Stability Analysis and Hopf Bifurcation of a Delayed Diffusive Predator–Prey Model with a Strong Allee Effect on the Prey and the Effect of Fear on the Predator

Abstract

In this paper, we propose a diffusive predator–prey model with a strong Allee effect and nonlocal competition in the prey and a fear effect and gestation delay in the predator. We mainly study the local stability of the coexisting equilibrium and the existence and properties of Hopf bifurcation. We provide bifurcation diagrams with the fear effect parameter (s) and the Allee effect parameter (a), showing that the stable region of the coexisting equilibrium increases (or decreases) with an increase in the fear effect parameter (s) (or the Allee effect parameter (a)). We also show that gestation delay ($\tau$) can affect the local stability of the coexisting equilibrium. When the delay ($\tau$) is greater than the critical value, the coexistence equilibrium loses its stability, and bifurcating periodic solutions appear. Whether the bifurcated periodic solution is spatially homogeneous or inhomogeneous depends on the fear effect parameter (s) and the Allee effect parameter (a). These results show that the fear effect parameter (s), the Allee effect parameter (a), and gestation delay ($\tau$) can be used to control the growth of prey and predator populations.

1. Introduction

Scholars have long been committed to using mathematical methods to explain and predict biological phenomena [1,2,3,4]. The analysis of predator–prey models is a research subject that has recently attracted considerable attention [5,6,7,8] from mathematicians and biologists. In order to better describe the law of changes in a population, many scholars have used differential equations to build predator–prey models and have introduced different parameters in order to consider biological factors. Considering that the internal mating of a population affects the law of change in that population when the population density is low, W. Allee proposed the famous Allee effect [9]. If the population density is too sparse, then mating between populations becomes difficult, and Allee effects may occur when the population density is under a specific threshold. Thus, Allee effects are strongly related to the vulnerability of populations to extinction [10,11,12]. For example, if pressure from the harvesting of bluefin tuna (Thunnus thynnus) is too strong, the population will collapse [11]. At a very small population size, the probability of finding an acceptable mate for some endangered species, such as lakapo (Strigos habroptilus), is very low [11].

The earliest single-population model exhibiting the Allee effect is as follows [13]:

$$\frac{du\left(t\right)}{dt}=r_{1}u\left(t\right)\left(1-\frac{u\left(t\right)}{K}\right)(u\left(t\right)-a_0),$$

where $u\left(t\right)$ represents the density of prey at time t, and $r_1$ and K are the prey’s intrinsic growth rate and carrying capacity, respectively. The parameter $a_0$ denotes the Allee threshold, and the term $(u\left(t\right)-a_0)$ denotes the Allee effect. It must be noted that the Allee threshold is the critical population size or density below which the per capita population growth rate becomes negative. A strong Allee effect is an Allee effect at the Allee threshold. Whether the Allee effect is weak or strong depends on the opposing strengths of positive and negative density dependence. After the introduction of this model, many researchers began to pay attention to predator–prey models with strong Allee effects.

In nature, the influence of predators on prey species is not mediated only by simple predatory behavior. Since the prey has memory, the presence of predators has an inevitable impact on the behavior and psychology of the prey. For example, when a predator appears, the prey will be vigilant and will stop eating and breeding. This indirect effect on prey populations is known as the fear effect, and it is found widely in nature. Many researchers have focused on predator–prey models with the fear effect [14,15,16]. However, these models describe the prey as having a fear effect in connection with the predator, which affects the growth law of the prey. In nature, predators also have fear effects. For example, scientists have used the barking of dogs on a tape to simulate a scene of fear in raccoons. In this way, raccoons reduce their frequency and time of foraging; this protects the raccoons’ prey to maintain a balance in the ecosystem. In [17], T. Liu et al. proposed a predator–prey model with a fear effect on the predator and a strong Allee effect on the prey:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\frac{du\left(t\right)}{dt}=r_{1}u\left(t\right)\left(1-\frac{u\left(t\right)}{K}\right)(u\left(t\right)-a_0)-\frac{\lambda u\left(t\right)v\left(t\right)}{1+kv\left(t\right)},\hfill \\ \frac{dv\left(t\right)}{dt}=\frac{r_{2}v\left(t\right)}{1+kv\left(t\right)}\left(1-\frac{v\left(t\right)}{qu\left(t\right)}\right).\hfill \end{array}\right.\end{array}$$

(1)

where $u\left(t\right)$ and $v\left(t\right)$ represent the densities of the prey and predator, respectively; and $r_1$, K, $a_0$, $\lambda$, k, $r_2$, and q are the prey’s intrinsic growth rate, the carrying capacity, the strong Allee effect, the capture rate, a measure of the fear effect, the predator’s intrinsic growth rate, and a measure of food quality for the predator, respectively. More explanations of the parameters can be found in [17]. By setting $\tilde{u}=\frac{u}{K}$, $\tilde{v}=\frac{v}{Kq}$, $\tilde{t}=\frac{t}{K{r}_1}$, $a=\frac{a_0}{K}$, $c=\frac{\lambda qK}{r_1}$, $s=kqK$, and $r=\frac{r_2}{r_{1}K}$ and dropping “$\tilde{\phantom{u}}$”, model (1) is changed into

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\frac{du\left(t\right)}{dt}=u\left(t\right)\left((1-u\left(t\right))(u\left(t\right)-a)-\frac{cv\left(t\right)}{1+sv\left(t\right)}\right),\hfill \\ \frac{dv\left(t\right)}{dt}=\frac{rv\left(t\right)}{1+sv\left(t\right)}\left(1-\frac{v\left(t\right)}{u\left(t\right)}\right).\hfill \end{array}\right.\end{array}$$

(2)

The authors mainly studied model (2) from the perspective of bifurcation, such as Hopf bifurcation and Bogdanov–Takens bifurcation [17]. Research has shown that increasing the fear effect on the predator is conducive to protecting prey populations.

We assume that the concentration distribution of species is uniform in model (2), but this is not always the actual situation in nature. In real nature, due to widespread self-diffusion phenomena, few populations of species have a homogeneous spatial distribution [18,19,20]. This is precisely because of the existence of diffusion phenomena; population models often show some more abundant dynamic phenomena, such as spatially inhomogeneous periodic solutions, spatial patterns, etc. In addition, time delays also exist [21,22,23], such as time delays in maturity, gestation, and predation. Time delays often affect the stability of the constant steady-state solution, and they cause periodic oscillations in the population density. Therefore, we introduce self-diffusion and time delay into model (2) as follows.

$$\left\{\begin{array}{c}\frac{\partial u(x,t)}{\partial t}=d_1\Delta u(x,t)+u(x,t)\left((1-u(x,t))(u(x,t)-a)-\frac{cv(x,t)}{1+sv(x,t)}\right),\hfill \\ \frac{\partial v(x,t)}{\partial t}=d_2\Delta v(x,t)+\frac{rv(x,t)}{1+sv(x,t)}\left(1-\frac{v(x,t-\tau )}{u(x,t-\tau )}\right).\hfill \end{array}\right.$$

(3)

where $d_1,d_2>0$ represent the self-diffusion coefficients of the prey and predator, respectively, and $\tau$ is the gestation delay in the predator. The growth law of the predator (the second equation in (3)) can be considered a logistic growth law, where $\left(u\right(t-\tau )$ is the carrying capacity of the environment. An increase in density of predators at time t already exists for predators at time $t-\tau$, where $\tau$ is the gestation time of predators. Therefore, the negative feedback of the density of the predator at time t is related to the relationship between the predator and the prey at time $t-\tau$.

