holds, then $E_*(u_*, v_*)$ is locally asymptotically stable.

(ⅱ) If ${\bf (H_2)}$ holds, then $E_*(u_*, v_*)$ is Turing instable.

Proof. Assume $\tau = 0$, and then (2.2) becomes to

and

When ${\bf (H_1)}$ holds, the roots of Eqs (2.4) and (2.5) are all with negative real parts. Therefore, the equilibrium $E_*(u_*, v_*)$ is locally asymptotically stable. When ${\bf (H_2)}$ holds, the roots of Eq (2.4) are all with negative real parts, but Eq (2.5) has at least one root with positive real part. Therefore, $E_*(u_*, v_*)$ is Turing unstable. □

Lemma 2.3. If ${\bf (H_1)}$ holds, then Eq (2.5) has a pair of purely imaginary roots $\pm \mathit{\mbox{i}}\omega_n$ at $\tau^j_n, \; j \in \mathbb{N}_0, \; n\in\mathbb{F}$, where

and

Proof. We suppose $\mbox{i}\omega$ ($\omega > 0$) is a solution of Eq (2.2), which leads to

Then, separating the real and imaginary parts, we have

Thus, we can obtain

and

Due to ${\text{cos}}^2\omega\tau+{\text{sin}}^2\omega\tau = 1$, we have

Let $m = \omega^2$, then Eq (2.8) becomes

Let $P_n = A_n^2-2 B_n-b_{22}^2$ and $Q_n = B_n^2-C_n^2$. The roots of Eq (2.9) are $m^{\pm} = \frac{1}{2}[-P_n \pm \sqrt{P_n^2-4 Q_n}]$. If ${\bf (H_1)}$ holds, then $B_n+C_n > 0\; (n \in \mathbb{N}_0)$.

Define

Then, we have

Based on the above analysis, we will discuss the existence of purely imaginary roots of Eq (2.5) in the following three cases.

Case 1: $a_{12} > a^*$. For $n\in \mathbb{M}_2$, we can obtain that $m^\pm > 0$ if $P_n^2-4 Q_n > 0$ and $m^+ = m^- > 0$ if $P_n^2-4 Q_n = 0$. Then, Eq (2.5) has either one or two pairs of purely imaginary roots $\pm \mbox{i}\omega_n$ at $\tau^j_n, \; j \in \mathbb{N}_0$, where $\mbox{i}\omega_n^\pm = \sqrt{m^\pm}$.

Case 2: $a_{12} = a^*$. For $n\in \mathbb{M}_2 \backslash \mathbb{M}_1$, we can obtain that $m^\pm > 0$ if $P_n^2-4 Q_n > 0$ and $m^+ = m^- > 0$ if $P_n^2-4 Q_n = 0$ under the condition $\mu_n \neq h^+$ and $\mu_n \neq h^-$. Then, Eq (2.5) has either one or two pairs of purely imaginary roots $\pm \mbox{i}\omega_n$ at $\tau^j_n, \; j \in \mathbb{N}_0$. For $\mu_n = h^+$ or $\mu_n = h^-$, the Eq (2.5) has a pair of purely imaginary roots $\pm \mbox{i}\omega_n$ at $\tau^j_n, \; j \in \mathbb{N}_0$ when $P_n < 0$ and $P_n^2-4 Q_n \geq 0$.

Case 3: $a_{12} < a^*$. For $n\in \mathbb{M}_2 \backslash \mathbb{M}_1$, we can obtain that $m^\pm > 0$ if $P_n^2-4 Q_n > 0$ and $m^+ = m^- > 0$ if $P_n^2-4 Q_n = 0$ under the condition $\mu_n \neq h^+$ and $\mu_n \neq h^-$. Then, Eq (2.5) has either one or two pairs of purely imaginary roots $\pm \mbox{i}\omega_n$ at $\tau^j_n, \; j \in \mathbb{N}_0$. For $\mu_n = h^+$ or $\mu_n = h^-$, Eq (2.5) has a pair of purely imaginary roots $\pm \mbox{i}\omega_n$ at $\tau^j_n, \; j \in \mathbb{N}_0$ when $P_n < 0$ and $P_n^2-4 Q_n \geq 0$. For $n\in \mathbb{M}_1$, Eq (2.5) has a pair of purely imaginary roots $\pm \mbox{i}\omega_n$ at $\tau^j_n, \; j \in \mathbb{N}_0$ when $P_n^2-4 Q_n \geq 0$.

