Abstract
A diffusive predator-prey system with both the additive Allee effect and the fear effect in the prey subject to Neumann boundary conditions is considered in this paper. Firstly, non-negative and non-trivial solution a priori estimations are shown. Furthermore, for specific parameter ranges, the absence of non-constant positive solutions is demonstrated. Secondly, we use the linearized theory to investigate the stability of non-negative constant solutions. The spatially homogeneous and non-homogeneous periodic solutions, as well as non-constant steady state solutions, are next investigated by using Allee effect parameters as the bifurcation parameter. Finally, numerical simulation is used to illustrate some theoretical results.
1. Introduction
The biodynamics of ecosystems are current hot issues in biology and ecology. The intense effort to understand the pattern formation and mechanisms of spatial diffusion during the late 20th century, especially in the context of biological and ecological contexts, has gradually raised more and more concerns. Especially, in biochemical reactions characterized by interactions of different species, the study on predator-prey types has been studied widely in [1–4].
Recently, Allee effect, which was initially introduced by Allee in 1931 [5], has been studied extensively [6–10]. With the development of the theory for reaction-diffusion equations, many scholars have done many mathematical research to better describe the relationship between different species. Especially, introducing the Allee effect into the model makes the dynamic behavior of the model closer to reality. The spatiotemporal complexity of a delayed predator-prey model with double Allee effect was given by [11]. In [12], P. J. Pal and S. Tapan consider a system with a double Allee effect in prey population growth, which are very sensitive to parameter perturbations and position of initial conditions. H. Molla and S. Sarwardi developed a predator-prey model that combines these phenomena, considering variable prey refuge with additive Allee effect on the prey species, and also investigated the appearance of Hopf bifurcations in a neighborhood of the unique interior equilibrium point of the dynamical system [13]. The rich behaviour of the dynamics suggests that both prey refuge and a strong Allee affect are important factors in ecological complexity. For a reaction-diffusion system with double Allee effect induced by fear factors subject to Neumann boundary conditions, for details, please refer to [2]. The dynamical behavior of a reaction-diffusion-advection model with weak Allee effect type growth has been studied in [9]. Han and Dai investigated the spatiotemporal pattern formation and selection driven by nonlinear cross-diffusion of a toxic-phytoplankton-zooplankton model with Allee effect. By taking cross-diffusion rate as bifurcation parameter, amplitude equations under nonlinear cross-diffusion are derived that describe the spatiotemporal dynamics [14].
Some researchers have indicated that predators can not only capture prey directly but also affect the behavior of prey, even that it could affect the prey more influential than predation [15, 16]. In fact, all animals show various kinds of antipredator responses, such as feeling of fear, habitat changes, vigilance, foraging, and different physiological changes ([17–21]).
The cost of fear is objective, and it should be taken into consideration when establishing predation and predation models. For example, Jana et al. [22] have explored the influence of habitat complexity on a predator-prey system under fear effect by incorporating self-diffusion. Tiwari et al. analyzed a predator-prey interaction model with Beddington-DeAngelis functional response (BDFR) and incorporating the cost of fear into prey reproduction. For the spatial system, the Hopf bifurcation around the interior equilibrium, stability of homogeneous steady state, direction, and stability of spatially homogeneous periodic orbits have been established [23]. For a plankton-fish model with both the zooplankton refuge and the fear effect, the local and global dynamics of such a model have been investigated in [24]. Moreover, the investigation in [25] has revealed the threshold behavior of a stochastic predator-prey system with fear effect, prey refuge, and non-constant mortality rate. Sasmal and Takeuchi studied the dynamics of a prey-predator interaction model using Monod–Haldane type functional response and provided detailed mathematical results, including basic dynamical properties, existence of positive equilibria, asymptotic stability of all equilibria, Hopf bifurcation, direction, and stability of bifurcated periodic solutions [26]. Furthermore, they also investigated the role of predation fear and its carry-over effects in the prey-predator model. Basic dynamical properties, as well as the global stability of each equilibrium, have been discussed [27].