In nature, animals in the same area usually compete for a common but limited resource; due to the depletion of resources, intraspecies competition effects should depend on the average population density in the neighborhood of the current location. In [24,25], the author suggested that internal competition within the population is often spatially inhomogeneous and measured this effect by weighting and integration, modifying the $\frac{u}{K}$ as $\frac{1}{K}{\int }_{\mathrm{\Omega }}G(x,y)u(y,t)dy$. $G(x,y)$ is a kernel function. In [26], Geng et al. studied Hopf, Turing, double-Hopf, and Turing–Hopf bifurcations of a diffusive predator–prey model with nonlocal competition. In [27], Liu et al. studied a diffusive predator–prey model with nonlocal competition and time delay. These works show that spatially inhomogeneous bifurcating periodic solutions are stable, in contrast to models without nonlocal competition. A predator–prey model with nonlocal competition can produce complex dynamics, such as spatiotemporal patterns and stably spatially inhomogeneous periodic solutions [26,27,28,29].

Based on the above considerations, we introduce nonlocal competition among prey into model (3) as follows.

$$\left\{\begin{array}{c}\frac{\partial u(x,t)}{\partial t}=d_1\Delta u(x,t)+u(x,t)\left((1-{\int }_{\mathrm{\Omega }}G(x,y)u(y,t)dy)(u(x,t)-a)-\frac{cv(x,t)}{1+sv(x,t)}\right),\hfill \\ \frac{\partial v(x,t)}{\partial t}=d_2\Delta v(x,t)+\frac{rv(x,t)}{1+sv(x,t)}(1-\frac{v(x,t-\tau )}{u(x,t-\tau )}),x\in \mathrm{\Omega },t>0\hfill \\ \frac{\partial u(x,t)}{\partial \overline{\nu }}=\frac{\partial v(x,t)}{\partial \overline{\nu }}=0,x\in \partial \mathrm{\Omega },t>0\hfill \\ u(x,\theta )=u_0(x,\theta )\ge 0,v(x,\theta )=v_0(x,\theta )\ge 0,x\in \overline{\mathrm{\Omega }},\theta \in [-\tau ,0].\hfill \end{array}\right.$$

(4)

The integral term ${\int }_{\mathrm{\Omega }}G(x,y)u(y,t)dy$ in the first equation of (4) accounts for nonlocal competition among the prey individuals. The kernel function is of the following form:

$$G(x,y)=\frac{1}{\left|\mathrm{\Omega }\right|}=\frac{1}{l\pi },x,y\in \mathrm{\Omega },$$

which can be regarded as a measurement of the competition pressure at location x from the individuals at another location (y), which is widely used by some scholars [26,27,28]. The region $\mathrm{\Omega }=(0,l\pi )$ with $l>0$ is used for the convenience of calculation. In this case, the strength of the competition among all prey individuals is the same across the habitat.

The aim of this paper is to consider the dynamics of model (4) from the perspective of stability and Hopf bifurcation and to study the effects of the Allee effect and fear effect on population growth law using numerical simulation. This article is structured as follows. In Section 2, we analyze the stability of the coexisting equilibrium and the existence of Hopf bifurcation. In Section 3, we analyze the properties of Hopf bifurcation. In Section 4, we perform some numerical simulations and analyze the results. In Section 5, we provide a brief conclusion.

2. Stability Analysis

In [17], the authors found that system (4) had no less than one equilibrium, which was noted as $E_{*}(u_{*},v_{*})$. We can obtain the concrete form of $u_{*}$ by calculating the positive root of the following equation:

$$s{u}^3-(as+s-1)u^2+(as+c-a-1)u+a=0=0.$$

(5)

For the completeness of the article, we provide the following lemma and a numerical simulation (see Figure 1). In Figure 1, we can see that with the increase in parameter a, the two positive equilibrium points degenerate into one positive equilibrium point. However, when parameter a is greater than a certain critical value, the positive equilibrium point disappears.

Lemma 1.

If support $\left({\mathbf{H}}_{\mathbf{1}}\right)$ holds, the following results are true for system (4).

$1.$ If $c<a(1-s)$ or $c>a(1-s)+1$, $s(a+1)>1$:

$\left(a\right).$ If $D<0$, there are two distinct positive equilibria;

$\left(b\right).$ If $D=0$,

$\left(ib\right).$ if $A>0$, there exists a unique positive equilibrium;

$\left(iib\right).$ if $A=0$, there exists no positive equilibrium;

$\left(c\right).$ If $D>0$, there exists no positive equilibrium.

$2.$ If $c\ge a(1-s)$, $s(a+1)\le 1$, there exists no positive equilibrium.

$$\begin{array}{cc}\hfill A& ={(as+s-1)}^2-3s(as+c-a-1),\hfill \\ \hfill B& =-(as+s-1)(as+c-a-1)-9as,\hfill \\ \hfill C& ={(as+c-a-1)}^2+3a(as+s-1),\hfill \\ \hfill D& =-B^2-4AC.\hfill \end{array}$$

(6)

According to [17], $(u_2,v_2)$ is always the saddle point under this set of parameters. We mainly study the stability and Hopf bifurcation of the equilibrium ($(u_1,v_1)$) in the following.

We linearize system (4) at $E_{*}(u_{*},v_{*})$:

$$\begin{array}{c}\hfill \frac{\partial }{\partial t}\left(\begin{array}{c}u(x,t)\hfill \\ u(x,t)\hfill \end{array}\right)=D\left(\begin{array}{c}\Delta u(x,t)\hfill \\ \Delta v(x,t)\hfill \end{array}\right)+L_1\left(\begin{array}{c}u(x,t)\hfill \\ v(x,t)\hfill \end{array}\right)+L_2\left(\begin{array}{c}u(x,t-\tau )\hfill \\ v(x,t-\tau )\hfill \end{array}\right)+L_3\left(\begin{array}{c}\widehat{u}(x,t)\hfill \\ \widehat{v}(x,t)\hfill \end{array}\right),\end{array}$$

(7)

where

$$D=\left(\begin{array}{cc}{d}_1& 0\\ 0& d_2\end{array}\right),L_1=\left(\begin{array}{cc}{a}_1& a_2\\ 0& 0\end{array}\right),L_2=\left(\begin{array}{cc}0& 0\\ b_1& -b_1\end{array}\right),L_3=\left(\begin{array}{cc}\widehat{a}& 0\\ 0& 0\end{array}\right),$$

$$a_1=(1-u_{*})u_{*}>0,a_2=-\frac{c{u}_{*}}{{(1+s{u}_{*})}^2}<0,b_1=\frac{r}{1+s{u}_{*}}>0,\widehat{a}=-u_{*}(u_{*}-a)<0,$$

(8)

and $\widehat{u}=\frac{1}{l\pi }{\int }_0^{l\pi }u(y,t)dy$.