Define

$\mathbb{F}$ is a finite set obviously, since

□

Lemma 2.4. If ${\bf (H_1)}$ is satisfied, then $\mathit{\mbox{Re}}[\frac{d \lambda} {d \tau}|_{\tau = \tau^j_n}] > 0$ for $n\in\mathbb{F}$, $j \in \mathbb{N}_0$ are true.

Proof. By Eq (2.2), we have

Then,

□

Then, we have the following theorem.

Theorem 2.5. Assume ${\bf (H_1)}$ are satisfied, the following statements are true for system (1.4).

(ⅰ) $E_*(u_*, v_*)$ is locally asymptotically stable for $\tau \in [0, \tau_*)$, where $\tau_* = min\{\tau^0_n|\; n\in\mathbb{F}\}$.

(ⅱ) $E_*(u_*, v_*)$ is unstable for $\tau \in [\tau_*, +\infty)$.

(ⅲ) The Hopf bifurcation values of system (2.1) are $\tau = \tau^j\; \mathit{\mbox{or}}\; \tau^j_n$ $(n\in \mathbb{F}, \; j \in \mathbb{N}_0)$.

3.   Property of Hopf bifurcation

In this section, we will give some conditions regarding the property of the Hopf bifurcation through the methods "the normal form theory" and "the center manifold theorem" in ^[40,41].

Denote $\tilde{\tau} = \tau^j_n$ for $j\in \mathbb{N}_0$ and $n\in \mathbb{F}$. Let $\bar{u}(x, t) = u(x, \tau t)-u_*$ and $\bar{v}(x, t) = v(x, \tau t)-v_*$. Then, system (1.4) (ignore the bar) becomes

System (3.1) can be rewritten in the following form:

where

Define a Hilbert space

The corresponding complexification $X_{\mathbb{C}}$ has the form $X_{\mathbb{C}}: = X \oplus \mbox{i}X = \left\{a+\mbox{i}b|\; a, \; b\in X\right\}$. The complex-valued $L^2$ inner product is provided by $\langle a, b\rangle: = \int_{0}^{l\pi} (\overline{a_1} b_1+\overline{a_2} b_2) dx$, for $a = (a_1, \; a_2)^{T}$, $b = (b_1, \; b_2)^{T} \in X_{\mathbb{C}}$. Define a notation $\mathscr{C}: = C([-1, 0], X_{\mathbb{C}})$, which means the phase space with the sup norm, and we could write $\phi_t \in \mathscr{C}$, $\phi_t(\rho) = \phi(t+\rho)$ for $\rho \in[-1, 0]$. Let $\chi_n^{(1)}(a) = (\gamma_n(a), 0)^T$, $\chi_n^{(2)}(a) = (0, \gamma_n(a))^T$ and $\chi_n = \{\chi_n^{(1)}(a), \chi_n^{(2)}(a)\}$, where $\{\chi_n^{(i)}(a) \}\; (i = 1, 2)$ is an orthonormal basis of $X$. Define the subspace of $\mathscr{C}$, which is, $\mathscr{B}_n: = \mbox{span}\{ \langle\phi(\cdot), \chi_n^{(j)} \rangle \chi_n^{(j)}| \phi \in \mathscr{C}, j = 1, 2 \}$, $n\in\mathbb{N}_0$. According to the Riesz representation theorem, there exists a $2\times2$ matrix function $\eta_n(\theta, \tilde{\tau})$ of the bounded variation for $-1\le \theta \le0$, such that $-\tilde{\tau} D\frac{n^2}{l^2} \phi(0)+\tilde{\tau}L(\phi) = \int_{-1}^{0}d \eta_n(\theta, \tilde{\tau}) \phi(\theta)$ for $\phi \in \mathscr{C}$. Define the bilinear form on $\mathscr{C}^* \times \mathscr{C}$, that is,