Allee effect comes in different forms, including multiplicative Allee effect and additive Allee effect. Furthermore, Dennis [6] first proposed the equation incorporating additive Allee effect:where and are constants, which reflect the degree of Allee effect; denotes the additive Allee effect; is the intrinsic growth rate of prey; presents capacity. We note that if , then (1) has the weak Allee effect and if , then it has the strong Allee effect.
Motivated by the previous works above, we further consider the following reaction-diffusion system with fear effect and additive Allee effect:where is the Laplace operator on domains. meanss the diffusion coefficients. The homogeneous Neumann boundary condition is imposed so that there is no population flow across the boundary, denotes the outward normal to the boundary . , stand for the density of the prey and predator, respectively; and are constants, which reflect the degree of Allee effect; is a constant, which reflects the degree of fear effect; and denote the fear effect and additive Allee effect, respectively; represents the modified capture rate; is the conversion coefficient; is the intrinsic growth rate of prey; is the death rate of predator. Then, the steady-state system corresponding to (2) is
The remainder of the paper is structured as follows. In Section 2, we carry out a priori estimates for (3) and the requirements for the nonexistence of non-constant positive solutions. In Section 3, we consider the stability of non-negative constant steady state solutions for system (3). In Section 4, we demonstrate the existence of Hopf bifurcation and steady state bifurcation. In Section 5, we show how the parameters affect the dynamical behavior of the system. Furthermore, we verify the analysis results with the numerical simulation results. In section 6, the paper ends with some conclusions.
2. Preliminaries
In this section, we first present some properties of equilibrium solutions of (3) including a priori estimates. Then, we discuss the nonexistence of non-constant positive solutions for certain parameter range. It is an essential part for analysis of the existence of non-constant positive steady states and the global bifurcation. We first recall the maximum principle in [28].
Lemma 1 (see [28]). We suppose that . If satisfiesand , then . Similarly, if the two inequalities are reversed and , then .
We note that is a bounded domain in with smooth boundary. Let be the eigenvalues of under Neumann boundary condition.
By Lemma 1, we have a prior estimates as follows:
Theorem 1. Let be non-negative and non-trivial solution of (3); we assume that . Then, satisfieswhere .
Proof. From the strong maximum principle, we have and . Then, by Lemma 1, it follows . The first equation of (3) is multiplied by and adding the two equations of (3), we obtainwhich leads to under the condition of . Then, by Lemma 1, we obtain which implies
Theorem 2. For any fixed , there exists such that if , then (3) has no non-constant positive solution.
Proof. Let be a non-negative solution of (3). We denote (10) asThen,Multiplying the first equation of (3) by and integrating on , applying Theorem 1 thatSimilarly, multiplying the second equation of (3) by , we obtainMultiplying the first equation of (3) by c, added to the second equations of (3), and integrating on , we obtainSubject it to the boundary conditions, we haveHence,By and (16), it follows from Theorem 1 and Young inequality thatSimilarly, we haveFrom (12), (13), (16)–(19) and the Poincaré inequality, we obtain thatwhereThis shows that ifthenand must be a constant solution.
3. Non-Negative Constant Steady-State Solutions
In this section, the stability of non-negative constant steady state solutions of (3) will be investigated by the standard linearization theory. By [17], under particular situations, (3) has the non-negative constant steady state solutions as follows.(1)the trivial solution always exists.(2)if , there is no boundary constant solution when .(3)if , then is unique boundary equilibria when .(4)if , there is two boundary constant solution and when .(5)if , there is unique boundary constant solution under the condition of .(6)if , there is unique boundary constant solution only when .(7)if , there is unique boundary constant solution when .(8)there is unique positive constant solution with when .