Then, the characteristic equations of (7) are

$${\lambda }^2+P_n\lambda +Q_n+(R_n+b_1\lambda )e^{-\lambda \tau }=0,n\in {\mathbb{N}}_0,$$

(9)

where

$$\begin{array}{cc}\hfill P_0& =-\widehat{a}-a_1,Q_0=0,R_0=-(\widehat{a}+a_1+a_2)b_1,\hfill \\ \hfill P_n& =(d_1+d_2)\frac{n^2}{l^2}-a_1,Q_n=d_2\frac{n^2}{l^2}\left(d_1\frac{n^2}{l^2}-a_1\right),\hfill \\ \hfill R_n& =b_1\left(d_1\frac{n^2}{l^2}-(a_1+a_2)\right),n\in \mathbb{N}.\hfill \end{array}$$

(10)

Let $\tau =0$; Equation (9) becomes

$${\lambda }^2+(P_n+b_1)\lambda +Q_n+R_n=0,n\in {\mathbb{N}}_0.$$

(11)

We make the following hypothesis:

$$\begin{array}{cc}\hfill \left({\mathbf{H}}_{\mathbf{1}}\right)& P_n+b_1>0,Q_n+R_n>0,\mathrm{for}n\in {\mathbb{N}}_0.\hfill \end{array}$$

Under this hypothesis $\left({\mathbf{H}}_{\mathbf{1}}\right)$, $E_{*}(u_{*},v_{*})$ is locally asymptotically stable when $\tau =0$.

Next, we will discuss the case of $\tau >0$.

Lemma 2.

If support $\left({\mathbf{H}}_{\mathbf{1}}\right)$ holds, the following results are true for Equation (9):

• There exists a pair of purely imaginary roots: $\pm i{\omega }_n^{+}$ at ${\tau }_n^{j,+}$ for $j\in {\mathbb{N}}_0$ and $n\in {\mathbb{W}}_1$;
• There are two pairs of purely imaginary roots: $\pm i{\omega }_n^{\pm }$ at ${\tau }_n^{j,\pm }$ for $j\in {\mathbb{N}}_0$ and $n\in {\mathbb{W}}_2$;
• There exists no purely imaginary root for $n\in {\mathbb{W}}_3$,

where $\pm i{\omega }_n^{\pm }$, ${\tau }_n^{j,\pm }$, ${\mathbb{W}}_1$, ${\mathbb{W}}_2$, and ${\mathbb{W}}_3$ are defined in (14) and (15).

Proof. 

Let $i\omega$ ($\omega >0$) be a solution of Equation (9). Then,

$$-{\omega }^2+\mathrm{i}\omega P_n+Q_n+(R_n+b_1\mathrm{i}\omega )(\mathrm{c}\mathrm{o}\mathrm{s}\omega \tau -\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{n}\omega \tau )=0,n\in {\mathbb{N}}_0.$$

Obviously, $\cos \omega \tau =\frac{{\omega }^2(R_n-b_1{P}_n)-Q_n{R}_n}{R_n^2+b_1^2{\omega }^2}$, $\sin \omega \tau =\frac{\omega \left(P_n{R}_n-b_1({\omega }^2-Q_n)\right)}{R_n^2+b_1^2{\omega }^2}$. This leads to

$${\omega }^4+{\omega }^2\left(P_n^2-2Q_n-b_1^2\right)+Q_n^2-R_n^2=0,n\in {\mathbb{N}}_0.$$

(12)

Let $z={\omega }^2$; then, (12) becomes

$$z^2+z\left(P_n^2-2Q_n-b_1^2\right)+Q_n^2-R_n^2=0,n\in {\mathbb{N}}_0.$$

(13)

Let $H_n=P_n^2-2Q_n-b_1^2$, $J_n=Q_n+R_n$, and $K_n=Q_n-R_n$. Then, $z^{\pm }=\frac{1}{2}[-H_n\pm \sqrt{H_n^2-4J_n{K}_n}]$ are the roots of (13). If $\left({\mathbf{H}}_{\mathbf{1}}\right)$ holds, $J_n>0(n\in {\mathbb{N}}_0)$. Then, we can obtain

$$\begin{array}{cc}\hfill H_0& ={(\widehat{a}+a_1)}^2-b_1^2,\hfill \\ \hfill H_k& ={\left(a_1-d_1\frac{k^2}{l^2}\right)}^2+d_2^2\frac{k^4}{l^4}-b_1^2,\mathrm{for}k\in \mathbb{N}\hfill \\ \hfill K_0& =b_1(\widehat{a}+a_1+a_2)<0,\hfill \\ \hfill K_k& =d_1{d}_2\frac{k^4}{l^4}-(b_1{d}_1+a_1{d}_2)\frac{k^2}{l^2}+(a_1+a_2)b_1,\mathrm{for}k\in \mathbb{N}.\hfill \end{array}$$

We define

$$\begin{array}{cc}\hfill & {\mathbb{S}}_1=\left\{n\right|K_n<0,n\in {\mathbb{N}}_0\},\hfill \\ \hfill & {\mathbb{S}}_2=\left\{n\right|K_n>0,H_n<0,H_n^2-4J_n{K}_n>0,n\in \mathbb{N}\},\hfill \\ \hfill & {\mathbb{S}}_3=\left\{n\right|K_n>0,H_n^2-4J_n{K}_n<0,n\in \mathbb{N}\},\hfill \end{array}$$

(14)

and

$$\begin{array}{cc}\hfill {\omega }_n^{\pm }& =\sqrt{z_n^{\pm }},{\tau }_n^{j,\pm }=\left\{\begin{array}{cc}\frac{1}{{\omega }_n^{\pm }}\arccos \left(V_{cos}^{(n,\pm )}\right)+2j\pi ,\hfill & V_{sin}^{(n,\pm )}\ge 0,\hfill \\ \frac{1}{{\omega }_n^{\pm }}\left[2\pi -\arccos \left(V_{cos}^{(n,\pm )}\right)\right]+2j\pi ,\hfill & V_{sin}^{(n,\pm )}<0.\hfill \end{array}\right.\hfill \\ \hfill V_{cos}^{(n,\pm )}& =\frac{{\left({\omega }_n^{\pm }\right)}^2(b_2{P}_n+R_n)-M_n{R}_n}{R_n^2+b_1^2{\left({\omega }_n^{\pm }\right)}^2},V_{sin}^{(n,\pm )}=\frac{{\omega }_n^{\pm }\left(P_n{R}_n+Q_n{b}_2+b_1{\left({\omega }_n^{\pm }\right)}^2\right)}{R_n^2+b_1^2{\left({\omega }_n^{\pm }\right)}^2}.\hfill \end{array}$$

(15)

It is easy to verify the conclusion in Lemma 2. □

Lemma 3.