Define $\tau = \tilde{\tau}+\mu$. When $\mu = 0$, system Eq (3.2) undergoes a Hopf bifurcation at equilibrium $(0, 0)$, and the eigenfunctions has a pair of purely imaginary roots $\pm \mbox{i}\omega_{n_0}$. $\mathscr{A}$ represents the infinitesimal generators of the semigroup with $\mu = 0$ and $n = n_0$. The formal adjoint of $\mathscr{A}$ is denoted by $\mathscr{A}^*$, which is under the bilinear pairing Eq (3.3). Then, define the following Boolean function:

Choose $\eta_{n_0}(0, \tilde{\tau}) = \tilde{\tau}(-n_0^2/l^2)D+\tilde{\tau}J_1+\tilde{\tau}J_3 \delta(n_{n_0})$, $\eta_{n_0}(-1, \tilde{\tau}) = -\tilde{\tau} J_2$, $\eta_{n_0}(\theta, \tilde{\tau}) = 0$ for $\theta \in (-1, 0)$. Let $p(\sigma) = p(0)e^{\mbox{i}\omega_{n_0} \tilde{\tau} \sigma}\; (\sigma \in [-1, 0])$ and $q(\theta) = q(0)e^{-\mbox{i}\omega_{n_0} \tilde{\tau} \theta}\; (\theta \in [0, 1])$ be the eigenfunctions of $\mathscr{A}$ and $\mathscr{A}^*$ corresponding to the eigenvalue $i \omega_{n_0} \tilde{\tau}$, respectively. By calculation, we choose $p(0) = (1, p_1)^T$ and $q(0) = W(1, q_2)$, where $p_1 = \frac{1}{a_{12}}\left(-a_{11}+d_1\frac{n_0^2}{l^2}-\hat{a}\delta(n_0)+\mbox{i}\omega_{n_0}\right)$, $q_2 = -\frac{a_{12}}{a_{22}+b_{22}e^{\mbox{i} \tau \omega_{n_0}}-d_2\frac{n_0^2}{l^2}-\mbox{i}\omega_{n_0}}$, and $W = (1+p_1 q_2+\tilde{\tau}(b_{21}q_2+b_{22}p_1q_2)e^{-\mbox{i}\omega_{n_0} \tilde{\tau}})^{-1}$. Thus, system (3.1) becomes

where

for $\phi = (\phi_1, \phi_2)^T \in \mathscr{C}$ and $\hat{\phi}_1 = \frac{1}{l\pi}\int_{0}^{l\pi}\phi_1 dx$. Then, we decompose the space $\mathscr{C}$ as $\mathscr{C} = \mathscr{P} \oplus \mathscr{Q}$, where $\mathscr{P} = \{a p \gamma_{n_0}(x) +\bar{a} \bar{p} \gamma_{n_0}(x)|a \in \mathbb{C}\}$, $\mathscr{Q} = \{\psi \in \mathscr{C} | (q\gamma_{n_0}(x), \psi) = 0\; \mbox{and}\; (\bar{q}\gamma_{n_0}(x), \psi) = 0 \}$. Thus, system (3.6) becomes $U_t = f(t)p(\cdot)\gamma_{n_0}(x) +\bar{f}(t)\bar{p}(\cdot)\gamma_{n_0}(x)+ \omega(t, \cdot)$ and $\hat{U_t} = \frac{1}{l\pi}\int_{0}^{l\pi} U_t dx$, where

Then, we get $\dot{f}(t) = \mbox{i}\omega_{n_0} \tilde{\tau} f(t)+\bar{q}(0) < G(0, U_t), \chi_{n_0} >$. There exists a center manifold $\mathscr{C}_0$ and we could write $\omega$ near $(0, 0)$ as follows:

Then, $\dot{f}(t) = \mbox{i}\omega_{n_0} \tilde{\tau} f(t)+\varpi(f, \bar{f})$ is the system restricted to the center manifold $\mathcal{C}_0$. Denote $\varpi(f, \bar{f}) = \varpi_{20}\frac{f^2}{2}+\varpi_{11}f\bar{f} +\varpi_{02}\frac{\bar{f}^2}{2}+\varpi_{21}\frac{f^2\bar{f}}{2}+\cdots$.