Under the no-flux boundary condition, has eigenvalues and . Let be the eigenspace generated by the eigenfunctions corresponding to . Let be the algebraic multiplicity of . Let be the normalized eigenfunctions corresponding to . Then, the set forms a complete orthonormal basis in .
Next, we consider the stability of constant steady state solutions.
Theorem 3. For all constants , we have that(1)For trivial solution, if, thenis locally asymptotically stable; if, thenis unstable(2)If, thenis unstable(3)If, thenis unstable(4)If,is stable and if,is unstable(5)exists if and only if. If, thenis stable. If, thenis unstable
Proof. We rewrite (3) asThe linearization matrix of (3) at a constant solution E = (u0,V0) can be expressed bywhereWe define that , , and . Let be a pair of eigenfunction of corresponding to an eigenvalue . Then, we haveWe setThen, we obtainFrom the chapter 5 of [29, 30], we know that if all the eigenvalues of have negative real parts, then the constant solution is locally asymptotically stable; is unstable if there is an eigenvalue of with positive real part; if all the eigenvalues have non-positive real parts while some eigenvalues have zero real parts, then the stability of cannot be determined by the linearization. Furthermore, is an eigenvalue of if and only if is an eigenvalue of the matrix for some . We have.where(1)For trival solution , If , then for all eigenvalues , we have and , which leads to . Hence, is locally asymptotically stable. If , then for , there exists a positive eigenvalue , which implies that is unstable. In addition, if , is stable, else if , is unstable.(2)For , with , For corresponding ordinary system, is unstable, so for any , is unstable.(3)For , with , For , there exists a positive eigenvalue . So, is always unstable.(4)For , is stable when , and in this case, for any . Additionally, in other cases, is unstable, so for any , is unstable.(5)For positive constant solution, . The Jacobi matrix of (3) at isIt is noted thatFor exists if and only if , so it is easy to conclude that if , which implies that is stable. If , for , we obtain that and , so it follows that there exist two of the eigenvalues with positive real parts, which implies that is unstable.
4. Existence of Non-Constant Positive Solutions
In this section, we consider the existence of non-constant positive solutions to (3) in . First, the existence of spatially homogeneous and non-homogeneous periodic solutions is studied by taking as the bifurcation parameter. Then, the structure and the stability of the bifurcation solutions that bifurcate from are shown. From Theorem 3, the stability of is determined by the trace and determinant of . Furthermore, we will restrict . To put out our discussion into the context of the Hopf bifurcation, we convert (3) into the following system by and and drop “” for simplicity. We have
Firstly, we define the real-valued Sobolev spaceand the corresponding complexification space is given by .
The linearized operator of the steady state system of (39) evaluated at iswhere is the domain of .
The adjoint operator of is defined bywhere the domain of is .
The following condition in [31] is crucial to ensure that the Hopf bifurcation occurs.
(H1) There exists a neighborhood of such that for , has a pair of complex, simple, conjugate eigenvalues , continuously differentiable in , with , and , all other eigenvalues of have non-zero real parts for .
Motived by [31], we apply the Hopf bifurcation theory to analyze our system. For the eigenvalue problem
we know that the corresponding (42) eigenvalues are , with corresponding eigenfunctions . Let
be a pair of eigenfunctions of corresponding to an eigenvalue , that is, . By a straightforward analysis, we havewhere
Hence, the eigenvalues of are given by the eigenvalues of . The characteristic equation of iswhere
Therefore, the eigenvalues are determined by
If the condition (H1) holds, has a pair of simple purely imaginary at , if and only if there exists a unique such that are the purely imaginary eigenvalues of . The related eigenvector is denoted by , with , such that .
We identify the Hopf bifurcation point m0 which satisfies the condition (H1): there exists such that
and for the unique pair of complex eigenvalues near the imaginary axis
It is easy to obtain and if , which implies that the steady state is locally asymptotically stable. Hence, any potential bifurcation points must be in the interval . This means that is essential for bifurcation condition. For any Hopf bifurcation point in , are the eigenvalues of , where
for in . Hence, the transversality condition is always satisfied.