Support $\left({\mathbf{H}}_{\mathbf{1}}\right)$ is satisfied. Then, $Re\left(\frac{d\lambda }{d\tau }\right){|}_{\tau ={\tau }_n^{j,+}}>0$, $Re\left(\frac{d\lambda }{d\tau }\right){|}_{\tau ={\tau }_n^{j,-}}<0$ for $n\in {\mathbb{S}}_1\cup {\mathbb{S}}_2$ and $j\in {\mathbb{N}}_0$.

Proof. 

According to (9), we have

$${\left(\frac{d\lambda }{d\tau }\right)}^{-1}=\frac{2\lambda +P_n+b_1{e}^{-\lambda \tau }}{(R_n+b_1\lambda )\lambda e^{-\lambda \tau }}-\frac{\tau }{\lambda }.$$

Then,

$$\begin{array}{cc}\hfill {\left[\mathrm{Re}{\left(\frac{d\lambda }{d\tau }\right)}^{-1}\right]}_{\tau ={\tau }_n^{j,\pm }}& =\mathrm{Re}{[\frac{2\lambda +P_n+b_1{e}^{-\lambda \tau }}{(R_n+b_1\lambda )\lambda e^{-\lambda \tau }}-\frac{\tau }{\lambda }]}_{\tau ={\tau }_n^{j,\pm }}\hfill \\ \hfill & ={\left[\frac{1}{R_n^2+b_1^2{\omega }^2}(2{\omega }^2+P_n^2-2Q_n-b_1^2)\right]}_{\tau ={\tau }_n^{j,\pm }}\hfill \\ \hfill & =\pm {\left[\frac{1}{R_n^2+b_1^2{\omega }^2}\sqrt{{(P_n^2-2Q_n-b_1^2)}^2-4(Q_n^2-R_n^2)}\right]}_{\tau ={\tau }_n^{j,\pm }}.\hfill \end{array}$$

Therefore, $\mathrm{Re}\left(\frac{d\lambda }{d\tau }\right){|}_{\tau ={\tau }_n^{j,+}}>0$, $\mathrm{Re}\left(\frac{d\lambda }{d\tau }\right){|}_{\tau ={\tau }_n^{j,-}}<0$. □

We denote ${\tau }_{*}=min\left\{{\tau }_n^0\right|n\in {\mathbb{S}}_1\cup {\mathbb{S}}_2\}$.

Naturally, we have the following theorem.

Theorem 1.

Assume that $\left({\mathbf{H}}_{\mathbf{1}}\right)$ holds; then, the following statements are true for system (4).

$1.$ $E_{*}(u_{*},v_{*})$ is locally asymptotically stable for $\tau >0$ when ${\mathbb{S}}_1\cup {\mathbb{S}}_2=⌀$;

$2.$ $E_{*}(u_{*},v_{*})$ is locally asymptotically stable for $\tau \in [0,{\tau }_{*})$ when ${\mathbb{S}}_1\cup {\mathbb{S}}_2\ne ⌀$;

$3.$ $E_{*}(u_{*},v_{*})$ is unstable for $\tau \in ({\tau }_{*},{\tau }_{*}+\epsilon )$ for some $\epsilon >0$ when ${\mathbb{S}}_1\cup {\mathbb{S}}_2\ne ⌀$;

$4.$ Hopf bifurcation occurs at $(u_{*},v_{*})$ when $\tau ={\tau }_n^{j,+}$ $(\tau ={\tau }_n^{j,-})$, $j\in {\mathbb{N}}_0$, $n\in {\mathbb{S}}_1\cup {\mathbb{S}}_2$.

Because stability switching is a highly concerned dynamic phenomenon [30,31,32], we provide the following remark about stability switching.

Remark 1.

According to lemma 3, we know that $Re\left(\frac{d\lambda }{d\tau }\right){|}_{\tau ={\tau }_n^{j,+}}>0$, $Re\left(\frac{d\lambda }{d\tau }\right){|}_{\tau ={\tau }_n^{j,-}}<0$. If ${\mathbb{W}}_2\ne ⌀$ and there exist ${\tau }_{*}={\tau }_{n_1}^{0,+}<{\tau }_{n_1}^{0,-}$, ${\tau }_{*}={\tau }_{n_1}^{0,+}<{\tau }_{n_2}^{0,+}<{\tau }_{n_2}^{0,-}<{\tau }_{n_1}^{0,-}$, ${\tau }_{*}={\tau }_{n_1}^{0,+}<{\tau }_{n_2}^{0,+}<\cdots <{\tau }_{n_j}^{0,+}<{\tau }_{n_j}^{0,-}<\cdots <{\tau }_{n_2}^{0,-}<{\tau }_{n_1}^{0,-}$, or other alternating forms of ${\tau }_n^{j,\pm }$. Then, stability switching may exist.

3. Properties of Hopf Bifurcation

From [33,34], we learned how to analyze the properties of Hopf bifurcation. For fixed $j\in {\mathbb{N}}_0$ and $n\in {\mathbb{S}}_1\cup {\mathbb{S}}_2$, we define $\tilde{\tau }={\tau }_n^{j,\pm }$. Let $\overline{u}(x,t)=u(x,\tau t)-u_{*}$ and $\overline{v}(x,t)=v(x,\tau t)-v_{*}$. By ignoring the bar, (4) becomes

$$\left\{\begin{array}{c}\frac{\partial u}{\partial t}=\tau [d_1\Delta u+(u+u_{*})\left(1-\frac{1}{l\pi }{\int }_0^{l\pi }(u(y,t)+u_{*})dy\right)(u+u_{*}-a)-\frac{c(v+v_{*})}{1+s(v+v_{*})}],\hfill \\ \frac{\partial v}{\partial t}=\tau [d_2\Delta v+\frac{r(v+v_{*})}{1+s(v+v_{*})}\left(1-\frac{v(t-1)+v_{*}}{u(t-1)+u_{*}}\right)].\hfill \end{array}\right.$$

(16)

Then, we rewrite (16) in the following form:

$$\left\{\begin{array}{cc}\hfill \frac{\partial u}{\partial t}=& \tau [d_1\Delta u+a_{1}u+a_{2}v-\widehat{a}\widehat{u}+{\alpha }_1{u}^2+(a-2u_{*})u\widehat{u}+{\alpha }_{2}uv+{\alpha }_3{v}^2+{\alpha }_{4}u{v}^2+{\alpha }_5{v}^3]+h.o.t.,\hfill \\ \hfill \frac{\partial v}{\partial t}=& \tau [d_2\Delta v-\eta \sigma v+b_{1}u(t-1)-b_{1}v(t-1)+{\beta }_1{u}^2(t-1)+{\beta }_{2}u(t-1)v+{\beta }_{3}u(t-1)v(t-1)+{\beta }_{4}vv(t-1)\hfill \\ \hfill & +{\beta }_5{u}^3(t-1)+{\beta }_6{u}^2(t-1)v+{\beta }_{7}u(t-1)v^2+{\beta }_8{u}^2(t-1)v(t-1)]+h.o.t.,\hfill \end{array}\right.$$