By direct computation, we have

where

Then, we should compute $\omega_{20}$ and $\omega_{11}$. Due to Eq (3.7), we have

where

Comparing the coefficients of Eq (3.8) with Eq (3.9), we will get

Then, we have

Denote $Q_{01} = e^{\mbox{i}\omega_{n_0} \tilde{\tau} \sigma} \gamma_{n_0}(x)$, $Q_{02} = e^{-\mbox{i}\omega_{n_0} \tilde{\tau} \sigma}\gamma_{n_0}(x)$, $Q_1 = \sum_{n = 0}^{\infty}Q_1^{(n)}\gamma_{n_0}(x)$ and $Q_2 = \sum_{n = 0}^{\infty}Q_2^{(n)}\gamma_{n_0}(x)$, and $Q_1$ and $Q_2$ are described as follows:

where

Therefore, we have

Theorem 3.1. For any critical value $\tau^j$ or $\tau^j_n$ $(n\in \mathbb{F}, \; j \in \mathbb{N}_0)$, the following statements are true for system (1.4).

(ⅰ) If $\mu_2 > 0\; (resp. < 0)$, the Hopf bifurcation is forward $($resp. backward$)$.

(ⅱ) If $\beta_2 < 0$ $($resp. $> 0$$)$, the bifurcation periodic solutions on the center manifold $\mathcal{C}_0$ are orbitally asymptotically stable $($resp. unstable$)$.

(ⅲ) If $T_2 > 0$ $($resp. $T_2 < 0$$)$, the Hopf bifurcation period increases $($resp. decreases$)$.

4.   Numerical simulations

4.1.   The influence of the weak Allee effect

We analyze the effect of the parameter $b$, which is related to the weak Allee effect.

Fix parameters $q = 0.9$, $p = 0.8$, $s = 0.29$, $d_1 = 0.18$, and $d_2 = 0.13$. The bifurcation diagram of system (1.4) is given in Figure 2. From this diagram, we can obtain the relationship between the curves $\tau_0$ and $\tau_1$, and we can also obtain the intersecting point $b^*$ $(b^*\approx 0.1727)$.

Referring to the results of Figure 2, we can also observe that the spatial homogeneous periodic solution appears when $b > b^*$ and $\tau > \tau_0$, which may be asymptotically stable. The spatial inhomogeneous periodic solution appears when $b \in (0, b^*)$ and $\tau > \tau_1$, which may also be asymptotically stable. The stable steady state $(u_*, \; v_*)$ will be reached at the rest.

Then, we select different parameter values to calculate and obtain some detailed values for properties of the Hopf bifurication (see Table 1). We found a phenomenon that a system with different parameters has different behaviors. Thus, we take six different sets of numbers and compare their different behaviors, which are summarized in Table 2. There will be three dynamic behaviors, namely asymptotically stable coexistence equilibrium (ASCE), stable spatially homogeneous periodic solutions (SSHPS), and stably spatially inhomogeneous periodic solutions (SSIPS).

Choose $b = 0.1$ and $\tau = 3$ $(\tau < \tau_1 < \tau_0)$. The coexistence equilibrium $(u_*, \; v_*)$ in system (1.4) is asymptotically stable (see the Figure 3).

Choose $b = 0.1$ and $\tau = 4.8$ $(\tau_1 < \tau < \tau_0)$. As we can see from Figure 4, the coexistence equilibrium $(u_*, \; v_*)$ is unstable, the spatial homogeneous periodic solution does not exist, and the stable spatial inhomogeneous periodic solution appears first. Therefore, we classify this equilibrium as a stable spatial inhomogeneous periodic solution.