From the discussion above, the determination of Hopf bifurcation points reduces to describing the set when a set of parameters are given.
In the following, for and fixed, we choose appropriately. is always an element of for any because of for any , and for any . This corresponds to the Hopf bifurcation of spatially homogeneous periodic solution. Apparently, is also the unique value for the Hopf bifurcation of spatially homogeneous periodic solution for any .
In the following, we search for spatially non-homogeneous Hopf bifurcation points for . As and for , we obtain that for . We define
Then for , and , we derive the root of as such that . Moreover, by in , we derive (56) and (57)
Since , now we discuss a condition to verify for . For , we have
The quadratic function is positive for all if the discriminant of is negative, which means that (60)
We note that
For , we can choose such that the discriminant of is negative. Then, for such that .
We summarize our analysis above and apply Theorem 2 in [31]. The existence of both spatially homogeneous and non-homogeneous periodic solutions bifurcation from can be obtained as follows:
Theorem 4. For anyinand, system (2) undergoes Hopf bifurcation at each. Moreover, the bifurcation periodic solutions nearcan be parameterized asso thatin the form offorfor some small, (61) and (62)where is the corresponding eigenvector, andFurthermore, we notice that(1)The bifurcating periodic orbits from are spatially homogeneous, which coincide with the periodic orbits of the corresponding ODE system(2)The bifurcating periodic orbits from are spatially non-homogeneous.
Then, we consider the direction and stability of spatially homogeneous Hopf bifurcation.
Theorem 5. For system (2), if all other eigenvalues of have negative real parts and , the spatially homogeneous periodic solutions bifurcating from are locally asymptotically stable (resp. unstable). Moreover, the Hopf bifurcation at is supercritical (resp. subcritical) if .
Proof. Here, the notations and calculations in [31] are used in the same way. For the sake of simplicity, we denote by . Then, we introducesuch that , and , whereAnd denotes the inner product in . Then, we get the derivatives at as follows:In addition, we note where are defined as the same with [31].For , by calculation, we derive (69)Then, we can obtain (70)And, we noteHence, , . By further calculation, we obtain thatFrom above analysis, we know that . Hence, by Theorem 2 in [31], the bifurcating solutions bifurcated from are locally asymptotically stable (resp. unstable) if and , for , and the Hopf bifurcation at is supercritical (resp. subcritical) if . The proof is complete.
Inspired by [31, 32], we take as the bifurcation parameter and also restrict . We suppose that . The non-negative steady state solutions of (72) satisfy the elliptic problem corresponding toFrom Theorem 3, we know that is locally asymptotically stable for and unstable for .
The steady state bifurcation point satisfies the steady state bifurcation condition (H2) in [31]:It means that the potential steady state bifurcation points shall satisfy conditions (74) and (75). Recall that for , and . Then, any potential bifurcation point must be in the interval . Hence, the steady state bifurcation points reduces to the setwhen a set of parameters are fixed.
Recall that , where . By solving , we haveWe define thatFor , we have and for .
Hence, there exists a unique root of denoted by , which implies that and exists only for . Therefore, the potential steady state bifurcation points reduces to the set (79).Then, the properties of can be summarized as follows:
Lemma 2. We assume that . Then, for any , exists. Moreover, is increasing and is decreasing.
Proof. The first limit equation is trivial, so we omit here. We mainly analyze the monotonicity result on with respect to for .
Differentiating with respect to , it follows that Hence, . It is easy to get and from (77). In addition, by calculation, we obtain that for , . The proof is completed.
It follows from Lemma 1 that the curve forms a smooth connected curve which connects , , and .
By the properties of , if then there exists such that or , and thus . We define . Then, for any , there exists a such that .
Next, we verify . We recall that . Moreover, we know that and . It follows that .