(17)

where ${\alpha }_1=1-u_{*}$, ${\alpha }_2=-\frac{c}{{(1+s{u}_{*})}^2}$, ${\alpha }_3=\frac{cs{u}_{*}}{{(1+s{u}_{*})}^3}$, ${\alpha }_4=\frac{cs}{{(1+s{u}_{*})}^3}$, ${\alpha }_5=-\frac{c{s}^2{u}_{*}}{{(1+s{u}_{*})}^4}$, ${\beta }_1=-\frac{r}{u_{*}(1+s{u}_{*})}$, ${\beta }_2=\frac{r}{u_{*}{(1+s{u}_{*})}^2}$, ${\beta }_3=\frac{r}{u_{*}(1+s{u}_{*})}$, ${\beta }_4=-\frac{r}{u_{*}{(1+s{u}_{*})}^2}$, ${\beta }_5=\frac{r}{u_{*}^2(1+s{u}_{*})}$, ${\beta }_6=-\frac{r}{u_{*}^2{(1+s{u}_{*})}^2}$, ${\beta }_7=-\frac{rs}{u_{*}{(1+s{u}_{*})}^3}$, and ${\beta }_8=-\frac{r}{u_{*}^2(1+s{u}_{*})}$.

We define a space, $X:=\left\{{(u,v)}^T:u,v\in H^2(0,l\pi ),(u_x,v_x){|}_{x=0,l\pi }=0\right\},$ which is called real-valued Sobolev. $X_{\mathbb{C}}$ is the complexification of X, which has the form $X_{\mathbb{C}}:=X\oplus \mathrm{i}X=\left\{u+\mathrm{i}v|u,v\in X\right\}.$ Then, we define the inner product: $<\tilde{u},\tilde{v}>:={\int }_0^{l\pi }\overline{u_1}{v}_{1}dx+{\int }_0^{l\pi }\overline{u_2}{v}_{2}dx$, where $\tilde{u}={(u_1,u_2)}^T\in X_{\mathbb{C}}$, $\tilde{v}={(v_1,v_2)}^T\in X_{\mathbb{C}}$.

We define the phase space, $\mathcal{C}:=C\left(\right[-1,0],X)$, which is with the sup norm. Then, we have ${\phi }_t\in \mathcal{C}$, ${\phi }_t\left(\sigma \right)=\phi (t+\sigma )$ for $\sigma \in [-1,0]$. To define the subspace of $\mathcal{C}$, we made the following definitions: ${\alpha }_n^{\left(1\right)}\left(u\right)={({\gamma }_n\left(u\right),0)}^T$, ${\alpha }_n^{\left(2\right)}\left(u\right)={(0,{\gamma }_n\left(u\right))}^T$, and ${\alpha }_n=\{{\alpha }_n^{\left(1\right)}\left(u\right),{\alpha }_n^{\left(2\right)}\left(u\right)\}$, where $\left\{{\alpha }_n^{\left(i\right)}\left(u\right)\right\}$ is an orthonormal basis of X. Then, we define the subspace of $\mathcal{C}$ as ${\mathbb{B}}_n:=\mathrm{span}\{<\phi (·),{\alpha }_n^{\left(j\right)}>{\alpha }_n^{\left(j\right)}|\phi \in \mathcal{C},j=1,2\}$, $n\in {\mathbb{N}}_0$. For $\theta \in [-1,0]$ and $\phi \in \mathcal{C}$, the $2\times 2$ matrix function (${\eta }^n(\theta ,\tilde{\tau })$) can satisfy the following: $-\tilde{\tau }D\frac{n^2}{l^2}\phi \left(0\right)+\tilde{\tau }L\left(\phi \right)={\int }_{-1}^{0}d{\eta }^n(\theta ,\tau )\phi \left(\theta \right)$. Then, Equation (18) defines the bilinear form on ${\mathcal{C}}^{*}\times \mathcal{C}$ for $\psi \in \mathcal{C}$, $\varphi \in {\mathcal{C}}^{*}$.

$$(\varphi ,\psi )=\varphi \left(0\right)\psi \left(0\right)-{\int }_{-1}^0{\int }_{\xi =0}^{\theta }\varphi (\xi -\theta )d{\eta }^n(\theta ,\tilde{\tau })\psi \left(\xi \right)d\xi ,$$

(18)

Let $\tau =\tilde{\tau }+\mu$. When $\mu =0$, the characteristic equation of the system has a pair of purely imaginary roots ($\pm i{\omega }_{n_0}$), and the system undergoes Hopf bifurcation at $(0,0)$. Assume that A represents the infinitesimal generators of the semigroup, and $A^{*}$ represents the formal adjoint of A under the bilinear form (18).

We define

$$\begin{array}{c}\hfill \delta \left(n_0\right)=\left\{\begin{array}{cc}1\hfill & n_0=0,\hfill \\ 0\hfill & n_0\in \mathbb{N}.\hfill \end{array}\right.\end{array}$$

(19)

Let ${\eta }_{n_0}(0,\tilde{\tau })=\tilde{\tau }[(-n_0^2/l^2)D+L_1+L_3\delta \left(n_{n_0}\right)]$, ${\eta }_{n_0}(-1,\tilde{\tau })=-\tilde{\tau }{L}_2$, and ${\eta }_{n_0}(\sigma ,\tilde{\tau })=0$ for $\sigma \in [-1,0]$. We define $p\left(\theta \right)=p\left(0\right)e^{\mathrm{i}{\omega }_{n_0}\tilde{\tau }\theta }(\theta \in [-1,0])$ as the eigenfunction of $A\left(\tilde{\tau }\right)$ for $i{\omega }_{n_0}\tilde{\tau }$, and $q\left(\vartheta \right)=q\left(0\right)e^{-\mathrm{i}{\omega }_{n_0}\tilde{\tau }\vartheta }(\vartheta \in [0,1])$ is the eigenfunction of $A^{*}$ for $i{\omega }_{n_0}\tilde{\tau }$. Let $p\left(0\right)={(1,p_1)}^T$, $q\left(0\right)=M(1,q_2)$, where $p_1=\frac{1}{a_2}(\mathrm{i}{\omega }_{n_0}+d_1{n}_0^2/l^2-a_1-\widehat{a}\delta \left(n_0\right))$, $q_2=\frac{a_2}{\mathrm{i}{\omega }_{n_0}+b_1{e}^{\mathrm{i}{\omega }_{n_0}}+d_2{n}^2/l^2}$, and $M={(1+p_1{q}_2+\tilde{\tau }{q}_2{b}_1(1-p_1)e^{-\mathrm{i}{\omega }_{n_0}\tilde{\tau }})}^{-1}$. Then, (16) becomes

$$\frac{dU\left(t\right)}{dt}=(\tilde{\tau }+\mu )D\Delta U\left(t\right)+(\tilde{\tau }+\mu )[L_{1}U\left(t\right)+L_{2}U(t-1)+L_3\widehat{U}\left(t\right)]+F(\mu ,U_t,{\widehat{U}}_t),$$