Choose $b = 0.1$ and $\tau = 5.2$ $(\tau_1 < \tau_0 < \tau)$. As we can see from Figure 5, the coexistence equilibrium $(u_*, \; v_*)$ is unstable, and the spatial homogeneous periodic solution appears first, though is not stable. Thus, system (1.4) has a stable spatial inhomogeneous periodic solution.

Choose $b = 0.3$ and $\tau = 3$ $(\tau < \tau_0 < \tau_1)$. As we can see from Figure 6, the coexistence equilibrium $(u_*, \; v_*)$ is asymptotically stable. Comparing detail diagrams between Figures 3(b) and 6(b), we can find that the coexistence equilibrium $(u^*, v^*)$ with a lager weak Allee effect becomes stable faster and has a smaller amplitude. Thus, we can conclude that the existence of the weak Allee effect term is beneficial to the stability of the coexistence equilibrium.

Choose $b = 0.3$ and $\tau = 4.2$ $(\tau_0 < \tau < \tau_1)$. As we can see from Figure 7, the coexistence equilibrium $(u_*, \; v_*)$ is unstable, the spatial inhomogeneous periodic solution does not exist, and the stable spatial homogeneous periodic solution appears first. Therefore, we classify this equilibrium as a stable spatial homogeneous periodic solution.

Choose $b = 0.3$ and $\tau = 5.2$ $(\tau_0 < \tau_1 < \tau)$. As we can see from Figure 8, the coexistence equilibrium $(u_*, \; v_*)$ is unstable, and the spatial inhomogeneous periodic solution appears first, though is not stable. The system (1.4) has stable spatial homogeneous periodic solution. Comparing detail diagrams between Figures 5(b) and 8(b), we can find that the weak Allee effect term has a slight effect on the solution but does not affect its stability. The period of the coexistence equilibrium $(u^*, v^*)$ with a lager weak Allee effect has increased and has a smaller amplitude. It can be seen that the weak Allee effect can affect the homogeneity of the periodic solution.

4.2.   The influence of fear effect

We analyze the effect of parameter $p$, which is related to the fear effect.

Fix parameters $q = 0.45$, $b = 0.1$, $s = 0.25$, $d_1 = 0.13$, and $d_2 = 0.2$. The bifurcation diagram of system (1.4) is given in Figure 9. In the diagram, the curves of $\tau_0$ and $\tau_1$ intersect at the points $p^*$, where $p^*\approx 0.3059$.

Referring to the results of Figure 9, we can also observe that the spatial homogeneous periodic solution appears when $p > p^*$ and $\tau > \tau_0$, which may be asymptotically stable. The spatial inhomogeneous periodic solution appears when $p \in (0, p^*)$ and $\tau > \tau_1$, which may also be asymptotically stable. The stable steady state $(u_*, \; v_*)$ will be reached at the rest.

Then, we select different parameter values to calculate and obtain some detailed values for the Hopf bifurcation properties (see Table 3). We took six different sets of numbers and compared their behavior, which are summarized into Table 4.

Choose $p = 0.05$ and $\tau = 3.9$ $(\tau < \tau_1 < \tau_0)$. As we can see from Figure 10, the coexistence equilibrium $(u_*, \; v_*)$ is asymptotically stable.

Choose $p = 0.05$ and $\tau = 4.15$ $(\tau_1 < \tau < \tau_0)$. As we can see from Figure 11, the coexistence equilibrium $(u_*, \; v_*)$ is unstable, the spatial homogeneous periodic solution does not exist, and the stable spatial inhomogeneous periodic solution appears first. Therefore, we classify this equilibrium as a stable spatial inhomogeneous periodic solution.

Choose $p = 0.05$ and $\tau = 4.2$ $(\tau_1 < \tau_0 < \tau)$. As we can see from Figure 12, the coexistence equilibrium $(u_*, \; v_*)$ is unstable, and the spatial homogeneous periodic solution appears first, though is not stable. Thus, system (1.4) has stable spatial inhomogeneous periodic solution.