5. Numerical Simulations
In this section, in order to reveal the influence of fear effect, Allee effect, and other factors on the predator-prey model, a numerical method is used to analyze the effect of parameters on the asymptotic behavior of system (2) so as to verify and supplement the theoretical results mentioned before.
In Figure 1, we choose . Varying the parameter and choosing the initial data near , we indicate the following numerical results on the effects of parameter :(1)Take . Since , is locally asymptotically stable by Theorem 3. The simulation results indicate that system (2) converges to the equilibrium (see Figures 1(a) and 1(b)).(2)Taking , there is no Allee effect on prey. Since , is locally asymptotically stable by Theorem 3. The simulation results indicate that system (2) converges to the equilibrium (see Figures 1(c) and 1(d)).(3)Taking , it satisfies the condition of the weak Allee effect and Hopf bifurcation condition by Theorem 4. The simulation results indicate that system (2) undergoes Hopf bifurcation (see Figures 1(e) and 1(f)).(4)Taking , it satisfies the condition of the strong Allee effect and Hopf bifurcation condition by Theorem 4. The simulation results indicate that system (2) undergoes Hopf bifurcation (see Figure 1(g) and 1(h)).
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In Figure 2, we choose . Varying the parameter and choosing the initial data near , it shows that when is small, system (2) has obviously periodic oscillation (see Figure 2(a)). When increases, the maximum norms of are almost the same. However, the maximum norms of decrease with increasing (see Figures 2(a)–2(d)). This means that the fear has a negative impact on predators. Moreover, the period of periodic solutions becomes larger as increases. (see Figures 2(a)–2(d)).
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In Figure 3, we choose . Varying the parameter and choosing the initial data near , it indicates that when increases, the period of periodic solution is decreasing. Furthermore, with increasing, the amplitude of periodic solutions is also decreasing. (see Figures 3(a)–3(c)). As continues to increase, system (2) converges to an equilibrium (see Figure 3(d)).
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6. Conclusion
In this paper, a diffusive predator-prey model with additive Allee effects induced by fear factors is considered, in which prey can represent the antipredator behavior due to fear factors. Analytical results indicate that the upper bound of depends on the diffusion rates , , the death rate of the predators , the Allee effects parameters , the conversion rate , and the level of fear . There exists , which depends on , such that if , the system have only constant positive solution. Furthermore, the dynamic behavior near is of more concerned. Then, we indicate the existence of non-constant positive solutions. Taking as a bifurcation parameter, system undergoes Hopf bifurcation at each . Furthermore, for , are fixed, and there is a smooth curve of non-constant positive solutions bifurcating from .
We observe that the Allee effect is essential to the dynamical behavior of system (2) by numerical simulations. The amplitude in the strong Allee effect is increasing with increasing. On the other hand, numerical simulations reveal that the fear effect have an impact on the dynamical behavior of system (2). With the fear effect increasing, the period of periodic solutions is increasing, but the maximum norm of is almost the same. On the contrary, the maximum norm of decreases with increasing. From a biological standpoint, the prey survives by adopting antipredator behavior as a result of the fear effect, and the predator is impacted by the prey’s antipredator behavior. At last, we show how the death rate affects system (2) with the strong Allee effect. As increases, the amplitude of periodic solution is decreasing, and the period of periodic solution is also decreasing.
It is extremely important to construct animal interaction models in the incorporation of these different types of factors. Considering the different ways of introducing the Allee effect and the interaction with the fear effect, further analysis of the bifurcating solutions of (2) remains a challenging problem. From our discussion before, we conjecture that Turing-Hopf bifurcation, Hopf-Hopf bifurcation is likely to exist in the system, which reveals more complex dynamic behavior and potential biological significance.
Data Availability
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation, to any qualified researcher.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This paper was supported by National Natural Science Foundation of China (Nos. 11861013, 11771444), and the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (No. 2018061).