(20)

where

$$\begin{array}{c}\hfill F(\varphi ,\mu )=(\tilde{\tau }+\mu )\left(\begin{array}{c}{\alpha }_1{\varphi }_1{\left(0\right)}^2-(2u_{*}-\beta ){\varphi }_1\left(0\right){\widehat{\varphi }}_1\left(0\right)+{\alpha }_2{\varphi }_1\left(0\right){\varphi }_2\left(0\right)+{\alpha }_3{\varphi }_2{\left(0\right)}^2+{\alpha }_4{\varphi }_1^3\left(0\right)\\ +{\alpha }_5{\varphi }_1^2\left(0\right){\varphi }_2\left(0\right)+{\alpha }_6{\varphi }_1\left(0\right){\varphi }_2^2\left(0\right)+{\alpha }_7{\varphi }_2^3\left(0\right)\\ {\beta }_1{\varphi }_1^2(-1)+{\beta }_2{\varphi }_1(-1){\varphi }_2(-1)+{\beta }_3{\varphi }_2^2(-1)+{\beta }_4{\varphi }_1^3(-1)+{\beta }_4{\varphi }_1^2(-1){\varphi }_2(-1)\\ +{\beta }_6{\varphi }_1(-1){\varphi }_2^2(-1)+{\beta }_7{\varphi }_2^3(-1)\end{array}\right)\end{array}$$

(21)

for $\varphi ={({\varphi }_1,{\varphi }_2)}^T\in \mathcal{C}$ and ${\widehat{\varphi }}_1=\frac{1}{l\pi }{\int }_0^{l\pi }\varphi dx$. Then, we decompose the space ($\mathcal{C}$) as $\mathcal{C}=P\oplus Q$, where $P=\{zp{\gamma }_{n_0}\left(x\right)+\overline{z}\overline{p}{\gamma }_{n_0}\left(x\right)|z\in \mathbb{C}\}$, $Q=\{\varphi \in \mathcal{C}|(q{\gamma }_{n_0}\left(x\right),\varphi )=0,and(\overline{q}{\gamma }_{n_0}\left(x\right),\varphi )=0\}$. Then, (21) is rewritten as $U_t=z\left(t\right)p(·){\gamma }_{n_0}\left(x\right)+\overline{z}\left(t\right)\overline{p}(·){\gamma }_{n_0}\left(x\right)+\omega (t,·)$, and $\widehat{U_t}=\frac{1}{l\pi }{\int }_0^{l\pi }{U}_{t}dx$, where

$$z\left(t\right)=(q{\gamma }_{n_0}\left(x\right),U_t),\omega (t,\theta )=U_t\left(\theta \right)-2\mathrm{Re}\left\{z\left(t\right)p\left(\theta \right){\gamma }_{n_0}\left(x\right)\right\}.$$

(22)

We found that $\dot{z}\left(t\right)=\mathrm{i}{\omega }_{n_0}\tilde{\tau }z\left(t\right)+\overline{q}\left(0\right)<F(0,U_t),{\beta }_{n_0}>$. Then, ${\mathcal{C}}_0$ and $\omega$ can have the following form near $(0,0)$:

$$\omega (t,\theta )=\omega (z\left(t\right),\overline{z}\left(t\right),\theta )={\omega }_{20}\left(\theta \right)\frac{z^2}{2}+{\omega }_{11}\left(\theta \right)z\overline{z}+{\omega }_{02}\left(\theta \right)\frac{{\overline{z}}^2}{2}+\cdots .$$

(23)

We restrict the system to ${\mathcal{C}}_0$ such that $\dot{z}\left(t\right)=\mathrm{i}{\omega }_{n_0}\tilde{\tau }z\left(t\right)+g(z,\overline{z})$. Let $g(z,\overline{z})=g_{20}\frac{z^2}{2}+g_{11}z\overline{z}+g_{02}\frac{{\overline{z}}^2}{2}+g_{21}\frac{z^2\overline{z}}{2}+\cdots$. By direct computation, we have

$$g_{20}=2\tilde{\tau }M({\varsigma }_1+q_2{\varsigma }_2)I_3,g_{11}=\tilde{\tau }M({\varrho }_1+q_2{\varrho }_2)I_3,g_{02}={\overline{g}}_{20},$$

$$g_{21}=2\tilde{\tau }M[({\kappa }_{11}+q_2{\kappa }_{21})I_2+({\kappa }_{12}+q_2{\kappa }_{22})I_4],$$

where $I_2={\int }_0^{l\pi }{\gamma }_{n_0}^2\left(x\right)dx$, $I_3={\int }_0^{l\pi }{\gamma }_{n_0}^3\left(x\right)dx$, $I_4={\int }_0^{l\pi }{\gamma }_{n_0}^4\left(x\right)dx$, ${\varsigma }_1=(a-2u_{*}){\delta }_{n_0}+({\alpha }_1+p_1({\alpha }_2+{\alpha }_3{p}_1))$, ${\varsigma }_2=e^{-2i\tau {\omega }_n}\left({\beta }_1+2{\beta }_8+e^{i\tau {\omega }_n}({\beta }_2+2{\beta }_3{\beta }_4)p_1\right)$, ${\varrho }_1=\frac{1}{4}(2{\alpha }_1+2(a-2u_{*}){\delta }_{n_0}+2{\alpha }_3\overline{p_1}{p}_1+{\alpha }_2(\overline{p_1}+p_1))$, ${\varrho }_2=\frac{1}{4}{e}^{-i\tau {\omega }_n}\left(2e^{i\tau {\omega }_n}({\beta }_1+2{\beta }_8)+({\beta }_2+2{\beta }_3{\beta }_4)\overline{p_1}+e^{2i\tau {\omega }_n}({\beta }_2+2{\beta }_3{\beta }_4)p_1\right)$, ${\kappa }_{11}=2W_{11}^{\left(1\right)}\left(0\right)(2{\alpha }_1+a(1+{\delta }_{n_0})-2u_{*}(1+{\delta }_{n_0})+{\alpha }_2{p}_1)+2$ $W_{11}^{\left(2\right)}\left(0\right)({\alpha }_2+2{\alpha }_3{p}_1)+W_{20}^{\left(1\right)}\left(0\right)(2{\alpha }_1+a(1+{\delta }_{n_0})-2u_{*}(1+{\delta }_{n_0})+{\alpha }_2\overline{p_1})+$ $W_{20}^{\left(2\right)}\left(0\right)({\alpha }_2+2{\alpha }_3\overline{p_1})$, ${\kappa }_{12}=2e^{-i\tau {\omega }_n}{W}_{11}^{\left(2\right)}\left(0\right)({\beta }_2+2{\beta }_3{\beta }_4)+e^{i\tau {\omega }_n}{W}_{20}^{\left(2\right)}\left(0\right)({\beta }_2+2{\beta }_3{\beta }_4)+2e^{-i\tau {\omega }_n}{W}_{11}^{\left(1\right)}(-1)\left(2{\beta }_1+4{\beta }_8+e^{i\tau {\omega }_n}({\beta }_2+2{\beta }_3{\beta }_4)p_1\right)+W_{20}^{\left(1\right)}(-1)\left(2e^{i\tau {\omega }_n}({\beta }_1+2{\beta }_8)+({\beta }_2+2{\beta }_3{\beta }_4)\overline{p_1}\right)$, ${\kappa }_{21}=\frac{1}{2}{p}_1(3{\alpha }_5\overline{p_1}{p}_1+{\alpha }_4(2\overline{p_1}+p_1))$, ${\kappa }_{22}=\frac{1}{2}{e}^{-2i\tau {\omega }_n}$ $(e^{3i\tau {\omega }_n}{\beta }_7{p}_1^2+\overline{p_1}({\beta }_6+3{\beta }_3{\beta }_4{p}_1)+e^{i\tau {\omega }_n}(3{\beta }_5+{\beta }_8\overline{p_1}+2{\beta }_8{p}_1+2{\beta }_7\overline{p_1}{p}_1)+e^{2i\tau {\omega }_n}{p}_1(2{\beta }_6+3{\beta }_3{\beta }_4(\overline{p_1}+p_1)))$.