Choose $p = 0.56$ and $\tau = 6.5$ $(\tau < \tau_0 < \tau_1)$. As we can see from Figure 13, the coexistence equilibrium $(u_*, \; v_*)$ is asymptotically stable. Comparing detail diagrams between Figures 10(b) and 13(b), we can find that the coexistence equilibrium $(u^*, v^*)$ with a lager fear effect becomes stable slower and has a larger amplitude. Thus, we can conclude that the existence of the fear effect term is not beneficial to the stability of the coexistence equilibrium.

Choose $p = 0.56$ and $\tau = 4.15$ $(\tau_0 < \tau < \tau_1)$. As can be seen in Figure 14, the coexistence equilibrium $(u_*, \; v_*)$ is unstable, the spatial inhomogeneous periodic solution does not exist, and the stable spatial homogeneous periodic solution appears first. Therefore, we classify this equilibrium as a stable spatial homogeneous periodic solution.

Choose $p = 0.56$ and $\tau = 4.2$ $(\tau_0 < \tau_1 < \tau)$. As can be seen in Figure 15, the coexistence equilibrium $(u_*, \; v_*)$ is unstable, and the spatial inhomogeneous periodic solution appears first, though is not stable. The system (1.4) has a stable spatial homogeneous periodic solution. We can find that the fear effect term has a slight effect on the solution and does not affect its stability. Additionally, it can affect the homogeneity of the periodic solution.

5.   Conclusions

In this work, we studied the Hopf bifurcation of a delayed diffusive predator-prey model with a weak Allee effect on prey and a fear effect on predator. By the qualitative analytical theory, we have obtained the conditions of local stability of a coexisting equilibrium and the existence of a Hopf bifurcation and Turing instable. By using the methods of the normal form theory and center manifold theorem, we have studied the property of the bifurcating periodic solutions. We found that the weak Allee effect and fear effect greatly affected the dynamical behaviour of the new Leslie-Gower model.

First, we discuss the influence of the weak Allee effect. On the whole, the weak Allee effect at a larger value has a great influence on the ecological extinction, controlling stable coexisting equilibrium, and periodic oscillation. Specifically, under the premise that the fear effect remains unchanged, when the weak Allee effect is small, a small increase is not beneficial to the stability of the coexisting equilibrium and will be easy to produce an inhomogeneous periodic solution, which is not conducive to the survival of the population. Thus, when the weak Allee effect increases, either improved defending or hiding of the prey species form the predator may be difficult, and the prey population is generally at a low density. After increasing to a certain extent, the effect will be the completely opposite, and the homogeneity of periodic solutions usually changes from inhomogeneous to homogeneous. The homogeneity is either an invariance or regularity under a particular transformation. Ecologically, the periodic solutions are homogeneous means that the members or parts of prey specie have the same dynamic behavior at all times. Thus, increasing the weak Allee effect to a certain extent undermines this invariance or regularity of the prey species, which is beneficial to the survival of the population. When the weak Allee effect is large, increasing the weak Allee effect will be beneficial to the stability of the coexisting equilibrium and will easily produce a homogeneous periodic solution, which will make the prey population increase their likelihood of extinction.

Finally, we will discuss the influence of the fear effect on population dynamics. Specifically speaking, under the premise that the weak Allee effect remains unchanged, an increase of the fear effect is not beneficial to the stability of the coexisting equilibrium. The homogeneity of periodic solutions usually changes from inhomogeneous to homogeneous, which is conducive to the survival of the population, since the fear effect on predators can protect prey and predators from being eliminated.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research is supported by the Fundamental Research Funds for the Central Universities (Grant No. 2572022DJ05), Harbin Science and Technology Bureau Manufacturing Innovation Talent Project (CXRC20221110393), Heilongjiang Science and Technology Department Provincial Key R & D Program Applied Research Project (SC2022ZX06C0025), Heilongjiang Science and Technology Department Provincial Key R & D Program Guidance Project (GZ20220088) and Postdoctoral program of Heilongjiang Province (No. LBHQ21060).

Conflict of interest

The authors have no conflicts of interest to declare. All co-authors have seen and agree with the contents of the manuscript and there is no financial interest to report.