Next, for $\theta \in [-1,0]$, we compute $W_{20}$ and $W_{11}$ to obtain $g_{21}$. According to (22), we have

$$\dot{\omega }={\dot{U}}_t-\dot{z}p{\gamma }_{n_0}\left(x\right)-\dot{\overline{z}}\overline{p}{\gamma }_{n_0}\left(x\right)=A\omega +H(z,\overline{z},\theta ),$$

(24)

where

$$H(z,\overline{z},\theta )=H_{20}\left(\theta \right)\frac{z^2}{2}+H_{11}\left(\theta \right)z\overline{z}+H_{02}\left(\theta \right)\frac{{\overline{z}}^2}{2}+\cdots .$$

(25)

By comparing the coefficients of (23) with those of (24), we have

$$\begin{array}{c}\hfill (A-2\mathrm{i}{\omega }_{n_0}\tilde{\tau }I){\omega }_{20}=-H_{20}\left(\theta \right),A{\omega }_{11}\left(\theta \right)=-H_{11}\left(\theta \right).\end{array}$$

(26)

Then, we have

$$\begin{array}{cc}\hfill {\omega }_{20}\left(\theta \right)& =\frac{-g_{20}}{\mathrm{i}{\omega }_{n_0}\tilde{\tau }}p\left(0\right)e^{\mathrm{i}{\omega }_{n_0}\tilde{\tau }\theta }-\frac{{\overline{g}}_{02}}{3\mathrm{i}{\omega }_{n_0}\tilde{\tau }}\overline{p}\left(0\right)e^{-\mathrm{i}{\omega }_{n_0}\tilde{\tau }\theta }+E_1{e}^{2\mathrm{i}{\omega }_{n_0}\tilde{\tau }\theta },\hfill \\ \hfill {\omega }_{11}\left(\theta \right)& =\frac{g_{11}}{\mathrm{i}{\omega }_{n_0}\tilde{\tau }}p\left(0\right)e^{\mathrm{i}{\omega }_{n_0}\tilde{\tau }\theta }-\frac{{\overline{g}}_{11}}{\mathrm{i}{\omega }_{n_0}\tilde{\tau }}\overline{p}\left(0\right)e^{-\mathrm{i}{\omega }_{n_0}\tilde{\tau }\theta }+E_2,\hfill \end{array}$$

(27)

where $E_1={\sum }_{n=0}^{\infty }{E}_1^{\left(n\right)}$, $E_2={\sum }_{n=0}^{\infty }{E}_2^{\left(n\right)}$,

$$\begin{array}{cc}\hfill E_1^{\left(n\right)}& ={(2\mathrm{i}{\omega }_{n_0}\tilde{\tau }I-{\int }_{-1}^0{e}^{2\mathrm{i}{\omega }_{n_0}\tilde{\tau }\theta }d{\eta }_{n_0}(\theta ,\overline{\tau }))}^{-1}<{\tilde{F}}_{20},{\beta }_n>,\hfill \\ \hfill E_2^{\left(n\right)}& =-{\left({\int }_{-1}^{0}d{\eta }_{n_0}(\theta ,\overline{\tau })\right)}^{-1}<{\tilde{F}}_{11},{\beta }_n>,n\in {\mathbb{N}}_0,\hfill \end{array}$$

$$<{\tilde{F}}_{20},{\beta }_n>=\left\{\begin{array}{cc}\frac{1}{l\pi }{\widehat{F}}_{20},\hfill & n_0\ne 0,n=0,\hfill \\ \frac{1}{2l\pi }{\widehat{F}}_{20},\hfill & n_0\ne 0,n=2n_0,\hfill \\ \frac{1}{l\pi }{\widehat{F}}_{20},\hfill & n_0=0,n=0,\hfill \\ 0,\hfill & other,\hfill \end{array}\right.<{\tilde{F}}_{11},{\beta }_n>=\left\{\begin{array}{cc}\frac{1}{l\pi }{\widehat{F}}_{11},\hfill & n_0\ne 0,n=0,\hfill \\ \frac{1}{2l\pi }{\widehat{F}}_{11},\hfill & n_0\ne 0,n=2n_0,\hfill \\ \frac{1}{l\pi }{\widehat{F}}_{11},\hfill & n_0=0,n=0,\hfill \\ 0,\hfill & other,\hfill \end{array}\right.$$

and ${\widehat{F}}_{20}=2{({\varsigma }_1,{\varsigma }_2)}^T$, ${\widehat{F}}_{11}=2{({\varrho }_1,{\varrho }_2)}^T$.

Thus, we can obtain

$$\begin{array}{c}{c}_1\left(0\right)=\frac{\mathrm{i}}{2{\omega }_n\tilde{\tau }}(g_{20}{g}_{11}-2|g_{11}{|}^2-\frac{|g_{02}{|}^2}{3})+\frac{1}{2}{g}_{21},{\mu }_2=-\frac{\mathrm{Re}\left(c_1\left(0\right)\right)}{\mathrm{Re}\left({\lambda }^{\prime }\left(\tilde{\tau }\right)\right)},\hfill \\ T_2=-\frac{1}{{\omega }_{n_0}\tilde{\tau }}[\mathrm{Im}\left(c_1\left(0\right)\right)+{\mu }_2\mathrm{Im}\left({\lambda }^{\prime }\left({\tau }_n^j\right)\right)],{\beta }_2=2\mathrm{Re}\left(c_1\left(0\right)\right).\hfill \end{array}$$

(28)

Theorem 2.

The following results are true for any critical value (${\tau }_n^j$ ($n\in \mathbb{S},j\in {\mathbb{N}}_0$)).

$1.$ If ${\mu }_2>0$ (or <0), the Hopf bifurcation is forward (or backward);

$2.$ If ${\beta }_2<0$ (or >0), the bifurcating periodic solutions on ${\mathcal{C}}_0$ are orbitally asymptotically stable (or unstable);

$3.$ If $T_2>0$ (or $T_2<0$), the period increases (or decreases).

4. Numerical Simulations

To analyze the influence of the fear effect parameter (s), the strong Allee effect parameter (a), and gestation delay ($\tau$) on model (4), we performed the following numerical simulations.

We fixed

$$c=0.1,r=0.25,d_1=0.5,d_2=0.1,l=1.5.$$

The existence of a positive equilibrium is provided in Figure 1. Obviously, the equilibrium $(u_2,v_2)$ is unstable through the following analysis, so we pay attention to the stability of the equilibrium $(u_1,v_1)$, which is positive. The bifurcation diagrams of model (4) with parameters s and a are provided in Figure 2 and Figure 3, respectively. In Figure 2, we can observe that increasing parameter s can increase the stable region of the equilibrium and eliminate the periodic oscillation. In Figure 3, we find that increasing parameter a can decrease the stable region of the equilibrium and induce periodic oscillation.

If we choose $a=0.05$ and $s=0.1$, then model (4) has two coexisting equilibria: $(u_1,v_1)\approx (0.9029,0.9029)$ and $(u_2,v_2)\approx (0.0559,0.0559)$. By direct calculation, the hypothesis $\left({\mathbf{H}}_{\mathbf{1}}\right)$ holds for $(u_1,v_1)$ and does not hold for $(u_2,v_2)$. Therefore, we mainly analyze the coexisting equilibrium $(u_1,v_1)$. By direct computation, we have ${\mathbb{S}}_1=\{0,1,2\}$ and ${\mathbb{S}}_2={\mathbb{S}}_3=\varnothing$, as well as ${\tau }_{*}={\tau }_1^0\approx 5.9478<{\tau }_0^0\approx 6.1053$. Theorem 1 shows that the coexisting equilibrium $(u_1,v_1)$ is locally asymptotically stable if $\tau \in [0,{\tau }_{*})$ (Figure 4). For model (4), Hopf bifurcation occurs if $\tau ={\tau }_{*}$. According to Theorem 2, we have

$${\mu }_2\approx 16.1916>0,{\beta }_2\approx -0.6295<0,T_2\approx 0.5672>0.$$

Therefore, when $\tau >{\tau }_{*}$, the bifurcating periodic solutions are stably spatially inhomogeneous (Figure 5). When we continue to increase parameter $\tau$, the bifurcating periodic solutions are still stably spatially inhomogeneous (Figure 6).

If we choose $a=0.2$ and $s=0.1$, then model (4) has two coexisting equilibria: $(u_1,v_1)\approx (0.8811,0.8811)$ and $(u_2,v_2)\approx (0.2290,0.2290)$. By direct calculation, the hypothesis $\left({\mathbf{H}}_{\mathbf{1}}\right)$ holds for $(u_1,v_1)$ and does not hold for $(u_2,v_2)$. Therefore, we mainly analyzed the coexisting equilibrium $(u_1,v_1)$. By direct computation, we have ${\mathbb{S}}_1=\{0,1,2\}$ and ${\mathbb{S}}_2={\mathbb{S}}_3=\varnothing$, as well as ${\tau }_{*}={\tau }_0^0\approx 5.8910<{\tau }_0^1\approx 5.9517$. Theorem 1 shows that the coexisting equilibrium $(u_1,v_1)$ is locally asymptotically stable if $\tau \in [0,{\tau }_{*})$. For model (4), Hopf bifurcation occurs if $\tau ={\tau }_{*}$. Theorem 2 shows that

$${\mu }_2\approx 43.8547>0,{\beta }_2\approx -1.6464<0,T_2\approx 10.3338>0.$$

Thus, when $\tau >{\tau }_{*}$, the bifurcating periodic solutions are stably spatially homogeneous (Figure 7). When we continue to increase parameter $\tau$, the bifurcating periodic solutions are still stably spatially homogeneous (Figure 8).

5. Conclusions and Discussion

In this work, we propose a delayed self-diffusive predator–prey model with a strong Allee effect on the prey and a fear effect on the predator. Unlike [17], in this paper, we added a time delay, self-diffusion, and nonlocal competition to the model, which makes the model more consistent with actual situations in nature and leads to homogeneous and inhomogeneous periodic solutions. By analyzing the eigenvalue spectrum, we studied the local stability of the coexisting equilibrium and the existence of Hopf bifurcation. By using the method of the center manifold theorem and the normal form method, we investigated the properties of Hopf bifurcation.

Next, we will discuss the influences of the fear effect and the strong Allee effect. The following conclusions can be drawn. Increasing the fear effect on the predator is beneficial to the uniform distribution of the prey and predator populations in space because the stable region of coexistence increases with the increase in the fear effect, and with the increase in the fear effect, a spatially inhomogeneous periodic solution appears first. However, when the fear effect is greater than a critical value, a spatially homogeneous periodic solution appears. However, increasing the strong Allee effect on the prey is not beneficial to the stability of the coexisting equilibrium because the stable region of coexistence decreases with the increase in the strong Allee effect. Whether the bifurcated periodic solution is spatially homogeneous or inhomogeneous depends on the strong Allee effect and the fear effect because with the increase in the strong Allee effect (or fear effect), a spatially inhomogeneous periodic solution appears first. However, when the strong Allee effect (or fear effect) is greater than a critical value, a spatially homogeneous periodic solution appears.

The main findings show that a strong Allee effect and the fear effect can be used to control the growth of prey and predator populations. For example, we could produce predation risk and affect the reproduction of sparrows by broadcasting their natural enemies’ sounds (such as those of magpies, shrikes, sparrow eagles, etc.) during sparrows’ entire breeding season. In this way, we can protect sparrows from direct killing and ensure that any effects on reproduction will only be ascribed to fear; this is the direction of our future research. Moreover, we found a Hopf–Hopf bifurcation point in the course of our research, which complicates the dynamic behavior of predator–prey systems and also requires further investigation.

However, questions remains as to whether emulating fear during an entire breeding season of a species is realistic, whether doing so would have other damaging consequences on the behavior of the species, and whether it would result in the species becoming acquainted with such sounds and no longer feeling fear (if the sounds perpetuate without any predation, the prey might consider that there is no danger after a while). This is also worth further research, especially in cooperation with biological experts.